Strong StabilityPreserving HighOrder Time Discretization Methods


 Kelly Lee
 2 years ago
 Views:
Transcription
1 SIAM REVIEW Vol. 43,No.,pp. 89 c 00 Society for Industrial and Applied Mathematics Strong StabilityPreserving HighOrder Time Discretization Methods Sigal Gottlieb ChiWang Shu Eitan Tadmor Abstract. In this paper we review and further develop a class of strong stabilitypreserving SSP highorder time discretizations for semidiscrete method of lines approximations of partial differential equations.previously termed TVD total variation diminishing time discretizations, these highorder time discretization methods preserve the strong stability properties of firstorder Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations.the new developments in this paper include the construction of optimal explicit SSP linear Runge Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge Kutta and multistep methods. Key words. strong stability preserving, Runge Kutta methods, multistep methods, highorder accuracy, time discretization AMS subject classifications. 65M0, 65L06 PII. S X. Introduction. It is a common practice in solving timedependent partial differential equations PDEs to first discretize the spatial variables to obtain a semidiscrete method of lines scheme. This is then an ordinarydifferential equation ODE system in the time variable, which can be discretized by an ODE solver. A relevant question here concerns stability. For problems with smooth solutions, usually a linear stabilityanalysis is adequate. For problems with discontinuous solutions, however, such as solutions to hyperbolic problems, a stronger measure of stability is usually required. Received by the editors February, 000; accepted for publication in revised form August, 000; published electronically February, 00. Department of Mathematics, University of Massachusetts at Dartmouth, Dartmouth, MA 0747 and Division of Applied Mathematics, Brown University, Providence, RI 09 umassd.edu. The research of this author was supported by ARO grant DAAG and NSF grant ECS Division of Applied Mathematics, Brown University, Providence, RI 09 The research of this author was supported by ARO grants DAAG and DAAD , NSF grants DMS , ECS , and INT , NASA Langley grant NAG070 and contract NAS while this author was in residence at ICASE, NASA Langley Research Center, Hampton, VA , and by AFOSR grant F Department of Mathematics, University of California at Los Angeles, Los Angeles, CA The research of this author was supported by NSF grant DMS and ONR grant N0004J
2 90 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR In this paper we review and further develop a class of highorder strong stabilitypreserving SSP time discretization methods for the semidiscrete method of lines approximations of PDEs. These time discretization methods were first developed in [0] and [9] and were called TVD total variation diminishing time discretizations. This class of methods was further developed in [6]. The idea is to assume that the firstorder forward Euler time discretization of the method of lines ODE is strongly stable under a certain norm when the time step t is suitablyrestricted, and then to tryto find a higher order time discretization Runge Kutta or multistep that maintains strong stabilityfor the same norm, perhaps under a different time step restriction. In [0] and [9], the relevant norm was the total variation norm: the forward Euler time discretization of the method of lines ODE was assumed to be TVD, hence the class of highorder time discretization developed there was termed TVD time discretization. This terminologywas also used in [6]. In fact, the essence of this class of highorder time discretizations lies in its abilityto maintain the strong stabilityin the same norm as the firstorder forward Euler version, hence SSP time discretization is a more suitable term, which we will use in this paper. We begin this paper bydiscussing explicit SSP methods. We first give, in section, a brief introduction to the setup and basic properties of the methods. We then move, in section 3, to our new results on optimal SSP Runge Kutta methods of arbitraryorder of accuracyfor linear ODEs suitable for solving PDEs with linear spatial discretizations. This is used to prove strong stabilityfor a class of wellposed problems u t = Lu, where the operator L is linear and coercive, improving and simplifying the proofs for the results in [3]. We review and further develop the results in [0], [9], and [6] for nonlinear SSP Runge Kutta methods in section 4 and for multistep methods in section 5. Section 6 of this paper contains our new results on implicit SSP schemes. It starts with a numerical example showing the necessity of preserving the strong stabilitypropertyof the method, then it moves on to the analysis of the rather disappointing negative results concerning the nonexistence of SSP implicit Runge Kutta or multistep methods of order higher than. Concluding remarks are given in section 7.. Explicit SSP Methods... Why SSP Methods? Explicit SSP methods were developed in [0] and [9] termed TVD time discretizations there to solve systems of ODEs. d dt u = Lu, resulting from a method of lines approximation of the hyperbolic conservation law,. u t = fu x, where the spatial derivative, fu x, is discretized bya TVD finite difference or finite element approximation; see, e.g., [8], [6], [], [], [9], and consult [] for a recent overview. Denoted by Lu, it is assumed that the spatial discretization has the propertythat when it is combined with the firstorder forward Euler time discretization,.3 u n+ = u n + tlu n, then, for a sufficientlysmall time step dictated bythe Courant Friedrichs LevyCFL condition,.4 t t FE,
3 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 9.5 u exact TVD.5 u exact nontvd x x Fig.. Secondorder TVD MUSCL spatial discretization. Solution after the shock moves 50 mesh points. Left: SSP time discretization;right: nonssp time discretization. the total variation TV of the onedimensional discrete solution u n := j u n j {xj x x j+ } does not increase in time; i.e., the following socalled TVD propertyholds:.5 TVu n+ TVu n, TVu n := j u n j+ u n j. The objective of the highorder SSP Runge Kutta or multistep time discretization is to maintain the strong stabilityproperty.5 while achieving higher order accuracyin time, perhaps with a modified CFL restriction measured here with a CFL coefficient, c.6 t c t FE. In [6] we gave numerical evidence to show that oscillations mayoccur when using a linearlystable, highorder method which lacks the strong stabilityproperty, even if the same spatial discretization is TVD when combined with the firstorder forward Euler time discretization. The example is illustrative, so we reproduce it here. We consider a scalar conservation law, the familiar Burgers equation.7 u t + u =0 with Riemann initial data.8 ux, 0 = x { if x 0, 0.5 if x>0. The spatial discretization is obtained bya second order MUSCL [], which is TVD for forward Euler time discretization under suitable CFL restriction. In Figure., we show the result of using an SSP secondorder Runge Kutta method for the time discretization left and that of using a nonssp secondorder Runge Kutta method right. We can clearlysee that the nonssp result is oscillatory there is an overshoot.
