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

Linhui Li, Xiong Meng, Boying Wu

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.

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

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 , th order superconvergence for the numerical flux and cell averages, th order superconvergence at generalized Radau points, and th order for error derivative at generalized Radau points. Moreover, a supercloseness result of order 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.
Paper Structure (16 sections, 8 theorems, 136 equations, 1 figure, 6 tables)

This paper contains 16 sections, 8 theorems, 136 equations, 1 figure, 6 tables.

Table of Contents

  1. Keywords
  2. AMS subject classifications
  3. Introduction
  4. The LDG scheme and preliminaries
  5. The LDG scheme
  6. Preliminaries
  7. Correction functions and interpolation functions
  8. Correction functions
  9. A superconvergent bound of interpolation errors
  10. Superconvergence
  11. The numerical initial condition
  12. Supercloseness
  13. Superconvergence
  14. Extension to mixed boundary conditions
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Lemma 2.1

Assume that $u_h, p_h, q_h, r_h$ are solutions to the LDG scheme num_lin_4_LDG with generalized numerical fluxes gflux_lin_4_LDG. Then, the following relationships hold

Figures (1)

  • Figure 6.1: Time evolution of the error for Example \ref{['ex2']} with different weights, $k=2$, $N=16$, $T=100$.

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 8 more