Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain

Abstract

This paper is concerned with two fast and efficient numerical methods to solve the multidimensional nonlinear fractional wave equation in unbounded domain. For the spatial discretization, a spectral-Galerkin method is adopted by using the Fourier-Like Mapped Chebyshev function approximation, which makes fractional Laplacian diagonalized. Then, for the temporal discretization, the second order accurate time-splitting method is proposed and it not only avoids the matrix exponential computation in view of the diagonalized structure of semi-discrete systems, but also results in an explicit and time symmetric scheme; Then, an exponential scalar auxiliary variable (E-SAV) method is developed for solving fractional wave equation, which preserves the original energy conservation and makes nonlinear term to be solved explicitly to obtain a linear system at each time step. In addition, the two schemes both can be implemented by the fast Fourier transform (FFT) related to Chebyshev polynomials. Numerical experiments are provided to test the numerical accuracy and the efficiency in long time simulations.