Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models

Abstract

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic direction, we also employ a random variable W based on the Q-Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

Keywords:

• Nonlinear system of advection–reaction–diffusion equation
• error estimate
• spectral element method (SEM)
• stochastic PDEs

2010 Mathematics Subject Classifications:

• 65M70
• 34A34

Acknowledgments

The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).