Semidiscretization in Time for Parabolic Problems

Abstract. We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.Mathematics Subject Classification. 60H15, 60F25, 60F99, 65C20, 60H35.

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“…For instance, in [4,17], the authors have shown that the infinite dimensional setting implies the restriction…”

Section: ) Where U Is a H-valued Random Process A : D(a) ⊂ H → H Dementioning

confidence: 99%

On the discretization in time of parabolic stochastic partial differential equations

Printems¹

2001

ESAIM: M2AN

127

177

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“…In order to prove Theorem 1, we follow the method of LeRoux [9]. (See also Theorem 4.1 of [7], Theorem 17.1 of [8], and Theorem 8.4 of [15]) On the other hand, the proof of Theorems 2 and 3 will be accomplished in the same line as [7] and [8].…”

Section: R(tn A)r(tt-a) R (1a)mentioning

confidence: 99%

“…Especially, the following stability result, which itself is worth stating separately, plays an important role. (Tr-iA) ... r(-r,A)11 <''atn a p (9) where Ca is a positive constant depending on a. Moreover Ca is independent of A, n and {r3 }.…”

Section: R(tn A)r(tt-a) R (1a)mentioning

confidence: 99%

Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions

Saito

2004

Japan J. Indust. Appl. Math.

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Some convergence and stability results for rational approximations of holomorphic semigroups are presented. The approximations are related to time discrete schemes with nonuniform stepsize to a linear evolution equation of parabolic type in a Banach space. Our theorems cover not only the backward Euler but also the Crank-Nicolson and forward Euler schemes.Let -A be the infinitesimal generator of a holomorphic semigroup {e -to }t>o in a Banach space X, and let r(z) be a rational function of order p > 1, that is,As is well-known, if r(z) is strongly A(0)-acceptable for some 0 E (0, ir/2), for example r(z) = (1 + z) -1 , we havewhere II -II denotes the operator norm in X; r a positive constant; n a positive integer; and C a positive constant which is independent of A, n and T. For more detail and other results, we refer to [7], [8] and [15]. The main purpose of the present paper is to extend (1) to the case where the stepsize T is not uniform. Specifically, taking positive constants Tl , T2,. . . ,^ , we introduce a general partition t, = Ti + T2 • + T^ of the time variable. Then we shall estimate Il r(TnA)r(Tn-iA) ... r (T1A) -e -t " A IIand derive convergence rate in terms of p, {Tj }^ 1 and tn . As will discuss below, in doing so, we meet a new issue. Namely, we can not avoid the ratio T/r0 , where T = max Tj, To = min Tj.1 0) under various assumptions on A and r(z).Our problem is closely related to error estimate for a time discretization to the evolution equationwhere u = u(t) E D(A) and a E X denote the unknown function and the initial value, respectively. The approximation un E D(A) of u(t) at the time level t = to is given by un = r(TA)r(Tr-1 A) ... r(Tl A)a. (4)Especially, when r(z) = (1 + z) -1 , (4) implies the backward Euler scheme with the nonuniform stepsize;un -un-1 + Au n = 0 (n>1), uo = a, Tom, and, by virtue of (2), we obtain the error estimatewhere II • 11 means the norm in X. Furthermore, when r(z) = (1 -z/2)(1 + z/2) -1 and r(z) = 1 -z, the scheme (4) corresponds to the Crank-Nicolson and forward Euler schemes with the nonuniform stepsize, respectively:un -un _1 1 + 2 (Au n + Au-i) = 0 (n > 1), uo = a; T n un -un-i + Aun-1 = 0 (n > 1), uo = a, TnRational Approximation of Semigroups 325 a...

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“…Most of them are concerned with the discretisation of problems with constant operator A (see, e.g., [5][6][7][8][9]). Other studies (see, e.g., [10][11][12][13]), study the general case where the operator A is time-dependent, under Hölder or Lipschitz-continuity assumptions.…”

Section: Introductionmentioning

confidence: 99%

Discretisation of abstract linear evolution equations of parabolic type

2012

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We investigate the discretisation of the linear parabolic equation du/dt = A(t)u + f(t) in abstract spaces, making use of both the implicit and the explicit finite-difference schemes. The stability of the explicit scheme is obtained, and the schemes' rates of convergence are estimated. Additionally, we study the special cases where A and f are approximated by integral averages and also by weighted arithmetic averages. MSC 2010: 65J10.

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Semidiscretization in Time for Parabolic Problems

Cited by 20 publications

References 0 publications

On the discretization in time of parabolic stochastic partial differential equations

On the discretization in time of parabolic stochastic partial differential equations

Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions

Discretisation of abstract linear evolution equations of parabolic type

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