Spigler, Renato

Vianello, Marco

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 8Renato Spigler; Marco VianelloSemi-implicit discretization of abstract evolution equationsIn: Pavol Brunovský and Milan Medveď (eds.): Equadiff 8, Czech - Slovak Conference on DifferentialEquations and Their Applications. Bratislava, August 24-28, 1993. Mathematical Institute, SlovakAcademy of Sciences, Bratislava, 1994. Tatra Mountains Mathematical Publications, 4. pp. 207--212.Persistent URL: http://dml.cz/dmlcz/700108Terms of use:© Comenius University in Bratislava, 1994Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czTATRA Tatra Mountains Math. Publ. 4 (1994), 207-212 -WDU"TdiNS Mathematical Publications SEMI-IMPLICIT DISCRETIZATION OF ABSTRACT EVOLUTION EQUATIONS RENATO SPIGLER — M A R C O VIANELLO ABSTRACT. Abstract evolution equations in an arbitrary Banach space X, like u — f(t,u,u), t G ( 0 , T ] , subject to the initial value u(0) — UQ , are discretized by a semi-implicit version of the Euler method . The basic assumptions being that / (£ , •, v) is one-sided Lipschitz, 1Z(l—hf(t, •, v)) — X for h > 0 sufficiently small, and f(t,u, •) is Lipschitz continuous, we show that the iterative scheme un+i — un -f hf((n -f l ) A t , t z n + i , i i n ) , n = 0 , 1 , . . . , N - 1 , At = T/N, is stable and consistent, and hence convergent. Applications to systems of evolutionary PDEs are presented and the computational advantages of the semi-implicit method are pointed out. 1. Introduction Discretizing first in time evolutionary problems (Rothe method, see [3, 7, 8], e.g.), presents the advantage of retaining a great flexibility in the choice of the subsequent space discretization algorithms. One of the most popular methods of this type is the implicit Euler method. In this paper, we propose and analyze a semi-implicit version of such a scheme, for a broad class of nonlinear abstract evolution problems in an arbitrary Banach space. Stability, consistency, and convergence are studied and precise error estimates obtained. Applications are given to semilinear parabolic (integro-) differential systems, in both reflexive and non-reflexive spaces, for the purpose of illustration. The main computational advantages in these cases are: (a) linearization at each time step; (b) decoupling of systems into independent stationary subsystems, thus allowing for parallel implementation] (c) handling local instead of global (e.g. integral) space operators. In order to contain the length of the paper, below we give only the main results, leaving all details to a future publication [11]. A M S S u b j e c t C l a s s i f i c a t i o n (1991): Primary 65J15; Secondary 35K50. K e y w o r d s : abstract evolution equations, Rothe method, semi-implicit Euler method, reaction-diffusion systems. This work has been supported by the Mathematical Analysis funds ("60%"-funds), the Nu-merical Analysis funds ("40%"-funds), the GNIM-CNR and the GNFM-CNR. 207 RENATO SPIGLER MARCO VIANELLO 2. Convergence analysis Consider the abstract evolution problem u = f(t,u,u), te ( 0 , T ] ; u(0) = uo, (1) where f(t, •, v): (Dt C X) —> X, X being a real or complex Banach space, t G (0, T ], v G D, with [J Dt C D C X, satisfies the one-sided Lipschitz condition te(o,T] \\u1-u2-h(f(t,u1,v)-f(t,u2,v))\\ >{l-hK1)\\u1"U2\\, (2) VMI , U2 G Dt, with hKx < 1, Kx G R, and f(t,u,-): (D C X) -> X sat-isfies a classical Lipschitz condition, | | /(t , ifc^i) — / ( £ , ^ ^ 2 ) | | < -K^H î — v2\\, uniformly in t,u. Moreover, suppose that , for every h > 0 sufficiently small, 7£(l - hf(t, • , ^;)) = X, that is the equation u = hf(t,u,v) + b has a (unique) solution u in Dt, for each fixed t G ( 0 , T ] , v G D, and b G X. Recall that, when / is (strongly) dissipative in u, i.e., K1 < 0, then the solvability of the previous equation for h = 1 suffices (cf. [14]). We assume that problem (1) has a (unique) "strict" solution, that is that u(t) G Dt for every t G ( 0 , T ] , u G C1([0,T]; .X), and ^x(t) solves (1). Our algorithm will approximate such a solution in the C°-norm. For n = 0 , 1 , . . . , N, we set tn=nh, h = At = T/N, and then consider the (ideal) iterative scheme un+1 =un + hf(tn+1,un+1,un) . (3) In order to take into account all relevant (and unavoidable) errors, we analyze the perturbed scheme, for n = 0 , 1 , . . . , N — 1, vn+1 = un + hf(tn+1,vn+1,un) + 6n+1 , (4) un+1 = vn+1 + an+1, un+1 G D . The scheme in (4) (and hence that in (3)) is well-defined since the first equation has a unique solution in Dtn+1, in view of the assumptions made on the operator f(t, •, v) for t > 0 and v G D. In (4), u0 = u0 + S0 G D, 60 denoting the error on the inital data, an+1 represents the overall error made in solving numerically the first equation, and Sn+1 takes into account, basically, the local truncation error. Note that when f(t, •, v) is discontinuous, the term an+1 cannot be embodied in 8n+1. This is the case of evolutionary partial differential equations, in contrast to that of ordinary differential equations. Comparing (3) and (4), one can find the stability estimate (cf. [11] N N maxv||«n-«n||<maX{l,piv}(||60|| + (l-/ iK1)-1£| |6 f e | | + ̂ | k f e | | ) , (5) ~ ~ fc=l k=l where hKx < 1, and p J V : = ( l - K i T / N ) ~ e xP{(*~i + * 2 ) T } , N-oo(/w0). (6) 208 SEMI-IMPLICIT DISCRETIZATION OF ABSTRACT EVOLUTION EQUATIONS Observe t ha t the solution u(i) to problem (1) solves an equation like u(tn+i) = u{tn) + hffa + x , H(/„ + 1 ),«/(/„)) i .v„. , • • / / ) . (7) for a sui table choice of cO?1 + i(/?), t ha t is H(/„) solves scheme (4) with c(f ~- 0, <5n+i = o ; n + i ( / i ) , and a n + i = 0, for n — 0,1 N - 1 . The stability es t imate N in (5) requires es t imat ing V] L;n+i(/j,)|L Using the regularity assumption u C ( ^ ( [ O / F ^ X ) , one obtains: 1v y ^ ||o;n_j_i(/i)|| < rF[osc(H;//.) -f- A~2 ose(H; /!,)] , (8) where osc(w;h) := sup{ | | i e ( / i ) — w(t<2)\\: /.,/.> G [0/F] , |/j •- / 2 | < //•} , and hence osc(u;h) = o ( l ) and osc(u; h) — O(h). Therefore max ||H..„ - //•(/„) II ~> 0, as 0<n<N h —> 0, which shows what is usually termed convergence of the algori thm . When u is Lipschitz continuous, the method turns out to be of the first order. More generally, we are interested in est imating the convergence of the al-gor i thm when per turba t ion terms are introduced implementing it in practice. Therefore, we consider the scheme in (4) with 6^ = 0, k — l , 2 , . . . , J V , and, resort ing again to (5), we get /v m a x \\un - u(tn)\\ < max { ! , / / ' }{ | | 6 0 | | + Y " ||crfc|| 0<n<NU n l L-J (9) + T ( l - hK{y\osc{u\ h) + K2 osc(H; h)]} . Such an es t imate has been obtained by triangle inequality, using the stability and consistency es t imates in (5) and (8), according to a Lax-type equivalence theorem (cf. [1], [11]). We stress again t ha t 6Q represents the error affecting the initial da t a , and a^ t he overall error made in solving the k-th equation of the scheme. Note t h a t , when for instance equat ion (1) represents an evolutionary part ial (integro-) differential equat ion, a^ embodies the space-discretization errors as wrell as t he errors (if any) on the boundary da ta . 3e Examples In th is Section, we conclude with some examples, for the purpose of illustrat-ing the me thod outl ined in §2. (A) Systems of ordinary differential equations. W h e n X = R m , equat ion (1) represents an m-dimensional system of ODEs. Assuming t h a t / satisfies all the relevant hypotheses in §2, the semi-implicit m e t h o d includes the so-called "decoupled implicit Euler method" recently stud-ied in [9]. In this case, one writes a given m-dimensional system u = F( t , u), ii(0) = tin a s u — f{t, M, u), u(fy = " 0 , where f (1, II, V) = ( F i ( t , Hi , V2, - . . , V r ) , • • • , F r ( ^ * 1 , • • • . V r - 1 , Ur)) , (10) 209 RENATO SPIGLER MARCO VIANELLO r Ffc, u/c, Vk being raj. -dimensional vectors, ^ ra& = ra. This approach leads fc=i to handling, at each time-step, r independent nonlinear