High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge-Kutta Methods with Asymptotic Preserving Properties

@article{Gottlieb2022HighOS,
  title={High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge-Kutta Methods with Asymptotic Preserving Properties},
  author={Sigal Gottlieb and Zachary J. Grant and Jingwei Hu and Ruiwen Shu},
  journal={SIAM J. Numer. Anal.},
  year={2022},
  volume={60},
  pages={423-449}
}
In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge–Kutta schemes, and SSP implicit-explicit (IMEX) multiderivative Runge–Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order p > 2 is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is… 

Figures from this paper

Semi-implicit high resolution numerical scheme for conservation laws
We present a novel semi-implicit scheme for numerical solutions of time-dependent conservation laws. The core idea of the presented method consists of exploiting and approximating mixed partial

References

SHOWING 1-10 OF 25 REFERENCES
Asymptotic-Preserving and Positivity-Preserving Implicit-Explicit Schemes for the Stiff BGK Equation
TLDR
A family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation is developed, which is asymptotic-preserving as well as positivity- Preserving --- a feature that is not possessed by any of the existing second or high order IMEX schemes.
Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes
TLDR
It is shown that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method, which is tested on simple scalar PDEs to verify the order of convergence.
A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions
TLDR
This work demonstrates sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation and proves that the maximal order of SSP-TS methods is p=6, and defines an optimization procedure that allows them to be found.
Strong Stability-Preserving High-Order Time Discretization Methods
TLDR
This paper reviews and further develops a class of strong stability-preserving high-order time discretizations for semidiscrete method of lines approximations of partial differential equations, and builds on the study of the SSP property of implicit Runge--Kutta and multistep methods.
Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation
TLDR
New implicit-explicit Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms are considered, and asymptotic preserving (AP) in the zero relaxation limit is proposed.
A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model
TLDR
A class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model are proposed.
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
TLDR
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken and ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.
Total-variation-diminishing time discretizations
TLDR
A class of m-step Runge–Kutta-type TVD time discretizations with large CFL number m, suitable for steady state calculations, and a class of multilevel type TVD high-order timeDiscretizations suitable for time-dependent problems are presented.
A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous
...
...