4 9 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR This simple numerical example illustrates that it is safer to use an SSP time discretization for solving hyperbolic problems. After all, they do not increase the computational cost and have the extra assurance of provable stability. As we have alreadymentioned above, the highorder SSP methods discussed here are not restricted to preserving not increasing the TV. Our arguments below relyon convexity, hence these properties hold for any norm. Consequently, SSP methods have a wide range of applicability, as theycan be used to ensure stabilityin an arbitrary norm once the forward Euler time discretization is shown to be stronglystable, i.e., u n + tlu n u n. For linear examples we refer to [7], where weighted L SSP higher order discretizations of spectral schemes were discussed. For nonlinear scalar conservation laws in several space dimensions, the TVD propertyis ruled out for highresolution schemes; instead, strong stabilityin the maximum norm is sought. Applications of L SSP higher order discretization for discontinuous Galerkin and central schemes can be found in [3] and [9]. Finally, we note that since our arguments below are based on convex decompositions of highorder methods in terms of the firstorder Euler method, anyconvex function will be preserved bysuch highorder time discretizations. In this context we refer, for example, to the cell entropystability propertyof highorder schemes studied in [7] and [5]. We remark that the strong stabilityassumption for the forward Euler u n + tlu n u n can be relaxed to the more general stabilityassumption u n + tlu n +O t u n. This general stabilitypropertywill also be preserved by the highorder SSP time discretizations. The total variation bounded TVB methods discussed in [8] and [] belong to this category. However, if the forward Euler operator is not stable, the framework in this paper cannot be used to determine whether a highorder time discretization is stable or not... SSP Runge Kutta Methods. In [0], a general mstage Runge Kutta method for. is written in the form.9 u 0 = u n, i u i = α i,k u k + tβ i,k Lu k, α i,k 0, i =,...,m, k=0 u n+ = u m. Clearly, if all the β i,k s are nonnegative, β i,k 0; then since byconsistency i k=0 α i,k =, it follows that the intermediate stages in.9, u i, amount to convex combinations of forward Euler operators, with t replaced by β i,k α i,k t. We thus conclude with the following lemma. Lemma. see [0]. If the forward Euler method.3 is strongly stable under the CFL restriction.4, u n + tlu n u n, then the Runge Kutta method.9 with β i,k 0 is SSP, u n+ u n, provided the following CFL restriction.6 is fulfilled:.0 t c t FE, c = min i,k α i,k β i,k. By the notion of strong stability we refer to the fact that there is no temporal growth, as opposed to the general notion of stability, which allows a bounded temporal growth, u n Const u 0, with any arbitrary constant, possibly Const >.
5 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 93 If some of the β i,k s are negative, we need to introduce an associated operator L corresponding to stepping backward in time. The requirement for L is that it approximate the same spatial derivatives as L, but that the strong stabilityproperty hold with u n+ u n with respect to either the TV or another relevant norm for the firstorder Euler scheme solved backward in time, i.e.,. u n+ = u n t Lu n. This can be achieved for hyperbolic conservation laws by solving the negativeintime version of.,. u t = fu x. Numerically, the only difference is the change of upwind direction. Clearly, L can be computed with the same cost as that of computing L. We then have the following lemma. Lemma. see [0]. If the forward Euler method combined with the spatial discretization L in.3 is strongly stable under the CFL restriction.4, u n + tlu n u n, and if Euler s method solved backward in time in combination with the spatial discretization L in. is also strongly stable under the CFL restriction.4, u n t Lu n u n, then the Runge Kutta method.9 is SSP, u n+ u n, under the CFL restriction.6,.3 t c t FE, c = min i,k α i,k β i,k, provided β i,k L is replaced by β i,k L whenever βi,k is negative. Notice that if, for the same k, both Lu k and Lu k must be computed, the cost as well as the storage requirement for this k is doubled. For this reason, we would like to avoid negative β i,k as much as possible. However, as shown in [6] it is not always possible to avoid negative β i,k..3. SSP Multistep Methods. SSP multistep methods of the form.4 u n+ = αi u n+ i + tβ i Lu n+ i, α i 0, i= were studied in [9]. Since α i =, it follows that u n+ is given bya convex combination of forward Euler solvers with suitablyscaled t s, and hence, similar to our discussion for Runge Kutta methods, we arrive at the following lemma. Lemma.3 see [9]. If the forward Euler method combined with the spatial discretization L in.3 is strongly stable under the CFL restriction.4, u n + tlu n u n, and if Euler s method solved backward in time in combination with the spatial discretization L in. is also strongly stable under the CFL restriction.4, u n t Lu n u n, then the multistep method.4 is SSP, u n+ u n, under the CFL restriction.6,.5 t c t FE, c = min i α i β i, provided β i L is replaced by β i L whenever βi is negative.
