The space–time generalized finite difference scheme for solving the nonlinear equal-width equation in the long-time simulation

Abstract

In this research, the space–time (ST) coupled generalized finite difference method (GFDM) was extended to cooperate with the Newton–Raphson and time-marching methods for stably solving the nonlinear equal-width equation in long-time simulation. The equal-width equation is a time-dependent nonlinear partial differential equation and has a third-order derivation that is mixed space and time. In such a case, the action of fluid flow is over a long period, and the numerical scheme needs to rely on the state-steady condition or set up a specific time to stop the simulation. In the ST-GFDM, the temporal and spatial discretizations can be performed using the GFDM in an ST-domain. That makes the proposed scheme can easily deal with the mixed derivative. By applying the GFDM, which is advanced from the Taylor series expansion and the moving-least square method, the process of the numerical discretizations is only related to nearby nodes on the central node. Thus, the Jacobian matrix is sparse and can be solved by the Newton–Raphson method efficiently. Furthermore, the time-marching method is used to proceed with an ST-domain along the time axis. In this paper, two numerical examples are solved to verify the capability of the proposed ST-GFDM.