Second order difference schemes for time-fractional KdV–Burgers' equation with initial singularity.

In this paper, we study the numerical method for time-fractional KdV–Burgers' equation with initial singularity. The famous L 2 - 1 σ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]