6 94 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR 3. Linear SSP Runge Kutta Methods of Arbitrary Order. 3.. SSP Runge Kutta Methods with Optimal CFL Condition. In this section we present a class of optimal in the sense of CFL number SSP Runge Kutta methods of anyorder for the ODE., where L is linear. With a linear L being realized as a finitedimensional matrix, we denote Lu = Lu. We will first show that the mstage, mthorder SSP Runge Kutta method can have, at most, CFL coefficient c = in.0. We then proceed to construct optimal SSP linear Runge Kutta methods. Proposition 3.. Consider the family of mstage, mthorder SSP Runge Kutta methods.9 with nonnegative coefficients α i,k and β i,k. The maximum CFL restriction attainable for such methods is the one dictated by the forward Euler scheme, t t FE ; i.e.,.6 holds with maximal CFL coefficient c =. Proof. We consider the special case where L is linear and prove that even in this special case the maximum CFL coefficient c attainable is. Any mstage method.9, for this linear case, can be rewritten as i u i = + A i,k tl k+ u 0, i =,...,m, where A i,k = k=0 i i A,0 = β,0, A i,0 = α i,k A k,0 + β i,k, i j=k+ k= k=0 i α i,j A j,k + β i,j A j,k, k =,...,i. j=k In particular, using induction, it is easyto show that the last two terms of the final stage can be expanded as A m,m = A m,m = m β l,l, l= β k,k k= m l=k+ k β l,l l= β l,l + α k,k k= m β l,l l=,l k For an mstage, mthorder linear Runge Kutta scheme, A m,k = k+!. Using A m,m = m l= β l,l = m!, we can rewrite α k,k m k A m,m = + β k,k β l,l β l,l. m!β k,k k= k= l=k+ With the nonnegative assumption on β i,k s and the fact A m,m = m l= β l,l = m! we have β l,l > 0 for all l. For the CFL coefficient c wemusthave α k,k β k,k for all k. Clearly, A m,m = m! is possible under these restrictions onlyif β k,k =0 and α k,k β k,k = for all k, in which case the CFL coefficient c. l=.
7 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 95 We remark that the conclusion of Proposition 3. is valid onlyif the mstage Runge Kutta method is mthorder accurate. In [9], we constructed an mstage, firstorder SSP Runge Kutta method with a CFL coefficient c = m which is suitable for steadystate calculations. The proof above also suggests a construction for the optimal linear mstage, mthorder SSP Runge Kutta methods. Proposition 3.. The class of mstage schemes given recursively by 3. u i = u i + tlu i, i =,...,m, u m = where α,0 =and m k=0 α m,k u k + α m,m u m + tlu m, 3. α m,k = k α m,k, k =,...,m, α m,m = m m!, α m,0 = k= α m,k is an mthorder linear Runge Kutta method which is SSP with CFL coefficient c =, t t FE. Proof. The firstorder case is forward Euler, which is firstorder accurate and SSP with CFL coefficient c = bydefinition. The other schemes will be SSP with a CFL coefficient c = byconstruction, as long as the coefficients are nonnegative. We now show that scheme is mthorder accurate when L is linear. In this case, clearly i u i =+ tl i u 0 i! = k!i k! tlk u 0, i =,...,m, k=0 hence scheme results in m j u m = j! m! α m,j k!j k! tlk + α m,m k!m k! tlk u 0. j=0 k=0 Clearly, by 3., the coefficient of tl m m! is α m,m m! = m!, the coefficient of tl m is α m,m = m!, and the coefficient of tl0 is m m! + α m,j =. j=0 It remains to show that, for k m, the coefficient of tl k is equal to k! : k=0 3.3 m k!m k! + j! α m,j k!j k! = k!. j=k
8 96 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR This will be shown byinduction. Thus we assume 3.3 is true for m, and then for m + we have, for 0 k m, that the coefficient of tl k+ is equal to m k +!m k! + = k +! j=k+ α m+,j j! k +!j k! m m k! + l=k l=k l +! α m+,l+ l k! m = k +! m k! + l + α l +! m,l l k! = = k +! k +!, m m k! + l=k l! α m,l l k! where in the second equalitywe used 3. and in the last equalitywe used the induction hypothesis 3.3. This finishes the proof. Finally, we show that all the α s are nonnegative. Clearly α,0 = α, = > 0. If we assume α m,j 0 for all j =0,...,m, then α m+,j = j α m,j 0, j =,...,m ; α m+,m = and, bynoticing that α m+,j α m,j for all j =,...,m,wehave m +! 0, α m+,0 = j= α m+,j α m,j =0. j= As the mstage, mthorder linear Runge Kutta method is unique, we have in effect proved that this unique mstage, mthorder linear Runge Kutta method is SSP under CFL coefficient c =. If L is nonlinear, scheme is still SSP under CFL coefficient c =, but it is no longer mthorder accurate. Notice that all but the last stage of these methods are simple forward Euler steps. We note in passing the examples of the ubiquitous third and fourthorder Runge Kutta methods, which admit the following convex, and hence SSP, decompositions: k=0 4 k=0 k! tlk = 3 + I + tl+ 6 I + tl3, k! tlk = I + tl+ 4 I + tl + 4 I + tl4. We list, in Table 3., the coefficients α m,j of these optimal methods in 3. up to m =8.
9 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 97 Table 3. Coefficients α m,j of the SSP methods Order m α m,0 α m, α m, α m,3 α m,4 α m,5 α m,6 α m, Application to Coercive Approximations. We now applythe optimal linear SSP Runge Kutta methods to coercive approximations. We consider the linear system of ODEs of the general form, with possibly variable, timedependent coefficients, 3.6 d ut =Ltut. dt As an example we refer to [7], where the farfromnormal character of the spectral differentiation matrices defies the straightforward von Neumann stabilityanalysis when augmented with highorder time discretizations. We begin our stabilitystudyfor Runge Kutta approximations of 3.6 with the firstorder forward Euler scheme with, denoting the usual Euclidean inner product u n+ = u n + t n Lt n u n, based on variable time steps, t n := n j=0 t j. Taking L norms on both sides one finds u n+ = u n + t n Re Lt n u n,u n + t n Lt n u n, and hence strong stabilityholds, u n+ u n, provided the following restriction on the time step, t n, is met: t n Re Lt n u n,u n / Lt n u n. Following Levyand Tadmor [3] we therefore make the following assumption. Assumption 3. coercivity. The operator Lt is uniformly coercive in the sense that there exists ηt > 0 such that 3.7 u ηt := inf Re Ltu, u = Ltu > 0. We conclude that for coercive L s, the forward Euler scheme is stronglystable, I + t n Lt n, if and onlyif t n ηt n.
