A Fully Spectral Space-Time Collocation Method for Two-Dimensional Allen-Cahn Equations with Tempered Space-Fractional Diffusion
We develop a fully spectral ultraspherical collocation method for the numerical solution of two-dimensional tempered space-fractional Allen-Cahn equations. The scheme combines a tensor-product ultraspherical collocation discretization in the spatial variables with an ultraspherical-Gauss-Radau collocation method in time, yielding a global high-order approximation for the nonlinear tempered fractional problem. To discretize the nonlocal diffusion terms, we construct left- and right-sided tempered Riemann-Liouville fractional differentiation matrices in the ultraspherical basis, thereby reducing the governing equation to a coupled nonlinear algebraic system for the ex- pansion coefficients. The resulting method accurately resolves the tempered fractional operators while retaining the high-order accuracy of spectral approximations. Numerical experiments for a benchmark problem with an exact solution demonstrate the accuracy, rapid convergence, and robustness of the proposed scheme over a range of fractional orders and tempering parameters. These results show that the method provides an effective and reliable computational framework for two-dimensional tempered fractional Allen-Cahn models.