High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights

Abstract

The equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the Lévy flights; and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the Lévy flights have divergent second order moments. However, in more practical problems, the physical domain is bounded and the involved observables have finite moments. Then the modified equation can be derived by tempering the probability of large jump length of the Lévy flights and the corresponding tempered space fractional derivative is introduced. This paper focuses on providing the high order algorithms for the modified equation, i.e., the equation with the time fractional substantial derivative and space tempered fractional derivative. More concretely, the contributions of this paper are as follows: 1. the detailed numerical stability analysis and error estimates of the schemes with first order accuracy in time and second order in space are given in {\textsl{complex}} space, which is necessary since the inverse Fourier transform needs to be made for getting the distribution of the functionals after solving the equation; 2. we further propose the schemes with high order accuracy in both time and space, and the techniques of treating the issue of keeping the high order accuracy of the schemes for {\textsl{nonhomogeneous}} boundary/initial conditions are introduced; 3. the multigrid methods are effectively used to solve the obtained algebraic equations which still have the Toeplitz structure; 4. we perform extensive numerical experiments, including verifying the high convergence orders, simulating the physical system which needs to numerically make the inverse Fourier transform to the numerical solutions of the equation.

28 pages, 4 figures

References in corpus (5)

• High order schemes for the tempered fractional diffusion equations
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• On distributions of functionals of anomalous diffusion paths
• Truncation effects in superdiffusive front propagation with Lévy flights
• Numerical algorithms for the forward and backward fractional Feynman-Kac equations

Cited by in corpus (10)

• A Series of High Order Quasi-Compact Schemes for Space Fractional Diffusion Equations Based on the Superconvergent Approximations for Fractional Derivatives
• A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation
• Third order quasi-compact schemes for space tempered fractional diffusion equations
• High Order Finite Difference Methods on Non-uniform Meshes for Space Fractional Operators
• Numerical schemes of the time tempered fractional Feynman-Kac equation
• Correction of BDFk for fractional Feynman-Kac equation with Lévy flight
• A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems
• High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Backward difference formula: The energy technique for subdiffusion equation