10 98 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR In a generic case, Lt n represents a spatial operator with a coercivitybound ηt n, which is proportional to some power of the smallest spatial scale. In this context the above restriction on the time step amounts to the celebrated CFL stabilitycondition. Our aim is to show that the general mstage, mthorder accurate Runge Kutta scheme is stronglystable under the same CFL condition. Remark. Observe that the coercivityconstant, η, is an upper bound in the size of L; indeed, bycauchy Schwartz, ηt Ltu u / Ltu and hence 3.8 Lt = sup u Ltu u ηt. To make one point, we consider the fourthorder Runge Kutta approximation of 3.6: k = Lt n u n, k = Lt n+ u n + t n k, k 3 = Lt n+ u n + t n k, k 4 = Lt n+ u n + t n k 3, u n+ = u n + t [ n k +k +k 3 + k 4]. 6 Starting with secondorder and higher, the Runge Kutta intermediate steps depend on the time variation of L, and hence we require a minimal smoothness in time, making the following assumption. Assumption 3. Lipschitz regularity. We assume that L is Lipschitz. Thus, there exists a constant K>0 such that 3.4 Lt Ls K t s. ηt We are now readyto make our main result, stating the following proposition. Proposition 3.3. Consider the coercive systems of ODEs , with Lipschitz continuous coefficients 3.4. Then the fourthorder Runge Kutta scheme is stable under CFL condition 3.5 t n ηt n, and the following estimate holds: 3.6 u n e 3Ktn u 0. Remark. The result along these lines was introduced bylevyand Tadmor [3, Main Theorem], stating the strong stabilityof the constant coefficients sorder Runge Kutta scheme under CFL condition t n C s ηt n. Here we improve in both simplicityand generality. Thus, for example, the previous bound of C 4 =/3 [3, Theorem 3.3] is now improved to a practical timestep restriction with our uniform C s =.
11 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 99 Proof. We proceed in two steps. We first freeze the coefficients at t = t n, considering here we abbreviate L n = Lt n j = L n u n, j = L n u n + t n j L n I + t n j 3 = L n u n + t n j Ln L n [ I + t n Ln j 4 = L n u n + t n j 3, v n+ = u n + t [ n j +j +j 3 + j 4]. 6 u n, I + t ] n Ln u n, Thus, v n+ = P 4 t n L n u n, where following 3.5, P 4 t n L n := 3 8 I + 3 I + tl+ 4 I + tl + 4 I + tl4. Since the CFL condition 3.5 implies the strong stabilityof forward Euler, i.e., I + t n L n, it follows that P 4 t n L n 3/8+/3+/4+/4 =. Thus, 3. v n+ u n. Next, we include the time dependence. We need to measure the difference between the exact and the frozen intermediate values the k s and the j s. We have k j =0, [ ] k j = Lt n+ Lt n I + t n Ln u n, k 3 j 3 = Lt n+ t [ ] n k j + Lt n+ Lt n tn j, k 4 j 4 = Lt n+ t n k 3 j 3 + [ Lt n+ Lt n ] t n j 3. Lipschitz continuity3.4 and the strong stabilityof forward Euler imply 3.7 k j K t n ηt n un K u n. Also, since L n ηt n, we find from 3.8 that j u n /ηt n, and hence 3.5 followed by3.7 and the CFL condition 3.5 imply 3.8 k 3 j 3 t n ηt n k j + K t n ηt n t n ηt n un tn K ηt n u n K u n.
12 00 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR Finally, since by 3.9 j 3 does not exceed j 3 < ηt n followed by3.8 and the CFL condition 3.5, 3.9 k 4 j 4 t n ηt n k3 j 3 + K t n ηt n K We conclude that u n+, t n ηt n tn ηt n un, we find from + t n ηt n 3 tn tn ηt n + ηt n u n K u n. u n+ = v n+ + t n 6 is upper bounded byconsult 3., u n+ v n+ + t n 6 u n [ ] k j +k 3 j 3 +k 4 j 4, [ K u n +4K u n +K u n +3K t n u n and the result 3.6 follows. 4. Nonlinear SSP Runge Kutta Methods. In the previous section we derived SSP Runge Kutta methods for linear spatial discretizations. As explained in the introduction, SSP methods are often required for nonlinear spatial discretizations. Thus, most of the research to date has been in the derivation of SSP methods for nonlinear spatial discretizations. In [0], schemes up to third order were found to satisfythe conditions in Lemma. with CFL coefficient c =. In [6] it was shown that all fourstage, fourthorder Runge Kutta methods with positive CFL coefficient c in.3 must have at least one negative β i,k, and a method which seems optimal was found. For largescale scientific computing in three space dimensions, storage is usuallya paramount consideration. We review the results presented in [6] about SSP properties among such lowstorage Runge Kutta methods. 4.. Nonlinear Methods of Second, Third, and Fourth Order. Here we review the optimal in the sense of CFL coefficient and the cost incurred by L if it appears SSP Runge Kutta methods of mstage, mthorder for m =, 3, 4, written in the form.9. Proposition 4. see [6]. If we require β i,k 0, then an optimal secondorder SSP Runge Kutta method.9 is given by 4. u = u n + tlu n, u n+ = un + u + tlu, with a CFL coefficient c =in.0. An optimal thirdorder SSP Runge Kutta method.9 is given by u = u n + tlu n, ] 4. u = 3 4 un + 4 u + 4 tlu, with a CFL coefficient c =in.0. u n+ = 3 un + 3 u + 3 tlu,
