A compact locally one-dimensional method for fractional diffusion-wave equations

Abstract

This paper is concerned with numerical methods for a class of multi-dimensional fractional diffusion-wave equations with a time fractional derivative of order $$\alpha $$ ( $$1<\alpha <2$$ ). A compact locally one-dimensional (LOD) finite difference method is proposed for the equations. The resulting scheme consists of one-dimensional tridiagonal systems, and all computations are carried out completely in one spatial direction as for one-dimensional problems. The unconditional stability and $$H^{1}$$ norm convergence of the scheme are rigorously proved for the three-dimensional case. The error estimates show that the proposed compact LOD method converges with the order $$(3-\alpha )$$ in time and $$4$$ in space. Numerical results confirm our theoretical analysis and illustrate the effectiveness of this new method.