Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable semi-implicit SDC methods for conservation laws

Joerg Stiller

TL;DR

This work introduces a robust family of semi‑implicit spectral deferred correction (SDC) methods for conservation laws, built from a Lax–Wendroff–inspired splitting that treats convection explicitly while handling diffusion and sources implicitly. The approach uses right‑sided Radau points, yielding high‑order accuracy and strong stability properties (notably $L$-stability up to order 11) with linear systems that are positive semidefinite. The methods are shown to be equivalent to Radau IIA collocation and to DG in time, and are validated with 1D DG spatial discretizations on convection–diffusion, Burgers, Euler, and Navier–Stokes problems, including shock scenarios and high CFL runs. Numerical experiments indicate excellent stability and accuracy, outperforming some IMEX methods at higher accuracies and remaining competitive with (and sometimes superior to) explicit RK schemes when diffusion plays a significant role. The work also notes avenues for acceleration (e.g., multilevel methods) and extension to multidimensional problems.

Abstract

Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$- or even $L$-stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are $L$-stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.

Stable semi-implicit SDC methods for conservation laws

TL;DR

This work introduces a robust family of semi‑implicit spectral deferred correction (SDC) methods for conservation laws, built from a Lax–Wendroff–inspired splitting that treats convection explicitly while handling diffusion and sources implicitly. The approach uses right‑sided Radau points, yielding high‑order accuracy and strong stability properties (notably -stability up to order 11) with linear systems that are positive semidefinite. The methods are shown to be equivalent to Radau IIA collocation and to DG in time, and are validated with 1D DG spatial discretizations on convection–diffusion, Burgers, Euler, and Navier–Stokes problems, including shock scenarios and high CFL runs. Numerical experiments indicate excellent stability and accuracy, outperforming some IMEX methods at higher accuracies and remaining competitive with (and sometimes superior to) explicit RK schemes when diffusion plays a significant role. The work also notes avenues for acceleration (e.g., multilevel methods) and extension to multidimensional problems.

Abstract

Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards - or even -stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are -stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.
Paper Structure (29 sections, 2 theorems, 70 equations, 10 figures, 2 tables)

This paper contains 29 sections, 2 theorems, 70 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conservation laws
  3. Conservation systems in one space dimension
  4. Scalar conservation laws
  5. Euler and Navier-Stokes equations
  6. Time integration
  7. Semi-explicit integrators of order 1 and 2
  8. Integrators of order 1
  9. Integrator of order 2
  10. Linear stability of semi-implicit integrators
  11. Spectral deferred correction method
  12. SDC with right-sided Gauss-Radau points
  13. Equivalence to Radau IIA collocation and discontinuous Galerkin methods
  14. Extension to other point sets and relation to Euler-based SDC methods
  15. Linear stability of SDC methods
  16. ...and 14 more sections

Key Result

Proposition 1

The time integration methods SI1(1) and SI1(2) are $L$-stable.

Figures (10)

  • Figure 1: Stability domains of the first-order integrators
  • Figure 2: Stability domain of the second-order integrator
  • Figure 3: Neutral stability curves for SDC methods with up to ${M\!=\!8}$ Radau points using different predictors and correctors
  • Figure 4: Convergence study for the convection-diffusion wave problem.
  • Figure 5: Convergence of the SDC-SI method for the inviscid Burgers wave problem
  • ...and 5 more figures

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6