Finite Difference Approximations


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1 Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip between various of its derivatives on some given region of space and/or time, along wit some boundary conditions along te edges of tis domain. In general tis is a difficult problem, and only rarely can an analytic formula be found for te solution. A finite difference metod proceeds by replacing te derivatives in te differential equations wit finite difference approximations. Tis gives a large but finite algebraic system of equations to be solved in place of te differential equation, someting tat can be done on a computer. Before tackling tis problem, we first consider te more basic question of ow we can approximate te derivatives of a known function by finite difference formulas based only on values of te function itself at discrete points. Besides providing a basis for te later development of finite difference metods for solving differential equations, tis allows us to investigate several key concepts suc as te order of accuracy of an approximation in te simplest possible setting. Let u.x/ represent a function of one variable tat, unless oterwise stated, will always be assumed to be smoot, meaning tat we can differentiate te function several times and eac derivative is a welldefined bounded function over an interval containing a particular point of interest Nx. Suppose we want to approximate u 0. Nx/ by a finite difference approximation based only on values of u at a finite number of points near Nx. One obvious coice would be to use u. Nx C / u. Nx/ D C u. Nx/ (.) for some small value of. Tis is motivated by te standard definition of te derivative as te limiting value of tis expression as! 0. Note tat D C u. Nx/ is te slope of te line interpolating u at te points Nx and Nx C (see Figure.). Te expression (.) is a onesided approximation to u 0 since u is evaluated only at values of x Nx. Anoter onesided approximation would be D u. Nx/ u. Nx/ u. Nx / : (.) 3
2 4 Capter. Finite Difference Approximations slope D C u. Nx/ slope u 0. Nx/ slope D u. Nx/ slope D 0 u. Nx/ Nx Nx Nx C u.x/ Figure.. Various approximations to u 0. Nx/ interpreted as te slope of secant lines. Eac of tese formulas gives a first order accurate approximation to u 0. Nx/, meaning tat te size of te error is rougly proportional to itself. Anoter possibility is to use te centered approximation u. Nx C / u. Nx / D 0 u. Nx/ D.D Cu. Nx/ C D u. Nx//: (.3) Tis is te slope of te line interpolating u at Nx and Nx C and is simply te average of te two onesided approximations defined above. From Figure. it sould be clear tat we would expect D 0 u. Nx/ to give a better approximation tan eiter of te onesided approximations. In fact tis gives a second order accurate approximation te error is proportional to and ence is muc smaller tan te error in a first order approximation wen is small. Oter approximations are also possible, for example, D 3 u. Nx/ Œu. Nx C / C 3u. Nx/ 6u. Nx / C u. Nx / : (.4) 6 It may not be clear were tis came from or wy it sould approximate u 0 at all, but in fact it turns out to be a tird order accurate approximation te error is proportional to 3 wen is small. Our first goal is to develop systematic ways to derive suc formulas and to analyze teir accuracy and relative wort. First we will look at a typical example of ow te errors in tese formulas compare. Example.. Let u.x/ D sin.x/ and Nx D ; tus we are trying to approximate u 0./ D cos./ D 0: Table. sows te error Du. Nx/ u 0. Nx/ for various values of for eac of te formulas above. We see tat D C u and D u beave similarly altoug one exibits an error tat is rougly te negative of te oter. Tis is reasonable from Figure. and explains wy D 0 u, te average of te two, as an error tat is muc smaller tan bot.
3 0 D C Copyrigt 007 by te Society for Industrial and Applied Matematics 5 Table.. Errors in various finite difference approximations to u 0. Nx/. D C u. Nx/ D u. Nx/ D 0 u. Nx/ D 3 u. Nx/.0e e e e e e 0.57e e 0.50e e 06.0e e e e e e e 03.04e 03.53e e 09.0e e e e e We see tat D C u. Nx/ u 0. Nx/ 0:4; D 0 u. Nx/ u 0. Nx/ 0:09 ; D 3 u. Nx/ u 0. Nx/ 0:007 3 ; confirming tat tese metods are first order, second order, and tird order accurate, respectively. Figure. sows tese errors plotted against on a loglog scale. Tis is a good way to plot errors wen we expect tem to beave like some power of, since if te error E./ beaves like E./ C p ; ten logje./j log jc jcplog : So on a loglog scale te error beaves linearly wit a slope tat is equal to p, te order of accuracy D D Figure.. Te errors in Du. Nx/ from Table. plotted against on a loglog scale.
