Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection-reaction-diffusion models. (English) Zbl 1487.65160
Summary: 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.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 86A05 | Hydrology, hydrography, oceanography |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
nonlinear system of advection-reaction-diffusion equation; error estimate; spectral element method (SEM); stochastic PDEsReferences:
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