A local discontinuous Galerkin method for nonlinear parabolic SPDEs

Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian

doi:10.1051/m2an/2020026

Li, Yunzhang ; Shu, Chi-Wang ; Tang, Shanjian

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S187-S223

Résumé

In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L²-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h^(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.

MR

DOI : 10.1051/m2an/2020026

Classification : 65C30, 60H35
Keywords: Local discontinuous Galerkin method, nonlinear parabolic stochastic partial differential equations, multiplicative noise, stability analysis, error estimates, stochastic viscous Burgers equation

@article{M2AN_2021__55_S1_S187_0,
     author = {Li, Yunzhang and Shu, Chi-Wang and Tang, Shanjian},
     title = {A local discontinuous {Galerkin} method for nonlinear parabolic {SPDEs}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {S187--S223},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {Suppl\'ement},
     doi = {10.1051/m2an/2020026},
     mrnumber = {4221304},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2020026/}
}

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Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian. A local discontinuous Galerkin method for nonlinear parabolic SPDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S187-S223. doi: 10.1051/m2an/2020026

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