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A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

Sigal Gottlieb, Zachary J. Grant

TL;DR

The paper surveys high-order SSP time discretizations that incorporate two derivatives, detailing explicit, implicit, IMEX, and general linear method (GLM) frameworks. It connects forward Euler stability with advanced time integrators via convex decompositions and introduces three second-derivative building blocks: second-derivative, Taylor-series, and implicit Taylor-type conditions, including their optimal methods and order barriers. It further extends SSP theory to IMEX two-derivative schemes and GLMs, presenting unconditional SSP implicit methods up to $p=4$ and new SSP IMEX GLMs with second- and third-order accuracy. The work demonstrates how two-derivative SSP methods can overcome explicit order barriers, enable asymptotic-preserving and positivity-preserving simulations for hyperbolic PDEs, and offers a rich set of practical schemes with computed coefficients and stability properties. The results provide a versatile toolkit for stable, high-order time integration in stiff and nonstiff hyperbolic problems, with implications for accurate, robust simulations in computational physics.

Abstract

High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade multiderivative time-stepping have been used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become increasingly relevant. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the SSP theory for explicit and implicit two-derivative Runge--Kutta schemes, and discuss a special condition on the second derivative under which these implicit methods may be unconditionally SSP. This condition is then used in the context of implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes, where the time-step restriction is independent of the stiff term. Finally, we present the SSP theory for implicit-explicit (IMEX) multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

TL;DR

The paper surveys high-order SSP time discretizations that incorporate two derivatives, detailing explicit, implicit, IMEX, and general linear method (GLM) frameworks. It connects forward Euler stability with advanced time integrators via convex decompositions and introduces three second-derivative building blocks: second-derivative, Taylor-series, and implicit Taylor-type conditions, including their optimal methods and order barriers. It further extends SSP theory to IMEX two-derivative schemes and GLMs, presenting unconditional SSP implicit methods up to and new SSP IMEX GLMs with second- and third-order accuracy. The work demonstrates how two-derivative SSP methods can overcome explicit order barriers, enable asymptotic-preserving and positivity-preserving simulations for hyperbolic PDEs, and offers a rich set of practical schemes with computed coefficients and stability properties. The results provide a versatile toolkit for stable, high-order time integration in stiff and nonstiff hyperbolic problems, with implications for accurate, robust simulations in computational physics.

Abstract

High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade multiderivative time-stepping have been used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become increasingly relevant. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the SSP theory for explicit and implicit two-derivative Runge--Kutta schemes, and discuss a special condition on the second derivative under which these implicit methods may be unconditionally SSP. This condition is then used in the context of implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes, where the time-step restriction is independent of the stiff term. Finally, we present the SSP theory for implicit-explicit (IMEX) multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

Paper Structure

This paper contains 22 sections, 6 theorems, 130 equations, 2 figures.

Table of Contents

  1. Overview
  2. SSP methods
  3. Explicit SSP Runge--Kutta methods
  4. Implicit SSP Runge--Kutta methods
  5. SSP two-derivative Runge--Kutta methods
  6. Second derivative condition
  7. Optimal methods based on the forward Euler and second derivative conditions
  8. Numerical example
  9. Taylor Series Condition
  10. Optimal methods and order barriers
  11. Negative derivative condition
  12. Unconditionally SSP implicit two-derivative Runge--Kutta methods up to order $p=4$
  13. SSP theory for Two-derivative IMEX methods
  14. SSP IMEX multi-derivative Runge--Kutta methods
  15. Multistep multi-stage two-derivative methods
  16. ...and 7 more sections

Key Result

Theorem 1

\newlabelthmSD0 Given spatial discretizations $F$ and $\dot{F}$ that satisfy FE and SDcond, a two-derivative multistage method of the form MSMDmatrix preserves the strong stability property $\| u^{n+1} \| \leq \|u^n \|$ under the time-step restriction $\Delta t \leq r \Delta t_{{FE}}$ for some $r> where and $\mathbf{e}$ is a vector of ones.

Figures (2)

  • Figure 1: Comparison of the rise in total variation as a function of the CFL number for the two-stage third order SSP \ref{['2s3p']} and non-SSP method \ref{['bad2s3p']}. On the left is the maximal per time-step rise and on the right the maximal TV rise above the initial TV.
  • Figure 2: Comparison of the rise in total variation as a function of the CFL number for the two-stage third order SSP \ref{['2s3p']} and non-SSP method \ref{['bad2s3p']}. On the left is the maximal per time-step rise and on the right the maximal TV rise above the initial TV.

Theorems & Definitions (14)

  • Remark 1
  • Theorem 1: SDpaper
  • Example 1
  • Example 2
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Remark 3
  • Example 3
  • ...and 4 more