13 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 0 In the fourthorder case we proved in [6] that we cannot avoid the appearance of negative β i,k, as demonstrated in the following proposition. Proposition 4. see [6]. The fourstage, fourthorder SSP Runge Kutta scheme.9 with a nonzero CFL coefficient c in.3 must have at least one negative β i,k. We thus must settle for finding an efficient fourthorder scheme containing L, which maximizes the operation cost measured by c 4+i, where c is the CFL coefficient.3 and i is the number of L s. This waywe are looking for an SSP method that reaches a fixed time T with a minimal number of evaluations of L or L. The best method we could find in [6] is 4.3 u = u n + tlun, u = u t Lu n u tlu, u 3 = un t Lu n u t Lu u tlu, u n+ = 5 un + 0 tlun u + 6 tlu u + 3 u3 + 6 tlu3, with a CFL coefficient c =0.936 in.3. Notice that two L s must be computed. The effective CFL coefficient, compared with an ideal case without L s, is =0.64. Since it is difficult to solve the global optimization problem, we do not claim that 4.3 is an optimal four stage, fourthorder SSP Runge Kutta method. 4.. Low Storage Methods. Storage is usuallyan important consideration for large scale scientific computing in three space dimensions. Therefore, lowstorage Runge Kutta methods [3], [], which onlyrequire two storage units per ODE variable, maybe desirable. Here we review the results presented in [6] concerning SSP properties among such lowstorage Runge Kutta methods. The general lowstorage Runge Kutta schemes can be written in the form [3], [] u 0 = u n, du 0 =0, 4.4 du i = A i du i + tlu i, i =,...,m, u i = u i + B i du i, i =,...,m, B = c, u n+ = u m. Only u and du must be stored, resulting in two storage units for each variable. Following Carpenter and Kennedy[], the best SSP thirdorder method found by numerical search in [6] is given bythe system z = 36c 4 +36c 3 35c +84c, z =c + c, z 3 =c 4 8c 3 +8c c +, z 4 =36c 4 36c 3 +3c 8c +4, z 5 =69c 3 6c +8c 8, z 6 =34c 4 46c 3 +34c 3c +,
14 0 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR B = cc 3z z 3z z 44c3c c, B 3 = A = 43c c 3z z cc 3z z, z 6c 4c ++3z 3 c +z 3c + c, A 3 = z z c c 5 3c z 5 4z cc 4 +7cz 6 +7c 6 c 3, with c = , resulting in a CFL coefficient c =0.3 in.6. This is of course less optimal than 4. in terms of CFL coefficient, but the lowstorage form is useful for largescale calculations. Carpenter and Kennedy[] have also given classes of fivestage, fourthorder lowstorage Runge Kutta methods. We have been unable to find SSP methods in that class with positive α i,k and β i,k. A lowstorage method with negative β i,k cannot be made SSP, as L cannot be used without destroying the lowstorage property Hybrid Multistep Runge Kutta Methods. Hybrid multistep Runge Kutta methods e.g., [0], [4] are methods that combine the properties of Runge Kutta and multistep methods. We explore the twostep, twostage method 4.5 u n+ = α u n + α 0 u n + t β 0 Lu n +β Lu n, α k 0, 4.6 u n+ = α 30 u n + α 3 u n+ + α3 u n + t β 30 Lu n +β 3 Lu n+ +β3 Lu n, α 3k 0. Clearly, this method is SSP under the CFL coefficient.0 if β i,k 0. We could also consider the case allowing negative β i,k s, using instead.3 for the CFL coefficient and replacing β i,k L by β i,k L for the negative βi,k s. For thirdorder accuracy, we have a threeparameter family depending on c, α 30, and α 3 : α 0 =3c +c 3, β 0 = c + c 3, α = 3c c 3, β = c +c + c 3, 4.7 β 30 = +α 30 3c +3α 30 c + α 3 c 3, 6 + c β 3 = 5 α 30 3α 3 c α 3 c 3 6c +6c, α 3 = α 3 α 30, β 3 = 5+α 30 +9c +3α 30 c 3α 3 c α 3 c 3. 6c
15 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 03 The best method we were able to find is given by c =0.4043, α 30 =0.0605, and α 3 =0.635 and has a CFL coefficient c Clearly, this is not as good as the optimal thirdorder Runge Kutta method 4. with CFL coefficient c =. We hoped that a fourthorder scheme with a large CFL coefficient could be found, but unfortunatelythis is not the case, as is proven in the following proposition. Proposition 4.3. There are no fourthorder schemes 4.5 with all nonnegative α i,k. Proof. The fourthorder schemes are given bya twoparameter familydepending on c, α 30 and setting α 3 in 4.7 to be α 3 = 7 α 30 +0c α 30 c c 3+8c +4c. The requirement α 0 enforces see 4.7 c. The further requirement α 0 0 yields 3 c. α 3 has a positive denominator and a negative numerator for <c<, and its denominator is 0 when c = or c = 3, thus we require 3 c<. In this range, the denominator of α 3 is negative, hence we also require its numerator to be negative, which translates to α c +c. Finally, we would require α 3 = α 3 α 30 0, which translates to α 30 c c+c+3+7 0c c+c c+. The two restrictions on α 30 give us the following inequality: 7+0c +c c c + c c c + c c +, which, in the range of 3 c<, yields c a contradiction. 5. Linear and Nonlinear Multistep Methods. In this section we review and further studyssp explicit multistep methods.4, which were first developed in [9]. These methods are rthorder accurate if 5. α i =, i= m i k α i = k i k β i, k =,...,r. i= i= We first prove a proposition that sets the minimum number of steps in our search for SSP multistep methods. Proposition 5.. For m, there is no mstep, mthorder SSP method with all nonnegative β i, and there is no mstep SSP method of order m +. Proof. Bythe accuracycondition 5., we clearlyhave for an rthorder accurate method 5. piα i = p iβ i i= i= for anypolynomial px of degree at most r satisfying p0 = 0. When r = m, we could choose px =xm x m.
