Yulong Xing, The Ohio State University

Chuchu Chen, Associate Professor

Structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations and stochastic Maxwell equations

9:30-10:30 June 27(Monday) Zoom ID: 933 3184 4016

Many partial differential equations (PDEs) can be written as a multi-symplectic Hamiltonian system, which obeys the multi-symplectic conservation law. In this presentation, we present and study semi-discrete discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian PDEs, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin--Bona--Mahony equation, the Camassa--Holm equation, the Korteweg--de Vries equation and the nonlinear Schrodinger equation are discussed. In the second part, we consider discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise. It is shown that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations.