LSSC 2019: Large-Scale Scientific Computing pp 95-104

# A Second Order Time Accurate Finite Volume Scheme for the Time-Fractional Diffusion Wave Equation on General Nonconforming Meshes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

## Abstract

SUSHI (Scheme Using Stabilization and Hybrid Interfaces) is a finite volume method has been developed at the first time to approximate heterogeneous and anisotropic diffusion problems. It has been applied later to approximate several types of partial differential equations. The main feature of SUSHI is that the control volumes can only be assumed to be polyhedral. Further, a consistent and stable Discrete Gradient is developed.

In this note, we establish a second order time accurate implicit scheme for the TFDWE (Time Fractional Diffusion-Wave Equation). The space discretization is based on the use of SUSHI whereas the time discretization is performed using a uniform mesh. The scheme is based on the use of an equivalent system of two low order equations. We sketch the proof of the convergence of the stated scheme. The convergence is unconditional. This work is an improvement of  in which a first order scheme, whose convergence is conditional, is established.

## Keywords

Finite volume Time Fractional Diffusion Wave Equation System Unconditional convergence Second order time accurate

## References

1. 1.
Alikhanov, A.A.: A new difference scheme for the fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
2. 2.
Bradji, A.: Convergence order of gradient schemes for time-fractional partial differential equations. C. R. Math. Acad. Sci. Paris 356(4), 439–448 (2018)
3. 3.
Bradji, A.: Some convergence results of a multi-dimensional finite volume scheme for a time-fractional diffusion-wave equation. In: Cancès, C., Omnes, P. (eds.) FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol. 199, pp. 391–399. Springer, Cham (2017).
4. 4.
Bradji, A.: An analysis of a second-order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes. Appl. Math. Comput. 219(11), 6354–6371 (2013)
5. 5.
Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)
6. 6.
Gao, G.-H., Sun, Z.-Z., Zhang, H.-W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
7. 7.
Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)
8. 8.
Sun, Z.-Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)