Superconvergence of Discontinuous Galerkin Solutions for a Nonlinear Scalar Hyperbolic Problem

Abstract

In this paper, we study the superconvergence of the discontinuous Galerkin solutions for nonlinear hyperbolic partial differential equations. On the first inflow element we prove that the p-degree discontinuous finite element solution converges at Radau points with an O(h p+2) rate. We further show that the solution flux converges on average at O(h 2p+2)) on element outflow boundary when no reaction terms are present. Globally, we prove that the flux converges at O(h 2p+1) on average at the outflow of smooth-solution regions for nonlinear conservation laws. Numerical computations indicate that our results extend to nonrectangular meshes and nonuniform polynomial degrees. We further include a numerical example which shows that discontinuous solutions are superconvergent to the unique entropy solution away from shock discontinuities.

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Document Details

Document Type
Technical Report
Publication Date
Jan 01, 2006
Accession Number
ADA449226

Entities

People

  • Slimane Adjerid
  • Thomas C. Massey

Organizations

  • United States Naval Research Laboratory

Tags

Communities of Interest

  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Boundaries
  • Computational Fluid Dynamics
  • Computational Science
  • Computations
  • Differential Equations
  • Discontinuities
  • Engineering
  • Equations
  • Error Analysis
  • Finite Element Analysis
  • Galerkin Method
  • Hyperbolic Differential Equations
  • Mathematics
  • Mechanics
  • Nonuniform
  • Partial Differential Equations
  • Polynomials

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)