L1/LDG method for the generalized time-fractional Burgers equation

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in (0, 1). If its solution u(x, t) has strong regularity, for example u(., t) is an element of C-2[0, T] for a given time T, then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u(x, t) likely behaves a certain regularity at the starting time, i.e.,. partial derivative u/partial derivative u and partial derivative(2)u/partial derivative(2)t can blow up as t -> 0(+) albeit u(., t) is an element of C[0, T] for a given time T. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.