High Order Compact Finite Difference Methods for Non-Fickian Flows in Porous Media

TL;DR

This paper develops and analyzes high-order compact finite difference schemes for simulating non-Fickian flows in porous media, achieving high accuracy and stability in both 1D and 2D cases.

Contribution

It introduces fourth-order compact finite difference schemes combined with Crank-Nicolson discretization for non-Fickian flows, with rigorous stability and convergence analysis.

Findings

Schemes are stable and convergent.

Achieved second-order temporal and fourth-order spatial accuracy.

Numerical results confirm theoretical predictions.

Abstract

In this work, fourth-order compact block-centered finite difference (CBCFD) schemes combined with the Crank-Nicolson discretization are constructed and analyzed for solving parabolic integro-differential type non-Fickian flows in one-dimensional and two-dimensional cases. Stability analyses of the constructed schemes are derived rigorously. We also obtain the optimal second-order convergence in temporal increment and the fourth-order convergence in spatial direction for both velocity and pressure. To verify the validity of the CBCFD schemes, we present some experiments to show that the numerical results are in agreement with our theoretical analysis.