[Paper Review] A finite difference approximation of a two dimensional time fractional advection-dispersion problem

Carlos E. Mejía, Alejandro Piedrahita|arXiv (Cornell University)|Jul 19, 2018

Fractional Differential Equations Solutions12 references3 citations

TL;DR

This paper presents an implicit finite difference scheme for solving a two-dimensional time-fractional advection-dispersion equation with variable coefficients, where the fractional derivative is in the Caputo sense and the dispersion terms are in nondivergence form. The method is proven consistent, unconditionally stable, and convergent under mild assumptions, with numerical experiments confirming second-order convergence in space and first-order in time for various fractional orders and diffusion coefficients.

ABSTRACT

Time fractional advection-dispersion equations arise as generalizations of classical integer order advection-dispersion equations and are increasingly used to model fluid flow problems through porous media. In this paper we develop an implicit finite difference method to solve a two-dimensional initial boundary value problem for the linear time fractional advection-dispersion equation with variable coefficients on a bounded domain. Consistency, stability and convergence of the method are proved in detail and the numerical experiments offer a good insight into the quality of the obtained approximations.

Motivation & Objective

• To develop a robust numerical method for modeling contaminant transport in groundwater using time-fractional advection-dispersion equations with variable coefficients.
• To construct an implicit finite difference scheme that handles nondivergence form dispersion terms and Caputo time-fractional derivatives.
• To rigorously prove consistency, unconditional stability, and convergence of the proposed scheme under mild assumptions on coefficient bounds.
• To provide a general and computationally efficient matrix framework for implementation.
• To validate the scheme through comprehensive numerical experiments with exact solutions for varying fractional orders and diffusion coefficients.

Proposed method

• An implicit finite difference scheme is formulated using the L1 approximation for the Caputo time-fractional derivative, as detailed in [6].
• Central finite differences are applied to spatial advection and dispersion terms, with variable coefficients a(x,y,t), b(x,y,t), c(x,y,t), and d(x,y,t).
• The scheme is constructed on a uniform rectangular grid with spatial steps Δx, Δy and time step Δt, and the solution is advanced in time using a linear system solve at each step.
• A matrix-based computational framework is developed to efficiently assemble and solve the resulting linear systems, enabling generalization to various problems.
• Stability and convergence are analyzed using energy methods, with proofs relying on the boundedness of coefficient functions and the properties of the L1 scheme.
• The method is applied to three numerical examples: one with a smooth exact solution and two with varying diffusion coefficients and fractional orders to test robustness.

Experimental results

Research questions

• RQ1 Can an unconditionally stable and convergent finite difference scheme be constructed for a two-dimensional time-fractional advection-dispersion equation with variable coefficients in nondivergence form?
• RQ2 How does the convergence rate of the scheme behave across different fractional orders α ∈ (0,1) and varying diffusion coefficients?
• RQ3 Does the proposed matrix-based framework allow for efficient and general implementation across different boundary and source conditions?
• RQ4 How does the scheme perform in the limit of small diffusion coefficients, approaching a degenerate parabolic regime?
• RQ5 What is the impact of the fractional order α on the solution profile, particularly in the absence of an exact solution?

Key findings

• The proposed finite difference scheme is proven to be consistent, unconditionally stable, and convergent under mild assumptions on the boundedness of coefficient functions.
• Numerical experiments confirm second-order convergence in space and first-order convergence in time for all tested fractional orders α = 0.1, 0.5, and 0.9.
• For Example 4.1 with smooth solution, the order of convergence approaches 2.0 in space and 1.0 in time as the mesh is refined, confirming theoretical expectations.
• In Example 4.2 with very small diffusion coefficient ε = 1e-5, the scheme maintains robust convergence behavior, demonstrating stability even in nearly degenerate cases.
• In Example 4.3, where no exact solution exists, the computed solutions show continuous dependence on the fractional order α, with solutions evolving smoothly from α = 0.3 to α = 0.9 at t = 1.
• The matrix framework enables efficient implementation and was successfully reused in a prior inverse source problem, confirming its practical utility.