# High order numerical schemes for solving fractional powers of elliptic operators

## TL;DR

This paper develops high-order finite difference schemes for numerically solving fractional powers of elliptic operators, improving accuracy and efficiency in modeling nonlocal diffusion processes.

## Contribution

It introduces a fourth-order finite difference scheme with an optimal weight parameter for pseudo-parabolic problems related to fractional elliptic operators.

## Key findings

- The scheme achieves fourth-order accuracy.
- Theoretical analysis confirms stability and convergence.
- Computational experiments demonstrate improved performance.

## Abstract

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Petr N. Vabishchevich, Journal of Computational Physics. 2015, Vol. 282, No.1, pp.289--302) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.