High-order exponential time differencing methods for third-order evolution equations with stiff linear terms

Efficient high-order exponential time differencing methods are proposed for third-order evolution equations with non-periodic boundary conditions in one space dimension. A dual-Petrov–Galerkin spectral method is used to discretize the spatial variables. The trial functions are selected to satisfy boundary conditions of the differential equation, and avoid handling boundary conditions at each time step. The spatial semi-discretization leads to a system of ordinary differential equations (ODEs) in time. The stiff system of ODEs is solved by two exponential time-differencing (ETD) schemes: ETD spectral deferred correction (ETDSDC) method and ETD Runge–Kutta (ETDRK) formula. Since the third-order differential operator is not symmetric, the stiff systems of ODEs have nondiagonal linear part. The coefficients of ETD schemes are numerically calculated by means of numerical integration of auxiliary problems. We rewrite the high-order ETDSDC and ETDRK in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for the generalized Korteweg–de Vries (gKdV) equations. Ample numerical results show that our proposed algorithms are particularly suitable for long time accurate simulations of various KdV-type equations.