On the explicit two-stage fourth-order accurate time discretizations

TL;DR

This paper introduces a new class of explicit two-stage fourth-order time discretization methods with variable weights, demonstrating improved stability regions and accuracy over classical Runge-Kutta methods through theoretical analysis and numerical experiments.

Contribution

It proposes a novel class of explicit two-stage fourth-order methods with variable weights, expanding the stability and accuracy capabilities beyond existing methods.

Findings

The new methods can achieve fourth-order accuracy conditionally.

Their stability regions can surpass those of classical Runge-Kutta methods.

Numerical experiments confirm improved performance and stability.

Abstract

This paper continues to study the explicit two-stage fourth-order accurate time discretiza- tions [5, 7]. By introducing variable weights, we propose a class of more general explicit one-step two-stage time discretizations, which are different from the existing methods, such as the Euler methods, Runge-Kutta methods, and multistage multiderivative methods etc. We study the absolute stability, the stability interval, and the intersection between the imaginary axis and the absolute stability region. Our results show that our two-stage time discretizations can be fourth-order accurate conditionally, the absolute stability region of the proposed methods with some special choices of the variable weights can be larger than that of the classical explicit fourth- or fifth-order Runge-Kutta method, and the interval of absolute stability can be almost twice as much as the latter. Several numerical experiments are carried out to demonstrate the performance and accuracy as well as the stability of our proposed methods