Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An efficient ADER discontinuous Galerkin scheme for directly solving Hamilton-Jacobi equation

Junming Duan, Huazhong Tang

TL;DR

This paper develops an efficient one-stage ADER-DG framework for directly solving the Hamilton-Jacobi equation $\varphi_t+H(\nabla_{\bm{x}}\varphi,\bm{x})=0$. It centers on a local spacetime continuous Galerkin predictor to produce a high-order spacetime representation $q_h$ within each cell, enabling explicit, low-cost volume and flux calculations in modal space. The 2D implementation on structured meshes is given at third order with detailed predictor and flux formulas, and the authors compare computational complexity to RKDG, demonstrating substantial speedups while preserving accuracy and stability, including for viscosity solutions. Extensive 1D and 2D numerical tests verify the expected order of accuracy and show favorable CPU times relative to RKDG, with the scheme adaptable to unstructured grids as well.

Abstract

This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulas of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

An efficient ADER discontinuous Galerkin scheme for directly solving Hamilton-Jacobi equation

TL;DR

This paper develops an efficient one-stage ADER-DG framework for directly solving the Hamilton-Jacobi equation . It centers on a local spacetime continuous Galerkin predictor to produce a high-order spacetime representation within each cell, enabling explicit, low-cost volume and flux calculations in modal space. The 2D implementation on structured meshes is given at third order with detailed predictor and flux formulas, and the authors compare computational complexity to RKDG, demonstrating substantial speedups while preserving accuracy and stability, including for viscosity solutions. Extensive 1D and 2D numerical tests verify the expected order of accuracy and show favorable CPU times relative to RKDG, with the scheme adaptable to unstructured grids as well.

Abstract

This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulas of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

Paper Structure

This paper contains 15 sections, 55 equations, 11 figures, 9 tables.

Table of Contents

  1. Introduction
  2. General formulation of the ADER-DG schemes
  3. One-dimensional ADER-DG scheme
  4. Two-dimensional ADER-DG scheme on structured meshes
  5. Local spacetime continuous Galerkin predictor
  6. General formulation of continuous Galerkin predictor
  7. Implementation details of 2D third order continuous Galerkin predictor
  8. Calculation of the volume integral and the numerical fluxes in the ADER-DG scheme
  9. The explicit formulae of the volume integral
  10. The explicit formulae of the numerical fluxes
  11. Comparison of the computational complexity between the ADER-DG scheme and the RKDG scheme
  12. Numerical results
  13. One-dimensional results
  14. Two-dimensional results
  15. Conclusion

Figures (11)

  • Figure 1: Example \ref{['ex:sign1D']}, $N=80$. Left: $P^2$, right: $P^3$.
  • Figure 2: Example \ref{['ex:burgers1D']}, $N=40$.
  • Figure 3: Example \ref{['ex:burgers_nonsmooth']}, $N=40$.
  • Figure 4: Example \ref{['ex:-cos1D']}, $N=40$.
  • Figure 5: Example \ref{['ex:nonconvex1D']}, $P^2,\text{CFL}=0.10$. Left: $N=80$, right: $N=81$.
  • ...and 6 more figures

Theorems & Definitions (1)

  • remark 1