An Explicit Sixth Order Runge-Kutta Method for Simple Lawson Integration

TL;DR

This paper introduces an explicit sixth order Runge-Kutta method designed for Lawson integration, enabling efficient handling of stiff linear operators common in fluid and plasma simulations.

Contribution

The paper presents a new explicit sixth order Runge-Kutta scheme that simplifies implementation for stiff problems using Lawson integration with ordered abscissae.

Findings

Provides a sixth order scheme compatible with Lawson integration

Uses Newton-Raphson iteration for simplicity

Facilitates efficient simulation of stiff linear operators

Abstract

Explicit Runge-Kutta schemes become impractical when a stiff linear operator is present in the dynamics. This failure mode is quite common in numerical simulations of fluids and plasmas. Lawson proposed Generalized Runge-Kutta Processes for stiff problems in 1967, in which the stiff linear operator is treated fully implicitly via matrix exponentiation. Any Runge-Kutta scheme induces valid Lawson integration, but a scheme is exceptionally simple to implement if the abscissa $c_i$ are ordered and equally spaced. Classical RK4 satisfies this requirement, but it is difficult to derive efficient higher order schemes with this constraint. Here I present an explicit sixth order method identified with Newton-Raphson iteration that provides simple Lawson integration.