A fast \(O(N\log N)\) finite difference method for the one-dimensional space-fractional diffusion equation
Summary: This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of \(O(N^2)\) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of \(O(N^3)\) per time step, where \(N\) is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of \(O(N)\) and a computational cost of \(O(N\log N)\) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.
scientific article; zbMATH DE number 6528860
scientific article