algebraic systems, which can be solved by parallel implementation. In the next two examples, the right-hand side of (1) is of the form f(t,u,v) = Au + B(t,v), (11) that is equation (1) is semilinear, and hence the stationary equations in (4) will be linear. (B) (Integro-) differential reaction-diffusion systems inL2 . Consider the system - ^ = MiAwi+gi(t,x,w)+ / Ki(t,x,£;w(Z,t))dt, x G SI C Rd, 0 < t < T, n (12) w(x,0) = w0(x) , x G fi ; Wi(x,t) = (f>i(x,t) on dQ if Mi > 0 , v } for the ra-dimensional vector w = (w\,..., w m ) , M = d i a g ( M i , . . . , M m ) with Mi > 0. Systems like that in (12) arise, e.g., from problems in epidemics and combustion theory. They include, as special cases, the classical reaction-diffusion systems (K = 0 ) , purely nonlocal reactions (g = 0 ) , as well as certain de-generate problems [some diagonal entries of M are zero], namely PDEs-ODEs couplings, cf. [4]. Assume that g(t, x, •) is Lipschitz continuous from lRm into Rm , uniformly in (t, x), with constant C9, and gi(t, •, 0) G L2(Q) for i = 1 ,2 , . . . , ra. Moreover, let K(t,x,£; •) be Lipschitz continuous from Rm into Rm , uniformly in t, with a Lipschitz constant C(x,£) G L2(Q x Q). Then, problem (12) can be recast into the abstract form (1), (11), with X = (L 2 ( f i ) )m , A = MA, Dt = {u G (L 2( f i ) )m : Ui G H2(Q), Ui(x) = (j>i(x,t) on dfi if Mj > 0} ; A is monotone and maximal (for suitable boundary data) for each fixed t > 0, B(t,-) is the sum of the substitution [or Nemickii] operator associated to g and the Uryshon operator associated to K, cf., e.g., [13]. It turns out that D = X, B(t, •) maps (L2(fi)) into (l/2(fi)) , and is Lipschitz continuous uniformly in t, with constant C9 + \\C(-, ')\\L2(QXQ\ • Therefore, when the abstract problem has a solution u G C f l([0,T];X), the method described in §2 converges to it, the estimate (9) holding. The required regularity is guaranteed, e.g., when <j>i(x, t) = 0 on d£l and w0 G Dt = D(A) = {u G (L2(Q))m: u{ G H2(tt) n F r } ^ ) if Mi > 0 } , without further regularity assumptions on g, since X is reflexive, cf. [12]. (C) An example in a non-reflexive space. Consider the following initial-boundary value problem for the system - ^ = Li(x)wi - Ci(x)wi + gi(t, x,w), x e O, C Rd, 0 < t < T , w(x,0) = Wo(x) , i 6 ( l ; Wi(x,t) = 0 on 9 f l if L j ^ O . (13) 210 SEMI-IMPLICIT DISCRETIZATION OF ABSTRACT EVOLUTION EQUATIONS In (13), w e Mm , and Lt(x) = a(£<)~iXk + b)l)<)Sl denotes, for each i = 1,2, . . . , r a , a linear strictly, elliptic operator, a+. b)1, . c, t ("n(0), ct > 0. Equations in (13) are parabolic unless the operator L} is replaced by 0 for some j ; in such a casev the corresponding equation reducers to an ODE and no boundary data are imposed on w-f . Moreover, assume that g is Lipschitz con-tinuous with respect to w1 uniformly in (x , / ) , g £ C'"1([0, F] x 0 x R,n) , and that gi(t,x,0) = 0 for x G 0 0 . Problems like that in (13), namely involving systems of parabolic-ordinary differential equations, arc? encountered in several applications (see [4, Ch. 8], for instance). Again, problem (13) can be recast into the abstract form (1), (11). in the non-reflexive space X = ( ( 7 Q ( 0 ) ) , with A -= diag(Lj — c l ? . . . , Ltn — cin). D(A) = {u e (W2*(n))m Vp < oc: LiUi € C°(0) . u, = 0, on Ml for Lt ^ ()} . £?(£, •) is the substitution operator associated to g , and I) -- X . It follows that 5 ( t , •) maps X into X , and is Lipschitz continuous uniformly in /. The oper-ator A turns out to be maximal dissipative on X , indeed strongly dissipative when Ci(x) > c > 0 for all i 's. However, X being non-reflexive, the regular-ity result used in Example (B) cannot be invoked to ensure the existence of a Cl ([0,T]; X)-solution. Nevertheless, the Cl regularity assumed for g implies that of £(• , • ) (cf. [13], e.g.), and hence, a C+ ([0, T] ; X) -solution to (1), (11) does exist by a classical result in semigroup theory (cf. [5, Theorem 1.5, p. 187]). Therefore, the semi-implicit