Numerical simulation methods and analysis for the dynamics of the time-fractional KdV equation

In this paper, two classes of efficient difference schemes for the simulation of the time-fractional Korteweg–de Vries equation are carried out. The temporal derivative is approximated with the help of the uniform/nonuniform L1 formula and the uniform L2-1σ formula, respectively. The spatial derivative is done with the uniform centered difference scheme. Unique solvability, boundedness and convergence of the corresponding difference schemes are rigorously proved at length. As far as we know, it achieves the highest convergence in space among all the difference methods for the time-fractional KdV equation. Several numerical examples confirm the theoretical results and demonstrate the dynamic behavior.