[Paper Review] Compact ADI method for solving two-dimensional Riesz space fractional diffusion equation

Sohrab Valizadeh, Alaeddin Malek+1 more|arXiv (Cornell University)|Feb 6, 2018

Fractional Differential Equations Solutions33 references3 citations

TL;DR

This paper proposes a compact alternating direction implicit (ADI) method for solving two-dimensional Riesz space fractional diffusion equations with second-order spatial accuracy and unconditional stability. By combining a compact finite difference scheme with the Crank-Nicolson method and operator-splitting techniques, the method achieves high accuracy and stability, validated through theoretical analysis and numerical experiments.

ABSTRACT

In this paper, a compact alternating direction implicit (ADI) method has been developed for solving two-dimensional Riesz space fractional diffusion equation. The precision of the discretization method used in spatial directions is twice the order of the corresponding fractional derivatives. It is proved that the proposed method is unconditionally stable via the matrix analysis method and the maximum error in achieving convergence is discussed. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed technique.

Motivation & Objective

• To develop a high-order, unconditionally stable numerical method for solving two-dimensional Riesz space fractional diffusion equations.
• To address the challenge of achieving second-order spatial accuracy in fractional diffusion problems with Riesz derivatives.
• To combine compact difference schemes with the alternating direction implicit (ADI) method to improve computational efficiency and stability.
• To theoretically analyze the unconditional stability and convergence order of the proposed scheme using matrix analysis.
• To demonstrate the method's validity and applicability through a numerical example with quantitative error analysis.

Proposed method

• Uses a compact finite difference scheme to discretize the Riesz space fractional derivatives, achieving second-order accuracy in space.
• Applies the Crank-Nicolson scheme in time to ensure second-order temporal accuracy and unconditional stability.
• Employs operator-splitting techniques via the ADI method to decompose the 2D problem into sequential 1D problems, reducing computational complexity.
• Represents the spatial discretization operators in matrix form using tridiagonal matrices $ A_x, B_x, A_y, B_y $, enabling efficient solution via diagonalization.
• Utilizes eigenvalue decomposition of the spatial operators to express the fractional derivative matrices as $ S_x = \frac{k_t}{2}\mathcal{C}_x P \mathbf{D}_x^{\alpha/2} P^{-1} $ and $ T_y = \frac{k_t}{2}\mathcal{C}_y Q \mathbf{D}_y^{\beta/2} Q^{-1} $.
• Ensures that the matrices $ S_x $ and $ T_y $ are symmetric and commute, enabling stable and efficient solution of the linear system at each time step.

Experimental results

Research questions

• RQ1 Can a compact ADI method achieve second-order spatial accuracy for two-dimensional Riesz space fractional diffusion equations?
• RQ2 Is the proposed compact ADI scheme unconditionally stable under the given formulation?
• RQ3 What is the convergence order of the numerical scheme in both space and time?
• RQ4 How does the method perform in terms of maximum error and computational efficiency compared to existing approaches?
• RQ5 Can the method effectively handle the non-local nature of Riesz fractional derivatives while maintaining high accuracy and stability?

Key findings

• The proposed compact ADI method achieves second-order spatial accuracy, as the compact difference scheme reduces truncation error to $ \mathcal{O}(h^4) $.
• The method is proven unconditionally stable using matrix analysis, based on the positive definiteness and commutativity of the spatial operators.
• Theoretical analysis confirms second-order convergence in both time and space, with the maximum error bounded by $ \mathcal{O}(k_t^2 + h_x^4 + h_y^4) $.
• Numerical experiments confirm the method's validity and applicability, showing consistent convergence rates and low error levels.
• The eigenvalues of the spatial operators $ -A_x^{-1}B_x $ and $ -A_y^{-1}B_y $ are positive and distinct, ensuring stability and diagonalizability.
• The matrices $ S_x $ and $ T_y $ are symmetric and commute, which enables efficient solution of the linear system at each time step via diagonalization.