A kind of adaptive variable stepsize embedded Runge–Kutta pairs coupled with the Sinc collocation method for solving the KdV equation

For solving the KdV equation, a novel numerical method with high order accuracy in both space and time is proposed. In the spatial direction, Sinc collocation method, which has the property of exponential convergence, is adopted. In the temporal direction, the variable stepsize Runge–Kutta-embedded pair RKq(p) is utilized. Sinc collocation method is applicable when the approximated function satisfies the exponential decay as the spatial variable tends to infinity, this characteristic is consistent with the one of the soliton solution of the KdV equation. For practical computation, a sufficiently large finite domain is taken, on which the differential matrices with respect to the discrete points are constructed. A new adaptive strategy is proposed to enhance the robustness of the variable stepsize algorithm. In the numerical experiment, four embedded pairs including RK5(4), RK6(5), RK8(7) and RK9(8) are investigated in terms of accuracy, CPU time, the minimum, average and maximum time stepsizes. The numerical results show that RK8(7) has a better performance in the computational efficiency, it achieves higher accuracy with significantly less CPU time. Besides, the KdV-Burgers equation with nonhomogeneous Dirichlet boundary condition imposed on a general interval is considered. The single-exponential transformation and double-exponential transformation are involved. We show that Sinc collocation method, enhanced by exponential transformations, provides an effective numerical approximation for this problem.