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An element-based convex limiting framework for continuous Galerkin methods with nonlinear stabilization

Dmitri Kuzmin, Hennes Hajduk, Joshua Vedral

Abstract

We equip a high-order continuous Galerkin discretization of a general hyperbolic problem with a nonlinear stabilization term and introduce a new methodology for enforcing preservation of invariant domains. The amount of shock-capturing artificial viscosity is determined by a smoothness sensor that measures deviations from a weighted essentially nonoscillatory (WENO) reconstruction. Since this kind of dissipative stabilization does not guarantee that the nodal states of the finite element approximation stay in a convex admissible set, we adaptively constrain deviations of these states from intermediate cell averages. The representation of our scheme in terms of such cell averages makes it possible to apply convex limiting techniques originally designed for positivity-preserving discontinuous Galerkin (DG) methods. Adapting these techniques to the continuous Galerkin setting and using Bernstein polynomials as local basis functions, we prove the invariant domain preservation property under a time step restriction that can be significantly weakened by using a flux limiter for the auxiliary cell averages. The close relationship to DG-WENO schemes is exploited and discussed. All algorithmic steps can be implemented in a matrix-free and hardware-aware manner. The effectiveness of the new element-based limiting strategy is illustrated by numerical examples.

An element-based convex limiting framework for continuous Galerkin methods with nonlinear stabilization

Abstract

We equip a high-order continuous Galerkin discretization of a general hyperbolic problem with a nonlinear stabilization term and introduce a new methodology for enforcing preservation of invariant domains. The amount of shock-capturing artificial viscosity is determined by a smoothness sensor that measures deviations from a weighted essentially nonoscillatory (WENO) reconstruction. Since this kind of dissipative stabilization does not guarantee that the nodal states of the finite element approximation stay in a convex admissible set, we adaptively constrain deviations of these states from intermediate cell averages. The representation of our scheme in terms of such cell averages makes it possible to apply convex limiting techniques originally designed for positivity-preserving discontinuous Galerkin (DG) methods. Adapting these techniques to the continuous Galerkin setting and using Bernstein polynomials as local basis functions, we prove the invariant domain preservation property under a time step restriction that can be significantly weakened by using a flux limiter for the auxiliary cell averages. The close relationship to DG-WENO schemes is exploited and discussed. All algorithmic steps can be implemented in a matrix-free and hardware-aware manner. The effectiveness of the new element-based limiting strategy is illustrated by numerical examples.

Paper Structure

This paper contains 25 sections, 4 theorems, 89 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stabilized CG discretization
  3. Galerkin scheme using Bernstein finite elements
  4. Dissipative WENO stabilization
  5. Low-order IDP discretization
  6. Intermediate cell averages
  7. Analysis in the one-dimensional case
  8. Analysis in the multidimensional case
  9. Monolithic convex limiting
  10. Flux limiting
  11. Slope limiting
  12. Summary and discussion
  13. Numerical results
  14. Scalar problems in 1D
  15. Linear advection
  16. ...and 10 more sections

Key Result

lemma 1

Let $\mathcal{G}$ be a convex invariant domain. Suppose that and the time step $\Delta t$ satisfies the CFL-like condition Then the forward Euler step festepbar is IDP in the sense that

Figures (9)

  • Figure 1: Intermediate states of convex decompositions on a triangular mesh.
  • Figure 2: Low-order (a,b) and slope-limited (c) solutions to the 1D linear advection problem with initial condition \ref{['eq:adv-init']} at $t=1$. All approximations use $N_h=128$ nodes. The number of employed time steps is denoted by #TS.
  • Figure 3: Results for the nonlinear 1D conservation law with the flux function \ref{['eq:1dkpp-flux']} and $u_0(x)=\chi_{x\ge1/4}$. Low-order (a,b) and slope-limited (c) solutions at $t=1$. All approximations use $N_h=129$ nodes. The number of employed time steps is denoted by #TS. The entropy evolution history is plotted in panel (d).
  • Figure 4: Modified shock tube of Sod, density at $t=0.2$ (a,b) and entropy evolution (c) computed with $N_h=129$ unknowns per variable.
  • Figure 5: Density profiles for the 1D blast wave problem at $t=0.038$ computed with $N_h=1001$.
  • ...and 4 more figures

Theorems & Definitions (13)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1
  • proof
  • remark 1
  • theorem 2
  • proof
  • remark 2
  • ...and 3 more