Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations

This paper presents a novel approach to preserving invariants in the time-implicit numerical discretization of high-order nonlinear wave equations. Many high-order nonlinear wave equations have an infinite number of conserved quantities. Designing time-implicit numerical schemes that preserve many conserved quantities simultaneously is challenging. The proposed method utilizes Lagrange multipliers to reformulate the local discontinuous Galerkin (LDG) discretization as a conservative discretization. Combining an implicit spectral deferred correction method for time discretization can achieve a high-order scheme in both temporal and spatial dimensions. We use the Korteweg-de Vries equations as an example to illustrate the implementation of the algorithm. The algorithm can be easily generalized to other nonlinear wave problems with conserved quantities. Numerical examples for various high-order nonlinear wave equations demonstrate the effectiveness of the proposed methods in preserving invariants while maintaining the high accuracy.