Properties of Runge-Kutta-Summation-By-Parts methods

TY - JOUR

T1 - Properties of Runge-Kutta-Summation-By-Parts methods

AU - Linders, Viktor

AU - Nordström, Jan

AU - Frankel, Steven H.

PY - 2020/10/15

Y1 - 2020/10/15

N2 - We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic approach that rests on the well-established theory of RK methods. The purpose of this paper is to provide a bottom-up overview of stability and convergence results for linear and non-linear problems that relate to general RK-SBP methods. To this end, we focus on the RK viewpoint, since this perspective so far is largely unexplored. This approach allows us to derive all results as simple consequences of the properties of SBP methods combined with well-known results from RK theory. In this way, new proofs of known results such as A-, L- and B-stability are given. Additionally, we establish previously unreported results such as strong S-stability, dissipative stability and stiff accuracy of certain RK-SBP methods. Further, it is shown that a subset of methods are B-convergent for strictly contractive non-linear problems and convergent for non-linear problems that are both contractive and dissipative.

AB - We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic approach that rests on the well-established theory of RK methods. The purpose of this paper is to provide a bottom-up overview of stability and convergence results for linear and non-linear problems that relate to general RK-SBP methods. To this end, we focus on the RK viewpoint, since this perspective so far is largely unexplored. This approach allows us to derive all results as simple consequences of the properties of SBP methods combined with well-known results from RK theory. In this way, new proofs of known results such as A-, L- and B-stability are given. Additionally, we establish previously unreported results such as strong S-stability, dissipative stability and stiff accuracy of certain RK-SBP methods. Further, it is shown that a subset of methods are B-convergent for strictly contractive non-linear problems and convergent for non-linear problems that are both contractive and dissipative.

KW - B-convergence

KW - Dissipative stability

KW - Runge-Kutta methods

KW - S-stability

KW - SBP in time

KW - Stiff accuracy

UR - http://www.scopus.com/inward/record.url?scp=85087591463&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2020.109684

DO - 10.1016/j.jcp.2020.109684

M3 - Article

AN - SCOPUS:85087591463

VL - 419

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

M1 - 109684

ER -