Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs

Dongfang Li, Xiaoxi Li, Zhimin Zhang

2022Mathematics of Computation66 citationsDOI

Abstract

Spatial discretizations of time-dependent partial differential equations usually result in a large system of semi-linear and stiff ordinary differential equations. Taking the structures into account, we develop a family of linearly implicit and high order accurate schemes for the time discretization, using the idea of implicit-explicit Runge-Kutta methods and the relaxation techniques. The proposed schemes are monotonicity-preserving/conservative for the original problems, while the previous linearized methods are usually not. We also discuss the linear stability and strong stability preserving (SSP) property of the new relaxation methods. Numerical experiments on several typical models are presented to confirm the effectiveness of the proposed methods.

Topics & Concepts

Runge–Kutta methodsMathematicsDiscretizationNumerical methods for ordinary differential equationsRelaxation (psychology)Stability (learning theory)Monotonic functionApplied mathematicsOrdinary differential equationPartial differential equationExplicit and implicit methodsLinear multistep methodNumerical analysisL-stabilityDifferential equationMathematical analysisDifferential algebraic equationComputer scienceMachine learningSocial psychologyPsychologyNumerical methods for differential equationsAdvanced Numerical Methods in Computational MathematicsComputational Fluid Dynamics and Aerodynamics