A fully discrete direct discontinuous Galerkin Method for the fractional diffusion-wave equation

Abstract

In this paper, we consider the numerical approximation for the fractional diffusion-wave equation, which is discretized by the finite-difference method in time and the direct discontinuous Galerkin (DDG) method in space. The DDG method is based on the direct weak formulation for solutions of parabolic equations in each computational cell, letting cells communicate via the numerical flux only. We prove that our scheme is unconditionally stable and get energy norm estimates of O under admissible numerical flux. The DDG method has the advantages of easier formulation and implementation as well as the high-order accuracy. Finally numerical examples are shown to illustrate the efficiency and the high-order accuracy of our scheme. Compared with the local discontinuous Galerkin method, the DDG method can reduce the storage and the computation.

Keywords:

• Fractional diffusion-wave equation
• DDG method
• stability
• error estimates

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are grateful to the National Natural Science Foundation of P.R. China [grant number 11571002]; the Natural Science and Technology Development Foundation of CAEP [grant number 2013A0202011], [grant number 2015B0101021].