Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

In this paper, we focus our attention on the numerical solution of a class of fractional-in-space nonlinear damped wave equations in two space dimensions. The fractional-in-space telegraph equation, sine–Gordon equation and Klein–Gordon equation can be regarded as particular cases of such equations. A compact difference scheme with accuracy of fourth-order in space and second-order in time is proposed. The solvability, stability and convergence of the scheme are shown under a common assumption. In order to reduce the computational burden, a compact alternating direction implicit (ADI) difference scheme is established. An ADI scheme with accuracy of second-order in both time and space is also derived. Both ADI schemes are applied to solve the aforesaid three kinds of equations. Numerical results are provided to verify the theoretical analysis.