Operator Splitting With Compact Differences and Second-Order Two-Stage Rosenbrock Method for the Gardner Equation

Abstract

We developed an accurate and reliable numerical method for a nonlinear partial differential equation called as Gardner equation which describes many important wave phenomena. The proposed numerical method employs a high-order compact difference scheme for space discretization which gives rise to a large system of ordinary differential equations for the Gardner equation. The obtained system of ordinary differential is usually stiff so to acquire reasonable results, time step size of time integrator should be very small. To loosen the restriction on the time step size, a splitting technique is used to split the equation into stiff and non-stiff parts. Then a second-order Rosenbrock method is employed for stiff part and a third-order strong stability-preserving Runge-Kutta method is used for non-stiff parts. To judge performance of the proposed method, a sequence of numerical simulations is performed, and obtained results are compared with exact solution and with existing numerical methods in the literature such as finite element and collocation methods. The numerical simulations and comparisons show accuracy and reliability of the proposed method.