Discontinuous Spectral Element Methods for Time- and Space-Fractional Advection Equations

Abstract

We develop spectral element methods for a time- and space-fractional advection equation of the form $\prescript{}{0}{\mathcal{D}}^{\tau}_{t} u(x,t) + \theta\,\, \prescript{}{0}{\mathcal{D}}^{\nu}_{x} u(x,t) = f(x,t)$, of order $\tau\in (0,1]$, $\nu \in (0,1)$, subject to Dirichlet initial/boundary conditions. We present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms, instead of employing traditional low-order time-integration methods. To this end, we first develop a Petrov--Galerkin in time and discontinuous Galerkin in space (PG-DG) method, where we carry out the time-integration using a single time-domain spectral method (SM), and we perform the spatial discretization using the discontinuous spectral/$hp$ element method (DSEM). This scheme also leads to a more efficient time-integration when the time-derivative is integer-order, i.e., $\tau=1$. We develop the SM-DSEM scheme based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), recently presented in J. Comput. Phys., 47 (2013), pp. 2108--2131. We choose the corresponding space-time bases from the span of tensor product of the introduced eigenfunctions. Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi polyfractonomials, as temporal bases. We also employ the corresponding asymptotic eigensolutions to FSLP-I, which are Jacobi polynomials, as the spatial basis. Next, we construct a different test function space, defined as the span of tensor product of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II), as the temporal test functions and the corresponding asymptotic eigensolutions to FSLP-II as the spatial ones. Subsequently, we extend PG-DG to a DG-DG scheme employing the DG method in both time and space. In this scheme, both time-integration and spatial discretization are performed in a DSEM fashion (DSEM-DSEM). Our numerical tests confirm the expected spectral/algebraic convergence, respectively, in corresponding $p$- and $h$-refinements in various test cases and show a four-order of magnitude speed-up compared to finite-difference discretizations.