The Strong or differential form is obtained by applying the divergence theorem

Transcription

1 JOp p ) The Heat Equation a Parabolic Conservation Law Derivation, preliminaries... 2 An initial-bounary value problem for the D Heat equation Solution estimates in L Continuous problem Growth estimate Maxima an Minima: Total variation is non-increasing Conservation Semi-iscretize problem Discretization by finite volumes The matrix formulation Growth estimate Maxima an minima Conservation Accuracy Error boun The fully iscrete system Time iscretization by the θ-metho Conservation Growth estimates Stability: Growth Factor Implicit methos Maxi-min properties... Derivation, preliminaries See L. pp x x Temperature T K) Internal energy per unit mass HT) J), for a soli or liqui) H = CT) T, incluing latent heat of phase change Specific heat C J/K kg)) Density mass per unit volume, ρ kg/m 3 ) Heat flux F, W/m 2 ) Volume heat source s W/m 3 ) from combustion, ohmic electric heating, ) Integral form of conservation law for thermal energy, Ω fixe volume with surface Ω, area element on Ω S, an outwars unit normal n: $ "HT)# $ F % ns = $ s# # &# # The Strong or ifferential form is obtaine by applying the ivergence theorem " #HT)) $ % F = sx,y,z,t) "t an letting the volume Ω shrink to the point x,y,z). The ivergence theorem, or Gauss theorem, says that " # F$ = F # ns & & $ %$ The ivergence operator in Cartesian coorinates is

2 JOp p 2 ) " # F = ivf e x F 2 e y F 3 e z ) = $F $x $F 2 $y $F 3 $z ; " # Fx,y,z) = lim * ' % & % F # n S- )* $%,- an the graient makes a column vector out of a scalar φ, "# = $# $x e x $# $y e y $# $z e z The Laplace operator is "# = $ %$# = &2 # &x 2 &2 # &x 2 &2 # &x 2 Fourier s law states that the heat flux F is in the irection of the negative temperature graient, F = "k#t; where k is the heat conuctivity, units W/mK). The heat equation finally becomes "C #T #t $ % & k%t ) = sx,y,z,t) obtaine by shrinking Ω to a point. This is the correct form also when the ata ρ, C an k vary with position an with T. When the coefficients are constant, "T "t = #$T s %C,# = s %C where α is calle the thermal iffusivity with units m 2 /s. It follows, that significant influence extens about a istance "x # $ % "t in time t. Actually, the influence is felt everywhere immeiately, but only very slightly for large istances. This is characteristic of parabolic equations, in contrast to the hyperbolic equations which moel finite propagation spees for isturbances. In the following we will often formulate the moels in non-imensional quantities. Here is an example: "C #T = k$t in a square [,L] 2 #t Initial conition: Tx,y,) = Bounary conitions : T = T e Choose temperature scale T e, length-scale L, an time scale k/ρcl 2 ). The moel with new non-imensional variables q = T/T e, ξ = x/l, η = y/l an τ = t/ becomes [ ] 2 q " = q ## q $$ in unit square, Initial conition: q#,$,) = Bounary conitions : q = 2 An initial-bounary value problem for the D Heat equation We will now analyze the solution properties of the initial-bounary value problem for a -D version with more general bounary conitions, an its finite volume iscretization. The finite volume metho works with fluxes, so B.C. in terms of flux are most natural. We change back to t,x) notation)

3 JOp p 3 ) q t = ".)q x ) x s in [,] Initial conition: qx,) = q x) Bounary conitions h, # ) : x = :")q x = h $ q g, x =:")q x = %h $ q % g where α is assume positive, αx) > δ >. The Raiation, or Robin, conition is often formulate as in imensional formulation) k"t # n = ht g where h is the heat transfer coefficient [W/m2 K) ]. Dirichlet conitions specification of the potential variable T itself are approximate for large h, T = T e O/h) by taking g = ht e. Neumann conitions specification of the flux g through the bounary h = Also finite element methos for the heat equation often implement Dirichlet conitions approximately by large heat transfer coefficients h, or exactly by introuction of Lagrange multipliers. Interesting properties: Existence an uniqueness of solution Growth of solution with time Smoothness of solution maxi/mini properties, e.g. positivity Conservation 2. Solution estimates in L 2. We will first look at the continuous problem, an the name of the game is the obtention of solution estimates. Inee, growth properties of the solution are the key also to showing existence an uniqeness. For a constant coefficient problem with simple bounary conitions like this one may write own the solution as a Fourier series or integral: Perioic problems an Cauchy problems i.e., pose on the whole real line "# < x < #) are pure initial value problems. The other class, Energy estimates, is applicable also to some variable coefficient problems, so we will treat the case with α = αx), an assume that αx) > δ >. The estimates here use the L 2 -norm, f 2 = f 2 x)x) /2 which is associate with the inner prouct f,g) = f x)gx)x " " for complex functions on [,]. The functions f which are square-integrable make up the linear norme function space{ f} = L 2 [,] ). If also their first erivatives are square-integrable, they form the Hilbert inner prouct function space H [,] ). The reaer is remine of the Cauchy-Schwarz inequality f,g) " f 2 # g 2 an the triangle inequality f g 2 " f 2 g 2.

