This paper deals with the convergence of stable and consistent one-step approximations for linear parabolic initial-boundary-value problems with non-smooth solutions. The proofs given may be extended to semilinear parabolic problems using H.B. Keller's stability concept. Finally an extension to Lax's convergence theorem is given

Markowich, P.

International Institute for Applied Systems Analysis (IIASA)

Numerical Solution of Parabolic Problems with Non-Smooth SolutionsMarkowich, P. IIASA Professional PaperMay 1979Markowich, P. (1979) Numerical Solution of Parabolic Problems with Non-Smooth Solutions. IIASA Professional Paper. Copyright © May 1979 by the author(s). http://pure.iiasa.ac.at/1207/ All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at NUMERICAL SOLUTION OF PARABOLICPROBLEMS WITH NON-SMOOTH SOLUTIONSP. MarkowichMay 1979PP-79-2Professional Papers do not report on work of theInternational Institute for Applied Systems Analysis,but are produced and distributed by the Institute asan aid to staff members in furthering their profes-sional activities. Views or opinions expressed arethose of the author(s) and should not be interpretedas representing the view of either the Institute orits National Member Organizations.INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSISA-2361 Laxenburg, AustriaPREFACEThis paper deals with the convergence of stable and consis-tent one-step approximations for linear parabolic initial-boundary-value problems with non-smooth solutions. The proofsgiven may be extended to semilinear parabolic problems usingH.B. Keller's stability concept. Finally an extension to Lax'sconvergence theorem is given.iiiNUMERICAL SOLUTION OF PARABOLICPROBLEMS WITH NON-SMOOTH SOLUTIONSP. MarkowichIn this paper we consider the problem:= a(x,t)U + b(x,t)Ux+ c(x,t)U + f(x,t),xx(x,t) E: (0,1) x (O,T)II) U(x,O) = U (x)o,xE:[0,1] , T > 0III) U(O,t) = YO(t), U(1,t) = Y1 (t), t E: (O,T](I) is called a linear inhomogenous parabolic differential equa-tion in one space ,variable x, (II) the initial condition and(III) the boundary conditions.For the following we make the assumptions:(A) a, b, c, f E: Cr ([0,1] x [O,T]), r sufficiently large(B) a(x,t) ｾ k > 0, (x,t) E: [0,1] x [O,T] stability condition(C) Uo(O) = Yo(O), Uo(1) = Y1 (0) continuity of initial andboundary functions.-2-We know that the initial and boundary functions determine thedifferentiability (smoothness) of the solution U in the points(0,0) and (1,0), which is important for the smallness of the localerror of a consistent numerical procedure.If U , Y and Y1 are continuous functions then a unique solu-o 0tion U exists, which is continuous on [0,1] x [O,T] and thereforebounded in the closed set [0,1] x [O,T], and if U E C3 ([0,1]);oYO' Y1 E c2 ([O,T]) and Y; (0) Ｈ ｹ ｾ (0)), u; (0), ｕ ｾ (0), Uo (0)(U" (1), U' (1), U (1)), set for Ut ' U , U , U into the differen-o 0 0 xx xtial equation I), fulfill I), then