Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise

Antoine Tambue,Jean Daniel Mukam

doi:10.1002/mma.7588

Abstract

In this paper, we investigate the numerical approximation of stochastic convection–reaction–diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately 1 for trace class noise and for space‐time white noise. These convergence orders are obtained under less regularity assumptions on the nonlinear drift function than those used in the literature for stochastic reaction–diffusion equations. Numerical experiments to illustrate our theoretical results are provided.

Highlights

This article is devoted to the space-time approximation of the following semilinear parabolic stochastic PDEs (SPDEs) dX(t) = [AX(t) + f (x, X(t))]dt + dW(t), t ∈ (0, T], x ∈ Λ, (1)with initial value X(0) = X0 and Dirichlet boundary conditions or Robin boundary conditions
In Lord and Tambue,[10,11] some exponential integrators and implicit schemes achieving convergence order 1 in time were introduced for stochastic convection–reaction–diffusion equations
We can observe that the schemes accurately in SETD1 (ASETD1) and SETD0 are still stable for large time steps and that the convergence rate is still called to the theoretical result in Theorem 2 as we have about 0.95 in Figure 2 for β = 1 and 1.05 for β = 2

Summary

INTRODUCTION

In Wang and Ruisheng,[9] an accelerated exponential integrator was investigated and proved to achieve convergence order 1 for trace class noise under more relaxed assumptions than in Jentzen et al.[7] Note that the works in other studies[6,7,9] are only for stochastic diffusion–reaction equations and heavily use the fact that the linear operator A is self-adjoint. In Lord and Tambue,[10,11] some exponential integrators and implicit schemes achieving convergence order 1 in time were introduced for stochastic convection–reaction–diffusion equations The idea behind such schemes consists of keeping the convection term in the nonlinear part F.

Notations and main assumptions

Fully discrete schemes and main results

Preparatory results

Proof of Theorem 2

NUMERICAL EXPERIMENTS

CONCLUSION