Linear energy-stable Runge–Kutta relaxation schemes for the Bi-flux diffusion model

Abstract

This paper conducts an in-depth study of nonlinear Bi-flux diffusion models with one energy stable linear relaxation with regularized energy reformulation numerical scheme. This novel scheme combines the single diagonal implicit Runge–Kutta method (SDIRK) in temporal dimension and a meshless generalized finite difference method (GFDM) in spatial dimension. Thus in terms of spatial discretization high quality grids are not required and in terms of time discretization large time step is potential compared with the existing methods. The rigorous proof of the unconditional energy stable property for the scheme is presented. According to different values of the coefficient in nonlinear Bi-flux model, it could degenerate to Allen–Cahn equation, Fisher–Kolmogorov equation and extended Fisher–Kolmogorov model. The accuracy and the effectiveness of the proposed scheme are presented. Moreover, a large number of evolution processes for the nonlinear Bi-flux model under different regimes are demonstrated.