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

Matthew Golden

TL;DR

The paper addresses stiff ODEs arising from linear operators by adapting explicit Runge-Kutta methods via Lawson integration. It introduces Simple Lawson Runge-Kutta (SLRK) and constructs an explicit eight-stage, sixth-order scheme with equally spaced abscissas ($\Delta c = 1/6$), verified to satisfy all $37$ order conditions and implementable with a single matrix exponential per step. The method is demonstrated on a 2D incompressible Navier–Stokes test with $\hat{A}=\nu\nabla^2$, showing favorable convergence over a lower-order Lawson scheme, and the coefficients are provided as rational numbers with symbolic verification. The work also provides practical MATLAB code for SLRK4 and SLRK6 and discusses the potential for extending this approach to higher orders, making stiff-explicit integration more accessible in fluid and plasma simulations.

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.

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

TL;DR

The paper addresses stiff ODEs arising from linear operators by adapting explicit Runge-Kutta methods via Lawson integration. It introduces Simple Lawson Runge-Kutta (SLRK) and constructs an explicit eight-stage, sixth-order scheme with equally spaced abscissas (), verified to satisfy all order conditions and implementable with a single matrix exponential per step. The method is demonstrated on a 2D incompressible Navier–Stokes test with , showing favorable convergence over a lower-order Lawson scheme, and the coefficients are provided as rational numbers with symbolic verification. The work also provides practical MATLAB code for SLRK4 and SLRK6 and discusses the potential for extending this approach to higher orders, making stiff-explicit integration more accessible in fluid and plasma simulations.

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 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.

Paper Structure

This paper contains 4 sections, 7 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main Results
  3. Conclusion
  4. Example Implementation

Figures (2)

  • Figure 1: (a) The analytic initial condition $\omega_0$. (b) The vorticity at $t=5$ after substantial nonlinear evolution. (c) The maximum pointwise error ($\ell_\infty$ norm) as a function of the number of timesteps $m$. A state evolved with SLRK6 and $m=2^{13}$ is treated as the ground truth. Power law fits to the error support the claimed convergence rates of SLRK. $\varepsilon_{\textrm{floor}}$ is an empirical error floor of order $10^{-10}$ likely arising from the dynamic amplification of roundoff error.
  • Figure 2: (a) Linear stability regions for RK1 (Euler's Method), RK4, and the provided RK6 scheme. (b) Linear stability regions for SLRK4 and SLRK6 with $z_2=-10$. The stability curves of subfigure (a) are shown in black to provide a sense of scale.