4 6 Capter. Finite Difference Approximations. Truncation errors Te standard approac to analyzing te error in a finite difference approximation is to expand eac of te function values of u in a Taylor series about te point Nx, e.g., u. Nx C / D u. Nx/ C u 0. Nx/ C u 00. Nx/ C 6 3 u 000. Nx/ C O. 4 /; (.5a) u. Nx / D u. Nx/ u 0. Nx/ C u 00. Nx/ 6 3 u 000. Nx/ C O. 4 /: (.5b) Tese expansions are valid provided tat u is sufficiently smoot. Readers unfamiliar wit te bigo notation O. 4 / are advised to read Section A. of Appendix A at tis point since tis notation will be eavily used and a proper understanding of its use is critical. Using (.5a) allows us to compute tat u. Nx C / D C u. Nx/ D u. Nx/ D u 0. Nx/ C u00. Nx/ C 6 u 000. Nx/ C O. 3 /: Recall tat Nx is a fixed point so tat u 00. Nx/; u 000. Nx/, etc., are fixed constants independent of. Tey depend on u of course, but te function is also fixed as we vary. For sufficiently small, te error will be dominated by te first term u00. Nx/ and all te oter terms will be negligible compared to tis term, so we expect te error to beave rougly like a constant times, were te constant as te value u00. Nx/. Note tat in Example., were u.x/ D sin x, weave u00./ D 0:407355, wic agrees wit te beavior seen in Table.. Similarly, from (.5b) we can compute tat te error in D u. Nx/ is D u. Nx/ u 0. Nx/ D u00. Nx/ C 6 u 000. Nx/ C O. 3 /; wic also agrees wit our expectations. Combining (.5a) and (.5b) sows tat u. Nx C / u. Nx / D u 0. Nx/ C 3 3 u 000. Nx/ C O. 5 / so tat D 0 u. Nx/ u 0. Nx/ D 6 u 000. Nx/ C O. 4 /: (.6) Tis confirms te second order accuracy of tis approximation and again agrees wit wat is seen in Table., since in te context of Example. we ave 6 u000. Nx/ D cos./ D 0: : 6 Note tat all te odd order terms drop out of te Taylor series expansion (.6) for D 0 u. Nx/. Tis is typical wit centered approximations and typically leads to a iger order approximation. To analyze D 3 u we need to also expand u. Nx / as u. Nx / D u. Nx/ u 0. Nx/ C./ u 00. Nx/ 6./3 u 000. Nx/ C O. 4 /: (.7)
5 .. Deriving finite difference approximations 7 Combining tis wit (.5a) and (.5b) sows tat D 3 u. Nx/ D u 0. Nx/ C 3 u.4/. Nx/ C O. 4 /; (.8) were u.4/ is te fourt derivative of u.. Deriving finite difference approximations Suppose we want to derive a finite difference approximation to u 0. Nx/ based on some given set of points. We can use Taylor series to derive an appropriate formula, using te metod of undetermined coefficients. Example.. Suppose we want a onesided approximation to u 0. Nx/ based on u. Nx/; u. Nx /, and u. Nx / of te form D u. Nx/ D au. Nx/ C bu. Nx / C cu. Nx /: (.9) We can determine te coefficients a; b, and c to give te best possible accuracy by expanding in Taylor series and collecting terms. Using (.5b) and (.7) in (.9) gives D u. Nx/ D.a C b C c/u. Nx/.b C c/u 0. Nx/ C.b C 4c/ u 00. Nx/ 6.b C 8c/3 u 000. Nx/ C: If tis is going to agree wit u 0. Nx/ to ig order, ten we need a C b C c D 0; b C c D =; (.0) b C 4c D 0: We migt like to require tat iger order coefficients be zero as well, but since tere are only tree unknowns a; b; and c, we cannot in general ope to satisfy more tan tree suc conditions. Solving te linear system (.0) gives so tat te formula is a D 3=; b D =; c D = D u. Nx/ D Œ3u. Nx/ 4u. Nx / C u. Nx / : (.) Tis approximation is used, for example, in te system of equations (.57) for a point boundary value problem wit a Neumann boundary condition at te left boundary. Te error in tis approximation is D u. Nx/ u 0. Nx/ D 6.b C 8c/3 u 000. Nx/ C D u 000. Nx/ C O. 3 /: (.)
