A Fast Difference Scheme For Solving Multi-term Time Fractional Mixed Diffusion And Wave Equations With Variable Coefficient

This paper discusses the establishment and theoretical analysis of the numerical scheme devised for solving space variable coefficient multi-term time fractional mixed diffusion and wave equations.The equations studied in this paper contain both multi-term fractional diffusion terms and multi-term wave terms which makes it challenging to achieve second-order accuracy in time.With the help of the method of order reduction,this paper converts the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively,and then uses the L1 formula and L2-1σ formula to discretize them separately,which can achieve second-order accuracy in time.In addition,this method is combined with SOE technology,and the resulting fast algorithm maintains almost the same numerical accuracy as traditional methods.In this way,the computational cost is significantly reduced and the computational efficiency is high in the long time simulation.The stability and convergence of these numerical schemes are rigorously analyzed by the energy method.Numerical experiments have verified that the convergence rates of the proposed method are second-order both in time and space,and the advantage of the SOE fast algorithm is demonstrated by comparing the CPU time with the direct scheme.Furthermore,this method is also applied to the multi-term time fractional mixed diffusion and wave equation with spatial fourth order variable coefficients.In order to avoid the difficulty of dealing directly with the fourth order derivative term,this paper also introduces auxiliary variables to transform the original problem into an equivalent low order coupling problem,and obtain the corresponding fast difference scheme.The stability and convergence of the proposed numerical scheme are theoretically analyzed by using the energy analysis method.Numerical experiments show that the convergence rate of the proposed method is second-order both in time and space.