A Composite Runge–Kutta Method for the Spectral Solution of Semilinear PDEs

Abstract

A new composite Runge–Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev–Petviashvili (KP), Kuramoto–Sivashinsky (KS), Cahn–Hilliard, and others having high-order derivatives in the linear term. The method uses Fourier collocation and the classical fourth-order RK method, except for the stiff linear modes, which are treated with a linearly implicit RK method. The composite RK method is simple to implement, indifferent to the distinction between dispersive and dissipative problems, and as efficient on test problems for KS and KP as any other generally applicable method.