U, Ut ' U , U are continuousx xxand -bounded on [°,1] x [0, T]. See [1] and [ 2] .We gain a numerical procedure by choosing numbers Nand M,and by forming the step sizes n = 1./N in x- direction andk = 1./M in t- direction, and by substituting appropriate differ-ence approximations for Ut ' U , U in the net-points (x., t )- x xx 1 nwith x. = ih and t = nk. So we can write our procedure in the1 nfollowing form assuming that h = h(k) with lim h(k) = °k-+O(*) n = 1 (1 ) H withUO = U0 = (U°(X 1 ),. . .. ,If BO(k,tn) = I, we call the scheme explicit, otherwise implicit.nThe U 's are (N-1)- vectors with the approximate solutions on then-th time level, R(k,t ) is the (N-1)- vector with worked-inn "-boundary-conditions on the n-th time level, f(t ) is the vectornwith the approximations for f(x.,t ), i = 1 (1)N-1, i.e.,"- 1 n TIlf(t) - (f(x 1 ,t), ... , f(xN 1,t) 11-+0 for k-+O with some appro-n n - npriate norm, and B (k,t ), B1 (k,t 1) are (N-1)- square matriceso n n-derived from the difference approximations for the derivatives.We define the local error of the procedure (*) for the para-bolic problem I), II), III) in the solution U as the sequence ofvectors.-3-- f (tn), n = 1 (1 ) Mwhere U(t.) are the vectors containing the solution U evaluatedｾin the net-points of the i-th-time-Ievel. Further we say that(*) is consistent with I), II), III) in U of order I ifIILn (u,k)11 ｾ C(U)k l , where C(U) is bounded and independent of n.We can show by Taylor's expansion that C (U) is a finitelinear com-:-bination of bounds of partial derivatives of U on the rectangle[0,1] x [O,T], if II-II is the maximum norm. The second importantconcept concerned with difference approximations is stability.We call the difference scheme (*) stable, if BO(k,tn) is invert-ible for k ｾ koand for all n ｾ N and if IIB O-1 (k,tn ) II ｾ P fork ｾ k and n ｾ N where P is independent of k and n and ifowith 1 ｾ m ｾ n, where L is independent of n, m and k. Furtherwe say that (*) is convergent to U, if for t = t = nk fixed,nlimIlUn(k) - U(tn)11 = 0 uniformly in t(Un(k) = Un).k+On+ooThe sequence of vectors En(k) = Un(k) - U(t ) is called globalnerror. We easily conclude convergence from stability and consis-tency. By solving the recursive relation (*) for Un = Un(k) wefind: IIUn (k)1I ｾ Llluoll + P(TL+1) max IIf(t.)1I presumingＱ ｾ ｩ ｾ ｾYO = Y1 = O. That means that Un(k) depends ｾ ｯ ｮ ｴ ｩ ｮ ｵ ｯ ｵ ｳ ｬ ｹ on theinitial condition UO and on the disturbance f (in the norm II II).For the following we set II xII = max Ix. I fori=1(1)N_tT N-1X = (X 1 ' . _., ｾ Ｍ Ｑ Ｉ e: JR - Now we can prove:Theorem 1: consider the parabolic problem I), II) and III) withthe assumptions (A), (B) and (C). Let (*) be a finit differenceapproximation to I), II) and III), which is stable and consistentof the order I with problems of the form 1*, 11*, III with-4-solutions in e m ( [0,1] x [O,T]) (problem-I), II), III) with inhomo-genity in e m- 2 ([0,1] x [O,T]) and changed initial function) andlet Uo' YO' Y1 of the given problem fulfill:a) YO (0)b) Y1 (0)" I= a(O,O)U (0) + b(O,O)U (0) + c(O,O)U (0) + f(O,O)00" I= a(1,0)Uo(1) + b(1,0)Uo(1) + c(1,0)Uo(1) + f(1,0)with YO' Y1 s em([O,T]), Uo s e3 ([0,1)], then the numerical pro-cedure (*) is convergent for the given problem I), II) and III)in the maximum norm.Proof: as mentioned before there exists a unique solution U ofthe given problem, so that U, Ut ' Ux' Uxx are continuous andbounded in [0,1] x [O,T]. (Proof in [1]).Now let s>o be fixed. We construct the sequence of Bernsteinpolynomials to U on [0,1] x [O,f]Bn(U,x,t)and know that: Bn(U, • , .) -+ Ua Bn(U, • , • ) -+ Utata Bn(U , . , . ) -+ Uax xa2---2 B (U,.,.) -+ Uax n xxuniformly on [0,1] x [O,T] for n -+ 00.1 2As Butzer has shown in [3] for functions U in e ([0,1] ), wecan prove it for our case.Now we set Us = Bn(U,.,.) with n > N(s) fixed so thatand define: Vs= Us-[(1-x) (UsP,t)-YO(t)) + x (Us(1,t)-Y 1 (t))].We have {vE (O,t) = YO (t)}v E (1 ,t) = Y1 (t)-5-and v E is a functionin Cm([0,1] x [O,T]), because· YO' Y1 are in Cm([O,T])00Bn(U, .. ) = UE is in C ([0,1] x [O,T]) and moreover:II U-v II 00 +11 Ut-v til 00 +11 U -v II 00 +11 U -v II 00 ｾ 2E+2E+2E+E = 7 EE E X EX XX EXXThat means, that we have constructed a function v in Cm ([0,1] xEx [O,T]) which has the boundary values as U and which approximatesU, Ut ' Ux and Uxx uniformly on the closed rectangle [0,1] x [O,T].We consider the neighboring problem:1*) v t = a(x,t)vxx + b(x,t)vx + c(x,t)v + f(x,t) +(x,t) E (0,1] x (O,T]11*) v(x,O) = v (x, 0), X £ [0 , 1 ]EIII) v(O,t) = YO(t), v(1,t) = Y1 (t), t E [O,T] [111* = III]which has the unique solution v = v .Em-2Z E C ([0,1] x [0,1]),Eand concludeliz Ｑ ｬ Ｐ Ｐ ｾ ｉ ｕ ｴ Ｍ ｡ Ｈ ｸ Ｌ ｴ Ｉ ｕ -b(x,t)U -c(x,t)U';"f(x,t}lI+E xx x+IIUt -v t-a(x,t) (U -v )-b(x,t) (U -v )-c(x,t) (U-v Ｉ ｉ ｉ ｾE xx EXX X EX EThe numerical procedure for 1*),11*), III) has the form, n = 1(1)MT= (v E: (x1 ' 0), - , V E: (x N- 1 ' 0) )and converges to v of oider 1, that means:sIIVn(k) - V (t )11 ｾ C(s)kl , because the order of convergence iss s nthe same as the order of consistency in the case of smooth solu-tions.The procedure for I), II), III) is:f(t )nWe subtract (v) from (vv) and get:T••• , Uo (xN - 1 ) - v s (xN_1'O»We use that the solution of a difference equation of this formdepends continuously on the initial condition and on the disturb-ance, if the boundary conditions are homogenous:We get for t = nk fixed in (O,T]:ｉ ｉ ｕ Ｈ ｴ Ｉ Ｍ ｕ ｮ Ｈ ｫ Ｉ ｉ ｉ ｾ ｉ ｕ Ｈ ｴ Ｉ Ｍ ｶ (t)II+lIv Ｈ ｴ Ｉ Ｍ ｖ ｮ Ｈ ｫ Ｉ ｉ ｉ Ｋ ｉ ｉ ｖ ｮ Ｈ ｫ Ｉ Ｍ ｕ ｮ Ｈ ｫ Ｉ ｬ ｬ ｾs s s s1 1ｾＷｳ + C(s)k + C2s = (7+C 2 )s + C(s)k1For k< (C td )I we get II U (t) _un (k) II ｾＨＸＫｃＲＩ E:, where C2 is independentof n, E: and k. If we start the proof with Ｘ Ｋ ｾ convergence follows.2Our second step is to neglect the conditions a) and b) inTheorem 1. So we prove:Theorem 2: consider the numerical procedure (*) for