4 JOp p 4 ) 3 Continuous problem 3. Growth estimate Multiply the ifferential equation by q an integrate by parts to obtain " qq t x = " #x)qq xx x " qsx $q 2 " x = q 2 2 = [#qq x ] 2 $t 2 % # qx q,s) The bounary terms become ")qq x ) #")qq x ) = #h q) 2 g ) # h q) 2 g ) < < #h q) 2 h q) 2 g g ) an the growth estimate now epens on the signs of g. If g >, the bounary terms are nonpositive, so: q 2 q " # $ q x q % s an the growth estimate q.,t) " q # t $ s follows. However, it neglects the action of iffusion: the q x - integral. To inclue that for the case g = g =, we can make use of the Poincaré inequality : For functions q in H [,] with ")q x ) = h q),")q x ) = #h q),h,h > there is a positive constant K such that " q x 2 # K q 2 will be shown for BC q) =, q x ) = ). Note that applications to imensional equations nee to inclue the size of the omain. This leas to q " #K 2 q s, q.,) = q an & q " e #Ct t ) q % e Ct s.,$ ) $ 2,C = #K ' * This L 2 - boun of the solution in terms of the ata initial ata q an riving function s) with a growth parameter C vali for all ata is a statement of the well-poseness of the initial bounary value problem: The solution is unique an epens continuously on ata. But we also have 2 " q x # $/2 q 2 q % s : t 2 & " q x #/2 q.,) 2 t & q % s so also the erivative is boune in terms of L 2 -norms of initial ata etc. The parabolic equation is smoothing. For s =, q.,t) is analytic for t > even if q is only in L 2 ).

5 JOp p 5 ) A more accurate estimate can use the eigenvalues of " #x)q x ) x = $q #)q x = h q),#)q x = "h q) because the minimal value of " q x 2 q 2 h q) 2 h q) 2 is just the minimal eigenvalue λ. This prouces q " #$ q s, q.,) = q We return to this in the analysis of the iscretize equation. 3.2 Maxima an Minima: Total variation is non-increasing The existence of maximum properties epen on the source function s. For s =, local maxima can t increase an local minima can t ecrease: The total variation of q is nonincreasing, # & " q x x % ) $ ' Sketch of proof: Assume that x*is a local maximum at t*: qx*,t*) < qx,t*) for all x in a neighborhoo of x * Since q is analytic, q x x*,t*) = an q xx x*,t*), so " "t qx*,t*) = #x*) x q x x*,t*) #x*)q!##"## $ xx x*,t*)!##" ## $ $ 3.3 Conservation Integration of the equation over [,] prouces # & " qx % = )q [ x ] " sx = *h q) * g * h q) * g " sx $ ' which is not so exciting since that is the conservation law in integral form. 4 Semi-iscretize problem 4. Discretization by finite volumes The iscretization is one in two steps, first in space then in time, a proceure calle semiiscretization or the metho of lines. The cells are numbere i =,2,, N. The q-noes are space x apart. The q i value associate with a cell i is the arithmetic mean over the cell. We a ghost cells numbere an N with values q an q N at x = x/2 an x/2, viz. The flux at interface i /2 is at x = i ) x an approximate by the central ifference F i"/2 = "#x i"/2 ) $q $x x i"/2 ) = "#x i"/2 ) qx i,t) " qx i",t) O%x 2 ) 2) %x The bounary conition nees to approximate both q an q x at x =. If only q were require we coul instea locate the mipoint of cell at x = so q woul be known an the formula above coul be use for interfaces 3/2, 5/2,. A secon orer accurate approximation to the bounary conition expression is

