Local Discontinuous Galerkin Methods for the $$\mu $$ μ -Camassa–Holm and $$\mu $$ μ -Degasperis–Procesi Equations

Abstract

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the $$\mu $$ -Camassa–Holm ( $$\mu $$ CH) and $$\mu $$ -Degasperis–Procesi ( $$\mu $$ DP) equations. The conservative schemes for both two equations can preserve discrete versions of their own first two Hamiltonian invariants, while the dissipative ones guarantee the corresponding stability. The error estimates of both LDG schemes for the $$\mu $$ CH equation are given. Comparing with the error estimates for the Camassa–Holm equation, some important tools are used to handle the unexpected terms caused by its particular Hamiltonian invariants. Moreover, a priori error estimates of two LDG schemes for the $$\mu $$ DP equation are also proven in detail. Numerical experiments for both equations in different circumstances are provided to illustrate the accuracy and capability of these schemes and give some comparisons about their performance on simulations.