A local discontinuous Galerkin method for the Burgers–Poisson equation

Abstract

] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The L 2 -stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

“…The numerical solutions in Figure 4.3 is consistent with the results in [16]. For the Fornberg-Whitham equation i.e.…”

“…Usually, the structure preserving schemes can help reduce the shape error of waves along with long time evolution. For example in [3,16,15,39], compared with dissipative schemes, the energy conservative or Hamiltonian conservative numerical schemes for the KdV equation have less shape error or amplitude damping for long time approximations, especially in the low-resolution cases.…”

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

“…Note that when the parameter θ is taken as 0 or 1, the projection P θ reduces to the standard local Gauss-Radau projection P + h or P − h as defined in [10]. Besides, the projection P θ satisfies the following optimal approximation property [22,23]…”

An Invariant-Preserving Scheme for the Viscous Burgers-Poisson System

“…Numerical solutions and exact solutions are compared to show the efficiency of the presented methods, but convergence and stability analysis is not provided. In [4,18], the Local Discontinuous Galerkin (LDG) method is applied to the inviscous problem and the viscous problem, respectively. The analysis on invariant-preserving and convergence is included in these LDG works.…”

“…The ability to preserve invariant helps retain accuracy after a long time. This claim is based on the numerical evidence in [18] where the numerical solution obtained from the non-preserving method fails to capture the essence of the exact solution in the long run. A similar comparison will be presented in this work.…”