16 04 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR Clearly pi 0 for i =,...,m, and equalityholds onlyfor i = m. On the other hand, p i =m im i m 0, and equalityholds onlyfor i = and i = m. Hence 5. would have a negative right side and a positive left side and would not be an inequalityif all α i and β i are nonnegative, unless the onlynonzero entries are α m, β, and β m. In this special case we have α m = and β = 0 to get a positive CFL coefficient c in.5. The first twoorder conditions in 5. now lead to β m = m and β m = m, which cannot be simultaneouslysatisfied. When r = m +, we could choose x m 5.3 px = qtdt, qt = i t. 0 Clearly p i =qi =0fori =,...,m. We also claim and prove below that all the pi s, i =,...,m, are positive. With this choice of p in 5., its righthand side vanishes, while the lefthand side is strictlypositive if all α i 0 a contradiction. We conclude with the proof of the following claim. Claim. pi = i 0 qtdt > 0, qt := m i= i t. Indeed, qt oscillates between being positive on the even intervals I 0 =0,,I =, 3,..., and being negative on the odd intervals, I =,,I 3 =3, 4,... The positivityof the pi s for i m +/ follows since the integral of qt over each pair of consecutive intervals is positive, at least for the first [m + /] intervals, pk + pk = qt dt qt dt I k I k+ = I k t t m t dt I k+ = t t m t m t t dt > 0, I k i= k + m +/. For the remaining intervals we note the symmetry of qt with respect to the midpoint m +/, i.e., qt = m qm + t, which enables us to write for i>m +/ m+/ i pi = qtdt + m qm + tdt 5.4 = 0 m+/ 0 m+/ m+/ qtdt + m qt dt. m+ i Thus, if m is odd, then pi =pm + i > 0 for i>m +/. If m is even, then the second integral on the right of 5.4 is positive for odd i s, since it starts with a positive integrand on the even interval I m+ i. And finally, if m is even and i is odd, then the second integral starts with a negative contribution from its first integrand on the odd interval I m+ i, while the remaining terms that follow cancel in pairs as before. A straightforward computation shows that this first negative contribution is compensated for bythe positive gain from the first pair, i.e., pm + i > 0 qtdt + This concludes the proof of our claim. m+ i m+ i qtdt > 0, m even, i odd.
17 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 05 Table 5. SSP multistep methods.4. # Steps Order CFL α i β i m r c , 5 3 4, 0, 4 8 9, 0, 0, , 7, , 35 50, , 0, 0, , 0, 0, 0, , 0, 0, 0, 0, , 7 4, 4, , , , , 0, 0, 0, 80 56, , 3000, 0, , , 4, 7 4, 6, , 3 50, 8 5, 7 50, , 3 0, 4 5, 0, 7 0, , 5 3, 0, 0 4, 0, 0, 0 3 5, 0, , 49 50, , 0, 0, , 0, 0, 0, , 0, 0, 0, 0, , , , , , , , 0, 0, 0, 8 8, , , , , , 85 88, 9 4, 5 96, , , 4 375, , , , , 0, , We remark that [4] contains a result stating that there are no linearlystable m step, m + storder methods when m is odd. When m is even such linearlystable methods exist but would require negative α i. This is consistent with our result. In the remainder of this section we will discuss optimal mstep, mthorder SSP methods which must have negative β i according to Proposition 5. and mstep, m storder SSP methods with positive β i. For twostep, secondorder SSP methods, a scheme was given in [9] with a CFL coefficient c = scheme in Table 5.. We prove this is optimal in terms of CFL coefficients. Proposition 5.. For twostep, secondorder SSP methods, the optimal CFL coefficient c in.5 is. Proof. The accuracycondition 5. can be explicitlysolved to obtain a oneparameter familyof solutions α = α, β = α, β = α. The CFL coefficient c is a function of α and it can be easilyverified that the maximum is c = achieved at α = 4 5. We move on to threestep, secondorder methods. It is now possible to have SSP schemes with positive α i and β i. One such method is given in [9] with a CFL coefficient c = scheme in Table 5.. We prove this is optimal in the CFL coefficient in the following proposition. We remark that this multistep method has the same efficiencyas the optimal twostage, secondorder Runge Kutta method 4.. This is because there is onlyone L evaluation per time step here, compared with two L evaluations in the twostage Runge Kutta method. Of course, the storage requirement here is larger.
18 06 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR Proposition 5.3. If we require β i 0, then the optimal threestep, secondorder method has a CFL coefficient c =. Proof. The coefficients of the threestep, secondorder method are given by α = 6 3β β + β 3, α = 3+β β 3, α 3 = β + β +3β 3. For CFL coefficient c>, we need α k β k > for all k. This implies α >β 6 4β β + β 3 > 0, α >β 6+4β β 4β 3 > 0. This means that β β 3 < 6 4β < β 4β 3 β < 3β 3. Thus, we would have a negative β. We remark that if more steps are allowed, then the CFL coefficient can be improved. Scheme 3 in Table 5. is a fourstep, secondorder method with positive α i and β i and a CFL coefficient c = 3. We now move to threestep, thirdorder methods. In [9] we gave a threestep, thirdorder method with a CFL coefficient c 0.74 scheme 4 in Table 5.. A computer search gives a slightlybetter scheme scheme 5 in Table 5. with a CFL coefficient c Next we move on to fourstep, thirdorder methods. It is now possible to have SSP schemes with positive α i and β i. One example was given in [9] with a CFL coefficient c = 3 scheme 6 in Table 5.. We prove this is optimal in the CFL coefficient in the following proposition. We remark again that this multistep method has the same efficiencyas the optimal threestage, thirdorder Runge Kutta method 4.. This is because there is onlyone L evaluation per time step here, compared with three L evaluations in the threestage Runge Kutta method. Of course, the storage requirement here is larger. Proposition 5.4. If we require β i 0, then the optimal fourstep, thirdorder method has a CFL coefficient c = 3. Proof. The coefficients of the fourstep, thirdorder method are given by α = 6 4 β β + β 3 β 4, α = 6+3β β β β 4, α 3 =4 3 β + β + β 3 3β 4, α 4 = 6 6+β β +β 3 +β 4. For a CFL coefficient c> 3 we need α k β k > 3 for all k. This implies 4 3β β + β 3 β 4 > 0, 36+8β 5β 6β 3 +9β 4 > 0, 4 9β +6β + β 3 8β 4 > 0, 6+β β +β 3 +9β 4 > 0. Combining these 9 times the first inequalityplus 8 times the second plus 3 times the third we get which implies a negative β. 40β 36β 3 > 0,