6 8 Capter. Finite Difference Approximations Tere are oter ways to derive te same finite difference approximations. One way is to approximate te function u.x/ by some polynomial p.x/ and ten use p 0. Nx/ as an approximation to u 0. Nx/. If we determine te polynomial by interpolating u at an appropriate set of points, ten we obtain te same finite difference metods as above. Example.3. To derive te metod of Example. in tis way, let p.x/ be te quadratic polynomial tat interpolates u at Nx, Nx and Nx, and ten compute p 0. Nx/. Te result is exactly (.)..3 Second order derivatives Approximations to te second derivative u 00.x/ can be obtained in an analogous manner. Te standard second order centered approximation is given by D u. Nx/ D Œu. Nx / u. Nx/ C u. Nx C / D u 00. Nx/ C u Nx/ C O. 4 /: (.3) Again, since tis is a symmetric centered approximation, all te odd order terms drop out. Tis approximation can also be obtained by te metod of undetermined coefficients, or alternatively by computing te second derivative of te quadratic polynomial interpolating u.x/ at Nx ; Nx, and Nx C, as is done in Example.4 below for te more general case of unequally spaced points. Anoter way to derive approximations to iger order derivatives is by repeatedly applying first order differences. Just as te second derivative is te derivative of u 0, we can view D u. Nx/ as being a difference of first differences. In fact, since Copyrigt 007 by te Society for Industrial and Applied Matematics D u. Nx/ D D C D u. Nx/ D C.D u. Nx// D ŒD u. Nx C / D u. Nx C / D D u. Nx/: D u. Nx/ u. Nx/ u. Nx/ u. Nx / Q3 Alternatively, D. Nx/ D D D C u. Nx/, or we can also view it as a centered difference of centered differences, if we use a step size = in eac centered approximation to te first derivative. If we define OD 0 u.x/ D u x C u x ; ten we find tat OD 0. OD 0 u. Nx// D u. Nx C / u. Nx/ u. Nx/ u. Nx / D D u. Nx/:
7 .4. Higer order derivatives 9 Example.4. Suppose we want to approximate u 00.x / based on data values U, U, and U 3, at tree unequally spaced points x ; x, and x 3. Tis approximation will be used in Section.8. Let D x x and D x 3 x. Te approximation can be found by interpolating by a quadratic function and differentiating twice. Using te Newton form of te interpolating polynomial (see Section B..3), p.x/ D U Œx C U Œx ; x.x x / C U Œx ; x ; x 3.x x /.x x /; we see tat te second derivative is constant and equal to twice te second order divided difference, p 00.x / D U Œx ; x ; x 3, U3 U U U D. C / (.4) were c D. C / ; D c U C c U C c 3 U 3 ; c D ; c 3 D. C / : (.5) Tis would be our approximation to u 00.x /. Te same result can be found by te metod of undetermined coefficients. To compute te error in tis approximation, we can expand u.x / and u.x 3 / in Taylor series about x and find tat c u.x / C c u.x / C c 3 u.x 3 / u 00.x / D 3. /u.3/.x / C 3 C! 3 u.4/.x / C: C (.6) In general, if, te error is proportional to max. ; / and tis approximation is first order accurate. In te special case D (equally spaced points), te approximation (.4) reduces to te standard centered approximate D u.x / from (.3) wit te second order error sown tere..4 Higer order derivatives Finite difference approximations to iger order derivatives can also be obtained using any of te approaces outlined above. Repeatedly differencing approximations to lower order derivatives is a particularly simple approac. Example.5. As an example, ere are two different approximations to u 000. Nx/. Te first is uncentered and first order accurate: D C D u. Nx/ D.u. Nx C / 3u. Nx C / C 3u. Nx/ u. Nx // 3 D u 000. Nx/ C u0000. Nx/ C O. /:
