A High-Order L2-1_σ Compact Exponential Difference Method For Multi-Term Time-Fractional Convection-Reaction-Diffusion Equations

In this paper,an interpolation approximation of linear combination of multi-term Caputo time-fractional derivatives with different orders between 0 and 1 is constructed on nonuniform time meshes.The approximation is based on piecewise quadratic and linear interpolation polynomials at a supper-convergence point,so it is called the L2-1_σapproximation on nonuniform time meshes in this paper.In order to find the supper-convergence point,a monotone iterative algorithm with second-order convergence rate is developed.It is shown that,when the mesh grading parameter is properly selected,the L2-1_σapproximation can achieve second-order convergence for non-smooth solu-tions with weak initial singularity.Combing the L2-1_σapproximation for multi-term Caputo time-fractional derivatives with the fourth-order compact exponential difference approximation for the spatial discretization,an L2-1_σcompact exponential difference method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion equations is developed.A proof of the unconditional stability and conver-gence of this method is given using a proper decomposition of the asymmetric coeffi-cient matrix of the discrete scheme,which is more complex than the case of uniform time meshes and the case of a single fractional derivative.Under the condition of weak initial singularity of the solution,the optimal error estimate of the numerical solution in the discrete L~2-norm is obtained.The error estimate shows that the method has spa-tial optimal fourth-order convergence and can achieve temporal optimal second-order convergence when selecting the proper mesh grading parameter.An extension of the method to two-dimensional problems is also simply discussed.Numerical results con-firm the theoretical convergence result and demonstrate the applicability of the method to convection dominated problems.