I), II) andIII) under the same assumptions as in Theorem 1. Let (A), (B)mand (C) be valid. If UoE:C([0,1]) and YO,Y 1 E:C ([O,T]), then thenumerical procedure (*) is convergent to the unique solution ofI), II) and III).Proof: Let E:>O be fixed. Then we chooseso that lIu -UE:ll oo<E:. The existence of UE:000approximation theorem of Weierstrass. Wea function UE: inois a consequencedefine:00C [(0,1]),of theｵ ｾ = ｕｾＭ｛ｸＨｙＱ ＨｏＩＭｕｾＨＱＩＩ + (1-x) Ｈ ｙ ｏ Ｈ ｏ Ｉ Ｍ ｕ ｾ Ｈ ｏ Ｉ Ｉ ｝We get: ｕ ｾ Ｈ ｏ Ｉ = YO(O) and ｕ ｾ Ｈ Ｑ Ｉ = Y1 (0) andE: -E: IIlu -U Ｑ ｉ ｾ ｬ ｵ -U 1I+lx ｅ Ｚ Ｋ Ｑ Ｑ Ｍ ｸ ｬ ｅ Ｚ ｾ ｅ Ｚo 0 0 0Now we choose a= yE:(1) = ° andshall satisfy:function yE:(x) E: C3 ([0,1]) fulfilling yE:(O) =II yE: 1I ｯｯｾ and form VE: = uE: + yE:. The function VE:o 0 . 02) Y1 (0)That means:= a(O,O)vE:" (0) + b(O,O)VE:' (0) + c(O,O)vE:(O) + f(O,O)00= a ( 1 , 0) ｶ ｾ Ｂ (1) + b ( 1 , 0) ｶ ｾ Ｇ (1) + c (1 , °)ｶｾ ( 1) + f (1 , °)1a) YO(O) - [f(O,O) + a(O,O)uE:" (0) + b(O,O)UE:' (0) +o 0E: " ,+ c(O,O)Uo(O)] = a(O,O)yE (0) + b(O,O)yE: (0)1b) Y1 (0) - [f(1,0) +a(1,0)uE:"(1) +b(1,0)UE:'(1) +o 0E: E:" ,+ c(1,0)Uo(1)] = a(1,0)y (1) + b(1,0)yE: (1)-8-E' E' E"We choose y (0) = Y (1) = 0 and compute y (0) = Y1 andE"y (1) = Y2 from the equations 1a) and 2a) and construct:EY (x) =Y 1 2 II--2 x (x-t 1 )2t 1oY2 2 4--(x-1) (x-t )2t 2 22ｏｾｾｴ 1t ｾｸｾｴ1 2t ｾｸｾＱ23EC ([0,1])1 4;----with 0 < t 1 < min Ｈ Ｒ Ｌ ｾ Ｒ Ｙ ｅ ) for Y1 f 0 and8TY11Otherwise there is no restriction on t 1 resp t 2 (only0<t 1<t 2 <1) .Now we consider:Ｈ ｾ )Vt = a(x,t)V +b(x,t)V +c(x,t)V+f(x,t)xx x= VE(x)o(x,t) E (0,1] x (0 ,T]xE[0,1]tE(O,T]E 3 rJn E EWe have: VOEC ([0,1]), YO' ｙ ｏ ｅ ｾ Ｍ Ｍ Ｍ Ｈ ｛ ｏ Ｌ ｔ ｝ Ｉ Ｌ Vo(O) = YO(O), Vo(1) =E= Y1 (0) and Vo' YO' Y1 fulfill the condition a) and b) in theorem1. So we can conclude, that this problem has a unique solutionVE ' so that VE' VEt' VEX' VEXX are continuous in [0,1]x[0,T].Also we can conclude that Z = U-V is the unique solution ofEZ(O,t) = Z(1,t) :: 0Zt = a(x,t)Z + b(x,t)Z + c(x,t)Zxx xZ (x, 0) = U0 (x) ｖ ｾ Ｈ ｸ Ｉ-9-(U is the unique solution of the given problem) .We know that the solution Z depends continuously on the initialdata Z(x,O), so we have:IIzll oo=IIU-V ｉｉｾＮｬｉｵ Ｍｶﾣｉｉｾﾣ£ 0 0The numerical procedure to the given problem has the form:f(t )nand to Ｈ ｬ ｾ Ｉf(t )nWe conclude by subtracting:-k1 [BO(k,t) (Un(k)-Vn(k» - B,(k,t 1) (Un-'(k)_Vn- 1 (k»] = 0n . £ n- £Applying theoremfor all k<k (£),oSo, ,1 we conclude, that there is a ko(£»O so thatBv (t) - ｖｮＨｫＩｬｉｾ for t = nk fixed in [O,T].£ £II U(t) _Un (k) II ｾｉ U(t) -v (t) II +11 v (t) _Vn (t) II Ｋｉｉｾ (t) _Un (k) ｉ ｉ ｾ£ £ £ £ｾﾣ + £ +3L£ = (C+1+3L)£And that means convergence.Putting the used proof-methods on a more formal level we canderive an extension to Lax's convergence theorem for stable ap-proximations to linear operator equations which are consistentfor data in a dense set. Consider the linear and invertibleoperator F : (A, II. II A) -+ (H, II II B where A, B are appropriate linear-1spaces and let II F II B be bounded by k 1 . That means that thesolution U of the equation FU = g depends continuously on thedata g. For the numerical computation of U we use approximationsFhUh = gh with the following properties:1)2)Fh : (Ah ,1I II A ) -+ (Bh ,1I liB ) for ｏ ＼ ｨ ｾ ｯ (step-size, gridh hparameter), where Ah , Bh are appropriate linear spaces.