6 JOp p 6 ) q#x /2) $ q$#x /2) q#x /2) q$#x /2) ") = h : #x 2 q = % q,% = h S BC) #x $ 2") h #x 2") ;% = 2h #x h #x 2") We shall see below that this is not really accurate enough, but it satisfies most other quantitative requirements. The flux expression 2) now is use also for the /2-interface. The equations written out are "x q = "x # $/2 % $)q $# /2 q $ q 2 )) "x & s "x q i = "x # i$/2 q i$ $ q i)$# i/2 q i $ q i )) "x & s i,i = 2,3,...,N $ "x q N = "x # N$/2 q N$ $ q N ) $# N /2 q N $% ) ) "x & s N Diviing by x prouces the final form, where the equations are scale to be compare with the terms of the ifferential equation an not OΔx) by integrating over cells 4.. The matrix formulation q = Aq b,q = q,q 2,...,q N )T,b = s,s 2,...,s N ) T & a $ /2... ) $ /2 %$ /2 $ 3/2 ) $ 3/2... "x 2 $ # A = 3/2 %$ /2 $ 3/2 ) $ 3/ $ N%3/2 %$ N%3/2 $ N%/2 ) $ N%/2 '... $ N%/2 a NN * 2h a = %$ /2 $ %/2 ) %, $ %/2 = %$ /2 %$ "x %/2 h "x 2$) a NN = %$ N%/2 $ N /2 ) %, $ N /2 = %$ N%/2 %$ N /2 2h "x h "x 2$) A is symmetric an negative efinite. Its largest eigenvalue µ is an approximation to the eigenvalue λ of the bounary value problem above. 4.2 Growth estimate Take the scalar prouct of the system of ODEs with q: q T q = qt Aq q T b q 2 = q q " µ q 2 q # b 2 q " µ q b : qt) " e µ t ' t q) e $µ * % & b % ), which mimics the continuous problem.

7 JOp p 7 ) 4.3 Maxima an minima To prove non-increase of the total variation: V = # q j " q j we show that a local maximum cannot increase: q j is a maximum if q i > q i",q i > q i. Then "x 2 q i = # i$/2 q i$ $ q i )# i/2 q i $ q i ) < an a maximum cannot increase. QED N j= 4.4 Conservation N The total amount of q is Q = "x # $ q i = "x # T q, =,,...,) T. Take the scalar prouct of the i= system of ODEs with : Q = "x #T q = "x #T Aq T b) = = "x #$% )& $/2 q $ % )& N /2 q N ) "x # T b which approximates the exact relation to O x). The b-integral is the mipoint quarature rule, accurate to secon orer, but the bounary terms are less accurate. 4.5 Accuracy In the sense of consistency, the truncation error e is the amount by which the ifference equations q t " Aq " b = are NOT satisfie by the gripoint values of the exact solution: q e t " Aq e " b e = e, q e i = qx i,t),b e i = sx i ) where qx,t) is the exact solution. It is easily seen by Taylor expansion that % "x 2 ' # i$/2 q e i$ $ q e i) $# i/2q e i $ q e i) * b e i = #x)q & ) x ) x ) xi sx i ) O"x 2 ) but that is NOT clear from the fact that the fluxes are approximate to O x 2 ), eq. 2), because the final equation is / x times the flux ifferences. There remains to estimate the errors in rows an N of the system. That is slightly more complicate because the bounary conitions must be invoke. The result is perhaps surprising: The error is O), an thus the l 2 error of qt) Qt) becomes O x /2 ), completely ominate by the bounary conition errors. It is easily seen for h very large an αx) constant so the BC becomes q) = : Then the first equation becomes h 2 "3q q 2 ) = h 2 "3 h 2 q # h2 q # h # # = 3 4 q # h 2 q # #...= q...) 3h 2 q # 9h2 8 q # 27h q # " h /2 q # #...) h /2 h 2 q # #" h /2qiv...) = 3 h /2 4 q # " 8 h q # #...) Oh 2 ) h /2 q # #...)) = For brevity of notation, we use h for x an omit the argument of the erivative values q p).), the two first lines, when it is. Thus, the error is /4q xx. It follows that the conition q) = must be approximate to O x 4 ) by q) = an the ata on the gri q, q 2, q 3, for secon orer accuracy.

