Energy Conserving Galerkin Approximation of Two Dimensional Wave Equations with Random Coefficients
Provides a provably energy-preserving and optimally convergent stochastic Galerkin method for wave propagation in heterogeneous media with uncertainty, relevant to physics and engineering applications.
This work develops an energy-conserving numerical method for two-dimensional wave equations with random coefficients, achieving optimal convergence rates and linear error growth in time, validated by numerical tests.
Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities of the media. This work considers a two-dimensional wave equation with random coefficients which may be discontinuous in space. Generalized polynomial chaos method is used in conjunction with stochastic Galerkin approximation, and local discontinuous Galerkin method is used for spatial discretization. Our method is shown to be energy preserving in semi-discrete form as well as in fully discrete form, when leap-frog time discretization is used. Its convergence rate is proved to be optimal and the error grows linearly in time. The theoretical properties of the proposed scheme are validated by numerical tests.