Euler method introduced in §2 converges and the estimate (9) holds. The algorithm (3) becomes, for the problem in Example (B), uiiU+1(x) = Ui,n(x) + hMiAuijn+1(x) + hgl(tn+1,x)u1/n(x)J. . . ,um , n(x)) + h i T z ( t n + i , X , c ; ; U l , n ( 0 ) - - - ^ m , n ( 0 ) ^ ) X ^ ^ > / ^ n for i = 1, 2 , . . . , m and n = 0 , 1 , . . ., N — 1, with the boundary conditions Ui(x) = </>i(x,tn+1), x E <90, if Mi > 0. Note that (14) represents, at each time-step, a system of (linear) inhomogeneous independent Helmholtz equations. If Mj = 0 for some j , the j -th equation in (14) yields explicitly uhn+1(x). Clearly, from the computational standpoint, all advantages mentioned in §1 have been obtained, namely linearization, decoupling, and removal of the nonlocal operator from the implicit part. In numerically solving each linear Helmholtz equation in (14), one could adopt, for instance, a probabilistic method, particularly convenient (if not the only feasible, in practice), when the space dimension is high. The direct applica-tion of a probabilistic method to the original semilinear system would be much more difficult (cf. [10], e.g.). Similar advantages are obtained in problem (C), the main difference being that the solution's components in this case is approximated (on the nodes tn) in the sup-norm, instead of in the L2 -norm. 211 RENATO SPIGLER MARCO VIANELLO We confined ourselves to reaction-diffusion systems for the purpose of illus­tration. The semi-implicit Euler method however could be applied to many other instances, such as systems of semilinear hyperbolic equations (cf. [2, 6], e.g., for the implicit Euler method in the abstract hyperbolic case). REFERENCES [1] C H A R T R E S , B .—STEPLEMAN , R. : A general theory of convergence for numerical methods, SIAM J. Numer. Anal. 9 (1972), 476-492. [2] KACUR, J.: Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czechoslovak Math. J. 3 4 (1984), 92-106. [3] KACUR, J.: Method of Rothe in Evolution Equations, Teubner, Leipzig, 1985. [4] PAO, C V.: Nonlinear Parabolic and Elliptic Equations, P lenum Press, New York, 1992. [5] PAZY, A.: Semigroups ofLinear Operators and Applications to Partial Differential Equa-tionsy Springer-Veгlag, New York, 1983. [6] P U L T A R , M . : Solution of abstract hyperbolic equations by Rothe method, Aplikace Matematiky 29 (1984), 23-39. [7] REKTORYS, K.: The Method of Discretization in Time, Reidel, Doгdrecht, 1982. [8] REKTORYS, K.: Numerical and theoretical treatinд of evolution problems by the method of discretization in time, in: Equadiŕľ 6, Proceed ings (Brno, 1985) (J. Vosmanský and M. Zlámal, eds.), Lecture Notes in Mathematics Vol. 1192, Springer-Verlag, Berlin, 1986, 70-84. [9] SAND, J.—SKELBOE, S.: Stability of backward Euler multirate methods and convergence of waveform relaxation, B I T 3 2 (1992), 350-366. [10] SHERMAN, A. S.—PESKIN, C S.: A Monte Carlo method for scalar reaction-diffusion equations, SIAM J. Sci. Statist . Comput . 7 (1986), 1360-1372. [11] SPIGLER, R .—VIANELLO, M. : Convergence analysis of the semi-implicit Euler method for abstract evolution equations, submitted. [12] VIANELLO, M. : On regularity at t = 0 for semilinear ADEs and its numerical applica-tions, Semigroup Forum (to appear) . [13] VILENKIN, N. Y. et al.: Functional Analysis, Wolters-Noordhoff, Groningen, 1972. [14] ZEIDLER, E.: Nonlinear Functional Analysis and its Applications II/B: Nonlinear Mono-tone Operators, Springer-Verlag, New York, 1990. Received December 8 1993 Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova Via Belzoni 7 35131 Padova ITALY E-mail: spiglerШpdudmsa.bitnet Dipartimento di Matematica Pura e Applicata Università di Padova Via Belzoni 7 35131 Padova ITALY 212 

Semi-implicit discretization of abstract evolution equations

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Semi-implicit discretization of abstract evolution equations

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Institute of Mathematics AS CR, v. v. i.