Sixth-order explicit one-step methods for stiff ODEs via hybrid deferred correction involving RK2 and RK4: Application to reaction-diffusion equations

TL;DR

This paper introduces a sixth-order explicit one-step method for stiff ODEs, combining RK2 and RK4 through hybrid deferred correction, demonstrating improved stability and efficiency in numerical experiments and applications to reaction-diffusion equations.

Contribution

A novel sixth-order explicit method using hybrid deferred correction with RK2 and RK4, offering better stability and performance for stiff ODEs compared to traditional implicit methods.

Findings

Performs well on stiff, nonlinear problems with long-term integration

Has a stability region larger than RK6 and contains significant parts of the imaginary axis

Outperforms implicit methods like BDF and DC methods in certain stiff scenarios

Abstract

In this paper, the fourth-order explicit Runge-Kutta method (RK4) is used to make a Deferred Correction (DC) on the explicit midpoint rule, resulting in an explicit one-step method of order six of accuracy, denoted DC6RK2/4. Convergence and order of accuracy of DC6RK2/4 are proven through a deferred correction condition satisfied by the RK4. The region of absolute stability of this method contains that of a RK6 and is tangent to the region [-5.626,0[x[-4.730,4.730] of the complex plane, containing a significant part of the imaginary axis. Numerical experiments with standard test problems for stiff systems of ODEs show that DC6RK2/4 performs well on problems regarding strong non-linearity and long-term integration, and this method does not require extremely small time steps for accurate numerical solutions of stiff problems. Moreover, this method is better than standard implicit methods like the Backward Differentiation Formulae and the DC methods for the implicit midpoint rule on stiff problems for which Jacobian matrices along the solution curve have complex eigenvalues where imaginary parts have larger magnitudes than real parts. An application of DC6RK2/4 to a class of test problems for reaction-diffusion equations in one dimensional is also carried out.