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Dissipation-dispersion analysis of fully-discrete implicit discontinuous Galerkin methods and application to stiff hyperbolic problems

Maya Briani, Gabriella Puppo, Giuseppe Visconti

Abstract

The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation-dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta impacts deeply on the quality of the solution. We analyze one-dimensional dissipation-dispersion to select the best combination of the space-time discretization for high Courant numbers. Then, we apply our findings to the integration of one-dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.

Dissipation-dispersion analysis of fully-discrete implicit discontinuous Galerkin methods and application to stiff hyperbolic problems

Abstract

The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation-dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta impacts deeply on the quality of the solution. We analyze one-dimensional dissipation-dispersion to select the best combination of the space-time discretization for high Courant numbers. Then, we apply our findings to the integration of one-dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.
Paper Structure (24 sections, 1 theorem, 99 equations, 14 figures, 13 tables, 1 algorithm)

This paper contains 24 sections, 1 theorem, 99 equations, 14 figures, 13 tables, 1 algorithm.

Table of Contents

  1. AMS subject classification
  2. Keywords
  3. Introduction
  4. Diagonally implict Runge-Kutta discontinuous Galerkin approximation
  5. Space approximation
  6. Implicit time-integration
  7. Dissipation-dispersion analysis of DIRK-DG schemes
  8. Numerical dissipation and dispersion of fully-discrete DG schemes
  9. The case of DIRK-DG approximations
  10. Numerical investigation of dissipation and dispersion in DIRK-DG schemes
  11. Space limiting in DIRK-DG schemes
  12. Freezing the nonlinearity with a low-order approximation of the space limiters
  13. The space-time first-order implicit predictor
  14. The space-time high-order semi-linearized implicit scheme
  15. Numerical simulations
  16. ...and 9 more sections

Key Result

Proposition 3.2

Let $M$ be the $(p+1)\times(p+1)$ matrix representing the fully-discrete numerical scheme applied to the linear problem eq:adv_lin, with exact solution eq:adv_lin_sol. Let $M$ be a diagonalizable matrix, and let $\{(\lambda_l,\boldsymbol{\theta}_l)\}_{l=0}^p$ denote the set of eigenvalues of $M$ and where the coefficients $\{\nu_l\}_{l=0}^p$ are defined in eq:coefCm with $\boldsymbol{\mu}(\mathrm{

Figures (14)

  • Figure 1: Numerical solution of a $2\times 2$ linear hyperbolic system obtained with the first-order backward Euler scheme. The dashed red line represents the approximation with a time-step computed on the fast wave, governed by the largest eigenvalue $\lambda_1 \approx -16$. The solid blue line shows the approximation computed with a time-step based on the slow wave, governed by the smallest eigenvalue $\lambda_2 \approx 1$.
  • Figure 2: We show the development of instability when the DIRK parameters do not yield A-stable schemes. Top row: The white area represents the stability regions of the DIRK methods of order $p+1=2,3$, whereas the $\star$ markers are obtained from the eigenvalues $\Omega_l$, $l=0,\dots,p$, of the matrix coming from the space $\mathbb{P}_p$ DG approximation as function of $K$. Bottom row: Measure of dissipation \ref{['eq:Mdiss']} computed for several Courant numbers.
  • Figure 3: We explore the sensitivity of second-order schemes to the choice of the parameter $\gamma$. Top row: Measure of dissipation \ref{['eq:Mdiss']} and dispersion \ref{['eq:Mdisp']} for second-order DIRK-DG schemes as function of $K$ and $\gamma$. Blue $\times$ markers and red $+$ markers denote the values of the DIRK parameter for which the measure is minimum and maximum, respectively. Bottom row: Dissipation and dispersion measures obtained with DIRK parameters $\gamma=0.25,1-\sqrt{2}/2,0.5$.
  • Figure 4: Measure of dissipation \ref{['eq:Mdiss']} and dispersion \ref{['eq:Mdisp']} for the two third-order DIRK-DG schemes given in Table \ref{['tab:dirk33']} and Table \ref{['tab:dirk43']}. Observe that the case $r=15$ in the top-left panel should be compared to the case $r=15$ of second-order schemes in Figure \ref{['fig:diss:disp:p1']}, since the ratio between the time-step used and the time-step required for explicit stability is the same.
  • Figure 5: Solutions to the linear advection equation with speed $a=1$, final time $t=2$ and initial conditions \ref{['eq:init:sinhf']} (left column), \ref{['eq:init:smoothhf']} (middle column), and \ref{['eq:init:doublestep']} (right column). All the solutions are obtained with $N=400$ cells.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Example 2.1: $\mathbb{P}_{p\leq2}$ DG approximation
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 4.1