Research On Fast Calculation Of Fractional Partial Differential Equations Based On Interpolation Method

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional problems.Here,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constantorder fractional operators,including contour integral method,Jacobi-Gauss quadrature method and trapezoidal rule method,and identify the current pitfalls of such methods.The global approximation method,for example Jacobi-Gauss quadrature method and trapezoidal rule method,does not work well when ? ? 1 and it cannot work for ? > 1.In order to overcome the pitfalls,an improved sum-of-exponentials is developed and verified.Then based on the sum-of-exponentials,we propose a unified framework for a fast timestepping method based on linear interpolation to solve time fractional partial differential equations.By using the discrete fractional Gr(?)nwall inequality,we establish the convergence analysis of the fast time-stepping method.Different from the exsiting methods,we prove the error bound between the numerical solution of the direct method and the fast method,which leads to a simplified error estimate.Finally,we demonstrate the efficiency and robustness of the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen–Cahn equation in two and three spatial dimensions,and the Schr(?)dinger equation with nonreflecting boundary conditions.The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.