Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations

Abstract

Abstract. We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order α(1 < α < 2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and fractional integrals of order 2 − α, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O(h k+1 ) for subdiffusion, and an order of convergence of O(h k+ 1 2 ) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method

“…[18,6,24,19,21,17,2,30,25,29,22,27,28,31,8]. In contrast to the classical diffusion equations, the fractional diffusion equations can be used to describe the anomalous diffusion phenomena such as super-diffusion or sub-diffusion.…”

“…A time-fractional diffusion equation occurs when replacing the standard time derivative with a time fractional derivative and can be applied in modeling of some problems in porous flows, rheology and mechanical systems, models of a variety of biological processes, control and robotics, transport in fusion plasmas, and many other areas of applications. The direct problems corresponding to the time-fractional diffusion equations have been studied extensively in recent years, including uniqueness and existence results [2], some analytical or numerical solutions [13,7,31], and numerical methods such as finite element methods or finite difference methods [12,14]. Here, we focus on an interesting inverse problem defined to the fractional inverse problem pioneered by Murio [18,16,17].…”

“…Of course, the forward fractional diffusion equation has been solved successfully by some DG methods. For example, Wei et al [28] have solved the time-fractional diffusion equation using a fully-discrete LDG method and some space-fractional diffusion equations have been solved by Hesthaven et al [7,31] using a local discontinuous Galerkin method in a semi-discrete setting.…”

A multi-domain spectral method for time-fractional differential equations

“…During the last decade, this approach has emerged as generalizations of many classic problems in mathematical physics, including the fractional Burgers' equation [11,25], the fractional Navier-Stokes equation [7,6], the fractional Maxwell equation [10], the fractional Schrödinger equation [8], the fractional Ginzburg Landau equation [8,22], etc. In parallel, numerical methods for classical differential equations, such as finite difference methods [15,14,23], finite element methods [3], spectral methods [13,2,12], and discontinuous Galerkin methods [19,24], have been developed, albeit this remains a relatively new topic of research.…”

A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation

“…While analytical methods, such as the Fourier transform method, the Laplace transform methods, and the Mellin transform method, have been developed to seek closed-form analytical solutions for fractional partial differential equations [6], there are very few cases in which the closed-form analytical solutions are available, just like in the case of integer-order partial differential equations. Numerical methods for the fractional partial differential equations, such as finite difference methods [7], finite element methods [8,9], spectral methods [10], and discontinuous Galerkin methods [11,12] have recently been developed and remains a relatively new topic of research because of the difficulties encountered.…”