Efficient Jacobi spectral-IRK method for one- and two-dimensional tempered fractional Allen-Cahn equations

Tempered fractional Allen-Cahn equations have many vital applications in science and engineering. The presence of a tempered fractional derivative enables these equations to efficiently explain more complex systems where long-range effects exist with an exponential decay. In this paper, we provide a numerical approach to tempered space-fractional Allen-Cahn equations. The spectral collocation method is implemented, based on Jacobi polynomials, to reduce one- and two-dimensional tempered space-fractional Allen-Cahn equations to a system of ordinary differential equations in the time direction. Then, the implicit Runge-Kutta method is applied to approximate the resulting system. This is the first work that uses the implicit Runge-Kutta method to solve one- and two-dimensional tempered space-fractional Allen-Cahn equations. High accuracy of the spectral collocation method, together with the simplicity and low computational cost of the implicit Runge-Kutta method, represent key advantages of the proposed scheme when applied to such a problem. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.