8 0 Capter. Finite Difference Approximations Te next approximation is centered and second order accurate: D 0 D C D u. Nx/ D.u. Nx C / u. Nx C / C u. Nx / u. Nx // 3 D u 000. Nx/ C 4 u Nx/ C O. 4 /: Anoter way to derive finite difference approximations to iger order derivatives is by interpolating wit a sufficiently ig order polynomial based on function values at te desired stencil points and ten computing te appropriate derivative of tis polynomial. Tis is generally a cumbersome way to do it. A simpler approac tat lends itself well to automation is to use te metod of undetermined coefficients, as illustrated in Section. for an approximation to te first order derivative and explained more generally in te next section..5 A general approac to deriving te coefficients Te metod illustrated in Section. can be extended to compute te finite difference coefficients for computing an approximation to u.k/. Nx/, te kt derivative of u.x/ evaluated at Nx, based on an arbitrary stencil of n k C points x ; :::; x n. Usually Nx is one of te stencil points, but not necessarily. We assume u.x/ is sufficiently smoot, namely, at least n C times continuously differentiable in te interval containing Nx and all te stencil points, so tat te Taylor series expansions below are valid. Taylor series expansions of u at eac point x i in te stencil about u. Nx/ yield u.x i / D u. Nx/ C.x i Nx/u 0. Nx/ CC k!.x i Nx/ k u.k/. Nx/ C (.7) for i D ; :::; n. We want to find a linear combination of tese values tat agrees wit u.k/. Nx/ as well as possible. So we want c u.x / C c u.x / CCc n u.x n / D u.k/. Nx/ C O. p /; (.8) were p is as large as possible. (Here is some measure of te widt of te stencil. If we are deriving approximations on stencils wit equally spaced points, ten is te mes widt, but more generally it is some average mes widt, so tat max in jx i Nxj C for some small constant C.) Following te approac of Section., we coose te coefficients c j so tat.i /! nx c j.x j Nx/.i / D jd if i D k; 0 oterwise (.9) for i D ; :::; n. Provided te points x j are distinct, tis n n Vandermonde system is nonsingular and as a unique solution. If n k (too few points in te stencil), ten te rigtand side and solution are bot te zero vector, but for n > k te coefficients give a suitable finite difference approximation.
9 .5. A general approac to deriving te coefficients Q4 How accurate is te metod? Te rigtand side vector as a in te i D k C row, wic ensures tat tis linear combination approximates te kt derivative. Te 0 in te oter component of te rigtand side ensures tat te terms 0 c j.x j Nx/.i / A u.i /. Nx/ jd drop out in te linear combination of Taylor series for i k. For i < k tis is necessary to get even first order accuracy of te finite difference approximation. For i > k (wic is possible only if n > k C ), tis gives cancellation of iger order terms in te expansion and greater tan first order accuracy. In general we expect te order of accuracy of te finite difference approximation to be at least p n k. It may be even iger if iger order terms appen to cancel out as well (as often appens wit centered approximations, for example). In MATLAB it is very easy to set up and solve tis Vandermonde system. If xbar is te point Nx and x(:n) are te desired stencil points, ten te following function can be used to compute te coefficients: function c = fdcoeffv(k,xbar,x) A = ones(n,n); xrow = (x(:)xbar) ; % displacements as a row vector. for i=:n A(i,:) = (xrow.ˆ (i))./ factorial(i); end b = zeros(n,); % b is rigt and side, b(k+) = ; % so k t derivative term remains c = A\b; % solve system for coefficients c = c ; % row vector If u is a column vector of n values u.x i /, ten in MATLAB te resulting approximation to u.k/. Nx/ can be computed by c*u. Tis function is implemented in te MATLAB function fdcoeffv.m available on te Web page for tis book, wic contains more documentation and data cecking but is essentially te same as te above code. A row vector is returned since in applications we will often use te output of tis routine as te row of a matrix approximating a differential operator (see Section.8, for example). Unfortunately, for a large number of points tis Vandermonde procedure is numerically unstable because te resulting linear system can be very poorly conditioned. A more stable procedure for calculating te weigts is given by Fornberg [8], wo also gives a FORTRAN implementation. Tis modified procedure is implemented in te MATLAB function fdcoefff.m on te Web page. Finite difference approximations of te sort derived in tis capter form te basis for finite difference algoritms for solving differential equations. In te next capter we begin te study of tis topic.
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