Dissipation-preserving Galerkin–Legendre spectral methods for two-dimensional fractional nonlinear wave equations

In this paper, we consider implicit and linearized dissipation-preserving Galerkin–Legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions. The full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case. Moreover, the stability and convergence analysis of the full discrete schemes are rigorously given. Furthermore, we get that two methods are convergent with second-order accuracy in time and optimal error estimates in space. In order to reduce the computational cost, we adopt matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Numerical results are presented to validate the theoretical analysis.