19 STRONG STABILITYPRESERVING TIME DISCRETIZATION METHODS 07 We again remark that if more steps are allowed, the CFL coefficient can be improved. Scheme 7 in Table 5. is a fivestep, thirdorder method with positive α i and β i and a CFL coefficient c =. Scheme 8 in Table 5. is a sixstep, thirdorder method with positive α i and β i and a CFL coefficient c = We now move on to fourstep, fourthorder methods. In [9] we gave a fourstep, fourthorder method scheme 9 in Table 5. with a CFL coefficient c A computer search gives a slightlybetter scheme with a CFL coefficient c 0.59, scheme 0 in Table 5.. If we allow two more steps, we can improve the CFL coefficient to c =0.45 scheme in Table 5.. Next we move on to fivestep, fourthorder methods. It is now possible to have SSP schemes with positive α i and β i. The solution can be written as the following fiveparameter family: α 5 = α α α 3 α 4, β = α +8α 3 +9α 4 +4β 5, β = α 45α 3α 3 37α 4 96β 5, β 3 = 4 5+3α +7α +40α 3 +59α β 5, β 4 = α 63α 64α 3 55α 4 96β 5. We can clearlysee that to get β 0 we would need α 5 64, and also β 55 4, hence the CFL coefficient cannot exceed c α β A computer search gives a scheme scheme in Table 5. with a CFL coefficient c = 0.0. The significance of this scheme is that it disproves the belief that SSP schemes of order four or higher must have negative β and hence must use L see Proposition 4. for Runge Kutta methods. However, the CFL coefficient here is probablytoo small for the scheme to be of much practical use. We finallylook at fivestep, fifthorder methods. In [9] a scheme with CFL coefficient c =0.077 is given scheme 3 in Table 5.. A computer search gives us a scheme with a slightlybetter CFL coefficient c 0.085, scheme 4 in Table 5.. Finally, by increasing one more step, one could get [9], a scheme with CFL coefficient c =0.30, scheme 5 in Table 5.. We list in Table 5. the multistep methods studied in this section. 6. Implicit SSP Methods. 6.. Implicit TVD Stable Scheme. Implicit methods are useful in that theytypicallyeliminate the stepsize restriction CFL associated with stabilityanalysis. For manyapplications, the backward Euler method possesses strong stabilityproperties that we would like to preserve in higher order methods. For example, it is easyto show a version of Harten s lemma [8] for the TVD propertyof implicit backward Euler methods. Lemma 6. Harten. The following implicit backward Euler method [ u n+ j = u n j + t C j+ u n+ j+ un+ j Dj u n+ j u n+ ] 6. j, where C j+ and D j are functions of un and/or u n+ at various usually neighboring grid points satisfying 6. C j+ 0, D j 0, is TVD in the sense of.5 for arbitrary t.
20 08 SIGAL GOTTLIEB,CHIWANG SHU,AND EITAN TADMOR Proof. Taking a spatial forward difference in 6. and moving terms, one gets [ ] u + t C j+ + D n+ j+ j+ un+ j = u n j+ u n j + tc j+ 3 u n+ j+ un+ j+ + tdj Using the positivityof C and D in 6. one gets [ ] + t C j+ + D j+ u n+ j+ un+ j u n j+ u n j + tc j+ 3 u n+ j+ un+ j+ + td j u n+ j u n+ j. u n+ j u n+ j, which, upon summing over j, would yield the TVD property.5. Another example is the cell entropyinequalityfor the square entropy, satisfied bythe discontinuous Galerkin method of arbitraryorder of accuracyin anyspace dimensions, when the time discretization is bya class of implicit time discretization including backward Euler and Crank Nicholson, again without anyrestriction on the time step t []. As in section for explicit methods, here we would like to discuss the possibility of designing higher order implicit methods that share the strong stabilityproperties of backward Euler, without anyrestriction on the time step t. Unfortunately, we are not as lucky in the implicit case. Let us look at a simple example of secondorder implicit Runge Kutta methods: 6.3 u = u n + β tlu, u n+ = α,0 u n + α, u + β tlu n+. Notice that we have onlya single implicit L term for each stage and no explicit L terms, in order to avoid timestep restrictions necessitated bythe strong stability of explicit schemes. However, since the explicit Lu term is contained indirectly in the second stage through the u term, we do not lose generalityin writing the schemes as the form in 6.3 except for the absence of the Lu n terms in both stages. To simplifyour example we assume L is linear. Secondorder accuracyrequires the coefficients in 6.3 to satisfy 6.4 α, = β β, α,0 = α,, β = β β. To obtain an SSP scheme from 6.4 we would require α,0 and α, to be nonnegative. We can clearlysee that this is impossible, as α, is in the range [4, + or, 0. We will use the following simple numerical example to demonstrate that a non SSP implicit method maydestroythe nonoscillatorypropertyof the backward Euler method, despite the same underlying nonoscillatory spatial discretization. We solve the simple linear wave equation 6.5 u t = u x with a stepfunction initial condition: 6.6 ux, 0 = { if x 0, 0 if x>0,
Advanced Computational Fluid Dynamics AA215A Lecture 5
Advanced Computational Fluid Dynamics AA5A Lecture 5 Antony Jameson Winter Quarter, 0, Stanford, CA Abstract Lecture 5 shock capturing schemes for scalar conservation laws Contents Shock Capturing Schemes
More information5.2 Accuracy and Stability for u t = c u x
c 006 Gilbert Strang 5. Accuracy and Stability for u t = c u x This section begins a major topic in scientific computing: Initialvalue problems for partial differential equations. Naturally we start with
More informationAdvanced CFD Methods 1
Advanced CFD Methods 1 Prof. Patrick Jenny, FS 2014 Date: 15.08.14, Time: 13:00, Student: Federico Danieli Summary The exam took place in Prof. Jenny s office, with his assistant taking notes on the answers.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationGrid adaptivity for systems of conservation laws
Grid adaptivity for systems of conservation laws M. Semplice 1 G. Puppo 2 1 Dipartimento di Matematica Università di Torino 2 Dipartimento di Scienze Matematiche Politecnico di Torino Numerical Aspects
More informationPerformance Comparison and Analysis of Different Schemes and Limiters
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:5, No:7, 0 Performance Comparison and Analysis of Different Schemes and Limiters Wang Wenlong, Li
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationFinite Difference Approach to Option Pricing
Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where
More informationFourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions
FourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of MichiganDearborn,
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationLinear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:
Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrixvector notation as 4 {{ A x x x {{ x = 6 {{ b General