-1Fh is linear and invertible and IIFh liB ｾＲ for allhｨｾ •oThe last property of F h is called stability:3) There exist linear and uniformly bounded operators,4) BIll:Ih(g) - gh"B = 0(1) for h-+o.h5) The scheme FhUh = gh is consistent with PO = g for allgEXCB, where X is dense in B, i.e.,ｉ ｉ ｆ ｨ Ｈ ｬ Ｚ ｉ ｾ Ｉ - gh"B = 0(1) for h-+Ohwhere U is the solution of FU = g.We can conclude:Theorem 3: under the given assumptions on F and F h the procedureFhUh = gh is convergent to the solution U of the equation FU = g,for all gEB, i.e.,Ill:IAh (U) _·U II = 0 (1) for h-+O.h AhProof: We have the following situation:At:.Ah-11-F ｾｂt:. BhFh... BhLet E fixed be greater o.For solving FU = g we consider the scheme FhUh = gh. Because Xis dense in B we can choose g EX so that IIg-g ｉ ｉ ｂ ｾ Ｎ Instead ofE E_ 1FU = g we now solve FUE = gE. We conclude ｉ ｉ ｕ Ｍ ｕ ｅ ｉ ｉ ａ ｾ ｉ ｆ II IIg-gEIi Bthat means:Now we consider FhUEh = gEh and we easily prove the conver-gence of U h to U for h+o and fixed E>O by the usual consistency -E Estability method:IIc (h)II BE h = 0(1) for h+o and fixedE>O because g EX.E-C (h)=>EB)cE(h)+O for h+oE fixed 9reater than o.Now we want to find a bound for UEh - Uh :C)= k 2d E (h) + 0 for h+O and E>O fixedbecause of the assumptions 2), 3) and 4). So we can concludefrom (A), (B) and (C):-12-We can find for every €>o a h<h (E:) -so that Ｑ Ｑ Ｖ ｾ Ｍ ｕ II A <CE: where Ch his independent of €, h and that means convergence.It is easy to extend Theorem 3 to cases where the differencescheme Fh is uniformly continuous in h (stable) in some componentsof the data-vector g, but not in all. The methods for doing thisare the sane as used in Theorem 2, because stability of one stepdifference - approximation means that the solutions Un(k) dependuniformly continuous (in the grid-parameter k) on the initialdata and on the disturbance but not on the boundary values.RemarkI am very grateful to Professor R. Weiss from the Institutefor Numerical Mathematics at the Technical University of Viennafor his aid during the work.-13-References[1} Partial Differential Equations of Parabolic Type. AnverFriedman, Prentice Hall, Inc. Englewood Cliffs, 1964.[2] Linear and quasilinear Equations of Parabolic Type.Ladyzenskaja, Solonnikov, Uralceva. AMS, Volume 23.[3] Bernstein Bolynomials, Butzer.[4] Approximation Methods for non-linear Problems with Applica-tion to two-Point-Boundary-Value Problems. H.B. Keller,Math. Comput. Vol. 29, April 1975.

Numerical Solution of Parabolic Problems with Non-Smooth Solutions

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