Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations

TL;DR

This paper demonstrates high-order superconvergence properties of the local discontinuous Galerkin method with generalized fluxes for 1D linear time-dependent fourth-order equations, enhancing accuracy and stability in long-term simulations.

Contribution

It introduces a superconvergence analysis for the LDG method with generalized fluxes, achieving higher order accuracy and stability for complex fourth-order equations.

Findings

Superconvergence of order (2k+1) for flux and cell averages.

Superconvergence of order (k+2) at generalized Radau points.

Supercloseness of order (k+2) between projection and numerical solution.

Abstract

In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations. The adjustable numerical viscosity of the generalized numerical fluxes is beneficial for long time simulations with a slower error growth. By using generalized Gauss--Radau projections and correction functions together with a suitable numerical initial condition, we derive, for polynomials of degree $k$, $(2k+1)$th order superconvergence for the numerical flux and cell averages, $(k+2)$th order superconvergence at generalized Radau points, and $(k+1)$th order for error derivative at generalized Radau points. Moreover, a supercloseness result of order $(k+2)$ is established between the generalized Gauss--Radau projection and the numerical solution. Superconvergence analysis of mixed boundary conditions is also given. Equations with discontinuous initial condition and nonlinear convection term are numerically investigated, illustrating that the conclusions are valid for more general cases.