We present and study an explicit exponential integrator for parabolic SPDEs in any dimension driven by a Gaussian noise which is white in time and with spatial correlation given by a Riesz kernel. Under assumptions on the coefficients of the SPDE, we prove strong error bounds and exhibit how the rate of convergence depends on the exponent in the Riesz kernel. Finally, numerical experiments in spatial dimensions $1$ and $2$ are provided in order to confirm our convergence results.
The analysis of convergence of numerical schemes for stochastic partial differential equations (SPDEs) started with the seminal paper [32]. In this reference, the authors showed the convergence in probability of an implicit numerical scheme to the solution of a stochastic heat equation with a nonlinear term and driven by additive space-time white noise defined on the interval r0, 1s. The literature on (strong) convergence of numerical schemes to solutions of SPDEs, in particular of the parabolic type, in dimension 1 is now well established, see e.g [49,31,54,28,39,35,44,3,62,5,22,50,51,61,59,48,24,9,10,30,53,8,29,16,57] and references therein.
Moving beyond the one dimensional case and focusing on the random field setting, convergence results of numerical methods for SPDEs in dimension larger than 1 are currently lacking. Without being exhaustive, we refer the interested reader to [12,51,63]. Indeed, the results presented in our paper are closely related to those obtained in [51]. More precisely, in the latter reference the authors study the speed of convergence of explicit and implicit (semi-) discretization schemes for the following semilinear stochastic heat equation driven by Riesz noise with parameter α P p0, 2 ^dq, where d ě 1 is the space dimension:
(1) # Bu Bt pt, xq " ∆upt, xq bpt, x, upt, xqq σpt, x, upt, xqq 9 F α pt, xq, up0, xq " u 0 pxq for x P p0, 1q d , with homogeneous Dirichlet boundary conditions. Here, ∆ is the Laplacian operator on r0, 1s d with Dirichlet boundary conditions, 9
F α denotes a Gaussian noise which is white in time and has a spatial correlation given by a Riesz kernel (see Section 2.2 for the precise definition), u 0 is a continuous function on r0, 1s d and the functions b, σ satisfy some conditions (see Section 2.4).
Our main objective is to analyze the rate of strong convergence of an explicit stochastic exponential integrator for the SPDE (1). The main contributions of the present work are twofold:
First, we prove that the strong rate of convergence of the stochastic exponential integrator is p 1 2 ´α 4 q (see Theorem 8 for the precise statement). This rate of convergence is optimal, in the sense that it coincides with the time regularity of the solution u of equation (1) (see Proposition 7). Moreover, the obtained rate of convergence coincides with those of the explicit and implicit Euler-Maruyama schemes considered in [51,Thm. 3.4. (ii) and (iii)]. On top of that, our error estimates are uniform with respect to time and space (see (35)) and require only the minimal assumption of the initial condition u 0 being a continuous function. We also point out that the considered stochastic exponential integrator provides two distinct benefits over the numerical schemes studied in [51]: it avoids the CFL-related step-size restriction of the explicit Euler-Maruyama scheme and it has advantages in implementation over the implicit Euler-Maruyama scheme (see Section 5 for numerical illustrations).
The second important contribution of the present paper is that we numerically confirm the theoretically derived strong convergence rates in dimensions d " 1 and d " 2 (see Section 5). To the best of our knowledge, no previous references in the literature have shown such numerical results in dimension 2 in the random field setting.
Before elaborating a bit more on the above described contributions, let us mention that exponential integrators have a long history in the numerical analysis of deterministic ordinary and partial differential equations, see the review [36]. It is also not the first time that this kind of time integrators have been applied in the context of SPDEs, see for instance [38,43,37,14,6,7,42,33] and the references above. In addition, we refer the reader to [3], where the authors consider a one-dimensional stochastic heat equation of the form (1) but driven by space-time white noise. The case of stochastic wave equations has been addressed in [19,60,58,20,2,55,18,21,46,11]. Furthermore, in the context of stochastic Schrödinger equations, there have been advances in, e.g., [52,17,1,34,41,15,23].
The proof of our main result (Theorem 8) on the strong rate of convergence for the stochastic exponential integrator is carried out using two important ingredients. On the one hand, we apply well-known estimates for stochastic integrals with respect to our Riesz noise. Here, we adapt the theory developed by Dalang in the seminal paper [25] to our setting, in which we consider our SPDE in the bounded domain r0, 1s d . On the other hand, most of the technical effort has been focused on proving precise estimates involving the Green function G d of the stochastic heat equation in r0, 1s d with Dirichlet boundary conditions (see Lemmas 3 and 4 for the precise statements), and using them sharply throughout the proof of our main theorem. The proofs of some of those results, which involve norms of G d in the Hilbert space determined by the space covariance of the noise F α , have
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