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Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems

David A. Kopriva, Andrew R. Winters, Jan Nordström

Abstract

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems

Abstract

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.
Paper Structure (19 sections, 9 theorems, 180 equations, 13 figures, 1 table)

This paper contains 19 sections, 9 theorems, 180 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. DGSEM Approximations to the Overset Grid Problem
  3. The DGSEM Approximation of the Scalar Characteristic Problem
  4. Approximation of Systems with Characteristic Interface Conditions
  5. Approximation of Systems with a Pure Penalty Formulation
  6. Energy Bounds
  7. Polynomial Approximation Summary
  8. Energy Bounds for the Scalar Problem
  9. Energy Bounds for the Characteristic Approximation of a System of Equations
  10. Energy Bounds for the Penalty Formulation
  11. Enhancing Dissipation with Additional Coupling within the Overlap Region
  12. Implementation of the Penalty Formulation
  13. Computed Examples
  14. Convergence with Equal Resolution
  15. Convergence with Unequal Resolution
  16. ...and 4 more sections

Key Result

Lemma 1

The DGSEM approximations to the solutions of the overset grid problem for the first order scalar wave equation satisfy the bounds

Figures (13)

  • Figure 1: Overset domain definitions in 1D
  • Figure 2: Base (B) overlap (O) subdomains divided into elements.
  • Figure 3: Eigenvalues for two elements for three approximations: No dissipation (\ref{['AppendixA']}), the upwind characteristic formulation \ref{['eqUDGSEMWeakForm']}, and the penalty formulation \ref{['eq:DGSEMChimeraU']}-\ref{['eq:DGSEMChimeraV']}. The graph on the right enlarges the region near the imaginary axis. Unstable eigenvalues are drawn in red.
  • Figure 4: Eigenvalues for the penalty formulation \ref{['eq:DGSEMChimeraU']}-\ref{['eq:DGSEMChimeraV']} ($\varepsilon=0$) compared to those for the penalty formulation plus two overlap penalty points, \ref{['eq:WeakFormsFonFullDomainsWOvrlap']}. Unstable eigenvalues are drawn in red.
  • Figure 5: Exact and computed solutions for the sinusoidal problem, \ref{['eq:ExactPeriodicSolution']}. Vertical dashed lines mark the element boundaries. The shaded area marks the overlap region.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 6
  • ...and 16 more