8 JOp p 8 ) Exercise a) Derive a formula for q xx x/2) using q) =,q x/2),q3 x/2), which has secon orer accuracy. How many points must be inclue? b) Extra challenge) The semi-iscrete system of a) is not symmetric. Is it negative efinite? Notes Dirichlet conitions are of course treate by shifting the gri half a step so x= is at the center of a cell, unless the present gri arrangement is manatory for other reasons other equations, other BC, etc. There remains to prove a growth estimate for the system with better bounary conition. Homogeneous conitions can sometimes be of higher orer than inhomogeneous because they imply symmetry which annihilates o or even erivatives, as the case may be. Ex. If q x ) = for q t = q xx, then all o space erivatives vanish at x =. 4.6 Error boun The boun for the error Et) = q e t) qt) of the solution with the simple BC follows immeiately from the growth estimate, because q e t " Aq e " b e = e, q t " Aq" b = : E t " AE = e b e " b), E) = q e ) " q) Theorem If i) the truncation error e, b b e, an the initial error E) are O x p ), ii) A is symmetric, negative efinite with a largest eigenvalue boune above by some c <, uniformly for all x small enough, then Et) 2 is also O x p ). 5 The fully iscrete system 5. Time iscretization by the θ-metho We will consier the θ-metho, which is a blening of explicit an implicit timeiscretization of the semi-iscrete system with the simple BC S BC) q n " q n = #ta $q n "$)q n )q n b n$ ) % I"$#tA)q n = I "$)#ta)q n #tb n$ $ = : Explicit Euler q n = I #ta)q n b n $ =/2 : Trapezoial or Crank " Nicholson I"/2#tA)q n = I #t /2A)q n #tb n/2 $ = : Implicit Euler I" #ta)q n = q n #tb n The non-integer subscript on b is implemente by interpolation. 5.2 Conservation The conservation properties follow from scalar proucts with, an Q n = T q n : The row sums of all rows of A except an N vanish,

9 JOp p 9 ) Q n " Q n = #t $c q "$)c N q N ) T b n$ where the c-coefficients are given in the escription of the A-matrix above. Note: A homogeneous Dirichlet conition q) = oes not give q =, etc.; but h =, zero flux, gives c = an inee Q n = const. 5.3 Growth estimates A is symmetric, negative efinite. All the matrices above share the same complete orthonormal system of eigenvectors S: AS = SΛ, S T S = I; Λ = iagµ, µ 2,, µ N ), an µ N µ N µ < -δ <. The growth estimate epens on Δt an θ. q n = Bq n "ti#$"ta) # b n$,b = I#$"tA) # I #$)"ta) q n % G q n "t b n$ G = &B) = max "tµ, "tµ N ) #$"tµ #$"tµ N Assume that the max. is the µ term an let max j b j be b max. Then q n " e µ % t n ' q #tb max & $ e #tµ *" e µ % t n q $ b max ' * ) & µ ) 5.3. Stability: Growth Factor A necessary conition is that the secon term in G has moulus, an that is guarantee if θ /2: for any t θ < /2: for Δt µ N < 2/ 2θ) We recognize here that the θ-metho is unconitionally stable for θ /2 the explicit scheme θ = has time-step restriction Δt µ N < 2. The latter has the perhaps more familiar form Δt/ x 2 /2 for αx) =, a time-step conition which is very restrictive. Most simulations prefer the freer choice of time step offere by the implicit schemes; the gain usually outweighs the extra work in solving equations in every timestep. When the coefficients o not vary with time, matrices etc. can be constructe once, an re-factore only as changes of time-step make it necessary. 5.4 Implicit methos The Crank-Nicholson scheme is secon orer accurate but gives slowly ecaying oscillations for large eigenvalues. It is unsuitable for parabolic problems with rapily ecaying transients. The θ = scheme amps all components, an shoul be use in the initial steps. The most use family of time-stepping schemes for parabolic problems are the Backwar Differentiation formulas, of orer through 5 which are Aα)-stable, also known as Gear s methos an available in matlab as ODE5S. For the system q/ = fq) with timestep Δt they are

10 JOp p ) "tf q n ) = #q n $/2# 2 q n / 3# 3 q n..., #q j = q j $ q j$, # 2 q j = # %#q j = #q j $ q j$ ) = q j $ 2q j$ $ q j$ Maxi-min properties Consier homogeneous Dirichlet conitions an no source. The θ-scheme gives q n = Bq n,b = I"#$tA) " I "#)$ta) n A scheme which gives q k = n " ckj q j,c kj # preserves monotonicity an can create no new j maxima because the conservation property makes the c kj sum to. For θ =, the positivity hols if "t "x 2 <. 2max#) For θ = it hols for any Δt, because B has positive elements. Thm. Let the real matrix B have non-positive off-iagonal elements an be iagonally ominant in the sense b ii " #% b ij,i =,2,...,N with strict inequality for at least one row. Then B is calle an M-matrix an B has non-negative elements. Proof Diviing each row by its iagonal element prouces a matrix with unit iagonal, an we can consier B = I F where # f ij $, f ii = an strict inequality in at least one row. The Gershgorin theorem guarantees that ρf) < an so j"i " F j j= sum obviously has only non-negative elements. QED. # converges to I$ F ) $. But the