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationLecture 5  Triangular Factorizations & Operation Counts
LU Factorization Lecture 5  Triangular Factorizations & Operation Counts We have seen that the process of GE essentially factors a matrix A into LU Now we want to see how this factorization allows us
More informationImplementation of the Bulirsch Stöer extrapolation method
Implementation of the Bulirsch Stöer extrapolation method Sujit Kirpekar Department of Mechanical Engineering kirpekar@newton.berkeley.edu December, 003 Introduction This paper outlines the pratical implementation
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationOptimization of Supply Chain Networks
Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal
More information1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easytocheck
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationABSTRACT FOR THE 1ST INTERNATIONAL WORKSHOP ON HIGH ORDER CFD METHODS
1 ABSTRACT FOR THE 1ST INTERNATIONAL WORKSHOP ON HIGH ORDER CFD METHODS Sreenivas Varadan a, Kentaro Hara b, Eric Johnsen a, Bram Van Leer b a. Department of Mechanical Engineering, University of Michigan,
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationHomework One Solutions. Keith Fratus
Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product
More informationModule MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions
Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective
More information2.4 Mathematical Induction
2.4 Mathematical Induction What Is (Weak) Induction? The Principle of Mathematical Induction works like this: What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want
More informationEnergy conserving time integration methods for the incompressible NavierStokes equations
Energy conserving time integration methods for the incompressible NavierStokes equations B. Sanderse WSC Spring Meeting, June 6, 2011, Delft June 2011 ECNM11064 Energy conserving time integration
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More information5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationMath 317 HW #5 Solutions
Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show
More information1 Polyhedra and Linear Programming
CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationNotes for AA214, Chapter 7. T. H. Pulliam Stanford University
Notes for AA214, Chapter 7 T. H. Pulliam Stanford University 1 Stability of Linear Systems Stability will be defined in terms of ODE s and O E s ODE: Couples System O E : Matrix form from applying Eq.
More informationOn computer algebraaided stability analysis of dierence schemes generated by means of Gr obner bases
On computer algebraaided stability analysis of dierence schemes generated by means of Gr obner bases Vladimir Gerdt 1 Yuri Blinkov 2 1 Laboratory of Information Technologies Joint Institute for Nuclear
More informationAlgebraic Flux Correction I. Scalar Conservation Laws
Algebraic Flux Correction I. Scalar Conservation Laws Dmitri Kuzmin 1 and Matthias Möller 2 1 Institute of Applied Mathematics (LS III), University of Dortmund Vogelpothsweg 87, D44227, Dortmund, Germany
More informationTD(0) Leads to Better Policies than Approximate Value Iteration
TD(0) Leads to Better Policies than Approximate Value Iteration Benjamin Van Roy Management Science and Engineering and Electrical Engineering Stanford University Stanford, CA 94305 bvr@stanford.edu Abstract
More informationDiscussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine GenonCatalot To cite this version: Fabienne
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationLecture 12 Basic Lyapunov theory
EE363 Winter 200809 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationA Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains
A Domain Embedding/Boundary Control Method to Solve Elliptic Problems in Arbitrary Domains L. Badea 1 P. Daripa 2 Abstract This paper briefly describes some essential theory of a numerical method based
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
MATHEMATICS OF COMPUTATION Volume 73, Number 247, Pages 1195 1 S 255718(3)15849 Article electronically published on July 14, 23 PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract.
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More information1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationSome Problems of SecondOrder Rational Difference Equations with Quadratic Terms
International Journal of Difference Equations ISSN 09736069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of SecondOrder Rational Difference Equations with Quadratic
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationMATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the padic absolute values for
More informationDomain Decomposition Methods. Partial Differential Equations
Domain Decomposition Methods for Partial Differential Equations ALFIO QUARTERONI Professor ofnumericalanalysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federale de Lausanne, Switzerland ALBERTO
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationProof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232  Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMath 317 HW #7 Solutions
Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationAn Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig
An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationDouble Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine Email: habil@iugaza.edu Abstract This research considers two traditional important questions,
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 1017 A MULTIWAVELET TYPE LIMITER FOR DISCONTINUOUS GALERKIN APPROXIMATIONS VANI CHERUVU AND JENNIFER K. RYAN ISSN 13896520 Reports of the Delft Institute of Applied
More informationThe distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña
The distance of a curve to its control polygon J. M. Carnicer, M. S. Floater and J. M. Peña Abstract. Recently, Nairn, Peters, and Lutterkort bounded the distance between a Bézier curve and its control
More informationLECTURE NOTES: FINITE ELEMENT METHOD
LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite
More informationA SemiLagrangian Approach for Natural Gas Storage Valuation and Optimal Operation
A SemiLagrangian Approach for Natural Gas Storage Valuation and Optimal Operation Zhuliang Chen Peter A. Forsyth October 2, 2006 Abstract The valuation of a gas storage facility is characterized as a
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More information