Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.