Efficient and Accurate Double Fast Algorithm for Solving High-Dimensional Time-Space Fractional Diffusion Problems with Spectral Fractional Laplacian

Core Concepts

This paper presents an efficient and accurate double fast algorithm to solve high-dimensional time-space fractional diffusion problems with spectral fractional Laplacian. The proposed scheme uses linear finite element or fourth-order compact difference method combined with matrix transfer technique to approximate the spectral fractional Laplacian, and introduces a fast time-stepping L1 scheme for time discretization. The algorithm can exactly evaluate the fractional power of matrices and perform matrix-vector multiplication efficiently using discrete sine transform, significantly reducing the computational cost and memory requirement.

Abstract

The paper focuses on developing an efficient and accurate numerical scheme for solving high-dimensional time-space fractional diffusion problems with spectral fractional Laplacian. The key highlights are: Spatial Semi-discretization: The authors establish a semi-discrete scheme using linear finite element or fourth-order compact difference method combined with matrix transfer technique to approximate the spectral fractional Laplacian. The proposed scheme can exactly evaluate the fractional power of matrices and perform matrix-vector multiplication efficiently using discrete sine transform, which reduces the computational cost and memory requirement compared to existing methods. Temporal Discretization: A fast time-stepping L1 scheme is introduced for time discretization, which can achieve optimal temporal convergence of O(N^-(2-α)) on graded temporal meshes, where N is the number of time steps. Stability and Convergence Analysis: Stability and error bounds of the full discrete scheme based on the fast time-stepping L1 method on graded time meshes are analyzed. The analysis shows that the choice of graded mesh factor ω = (2-α)/α leads to the optimal temporal convergence. Numerical Examples: Extensive numerical experiments are provided to illustrate the theoretical analysis and the efficiency of the suggested scheme. The proposed double fast algorithm is concise, implementation-friendly, and significantly reduces the computational cost and memory requirement compared to existing methods, making it an efficient tool for solving high-dimensional time-space fractional diffusion problems.

Stats

The second moment or mean squared displacement of a particle in anomalous diffusion follows a nonlinear function of time t, i.e., <|x(t)|^2> ~ t^β with β = [0, 1) ∪ (1, 2]. The Caputo derivative of u(x, t) is defined as: ∂^α_t u(x, t) = 1/Γ(1-α) ∫_0^t ∂u(x, τ)/∂τ dτ / (t-τ)^α. The spectral fractional Laplacian (−Δ + γI)^s is defined as: (−Δ + γI)^s u(x, t) = Σ_j^∞ (u, φ_j) (λ_j + γ)^s φ_j.

Quotes

"Anomalous diffusion phenomena are ubiquitous in natural world, such as diffusive transport of solutes in heterogeneous porous media, RNA movement in bacterial cytoplasm, animals' food-seeking, contaminants in groundwater, material with thermal memory and so on." "To this end, it is necessary to establish concise and implementable solutions for time-space nonlocal model problems."

Key Insights Distilled From

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

by Yi Yang,Jin ... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2212.03493.pdf

Deeper Inquiries

How can the proposed double fast algorithm be extended to solve time-space fractional diffusion problems with more general nonlinear or time-dependent coefficients?

The proposed double fast algorithm can be extended to solve time-space fractional diffusion problems with more general nonlinear or time-dependent coefficients by incorporating these variations into the spatial and temporal discretization schemes. For nonlinear coefficients, the matrix transfer technique and fast time-stepping L1 scheme can be adapted to handle the nonlinearity in the diffusion equation. This may involve iterative methods or nonlinear solvers to handle the nonlinearity efficiently. For time-dependent coefficients, the algorithm can be modified to account for the varying coefficients at different time steps. This may require updating the matrix transfer technique and the fast time-stepping L1 scheme to accommodate the time-dependent nature of the coefficients. Additionally, the stability and convergence analyses may need to be adjusted to ensure the accuracy and efficiency of the algorithm in the presence of time-dependent coefficients. Overall, by incorporating these variations into the algorithm, the double fast algorithm can be extended to handle more general nonlinear or time-dependent coefficients in time-space fractional diffusion problems.

What are the potential applications of the efficient numerical scheme developed in this paper beyond the field of fractional diffusion problems?

The efficient numerical scheme developed in this paper has potential applications beyond the field of fractional diffusion problems. Some of the potential applications include: Fluid Dynamics: The numerical scheme can be applied to simulate fluid flow problems, such as Navier-Stokes equations, where efficient and accurate numerical methods are essential for modeling complex fluid behavior. Heat Transfer: In the field of heat transfer, the numerical scheme can be used to solve transient heat conduction problems with varying thermal properties, boundary conditions, and geometries. Electromagnetics: The scheme can be extended to solve time-dependent electromagnetic field problems, such as Maxwell's equations, in materials with complex properties and geometries. Biomedical Modeling: The numerical scheme can be applied to model and simulate biological processes, such as drug diffusion in tissues, neural signal propagation, or bioheat transfer in living organisms. Financial Modeling: In the realm of finance, the scheme can be utilized for pricing financial derivatives, risk management, and analyzing time-dependent market data. Material Science: The scheme can be employed to study the diffusion of particles in porous media, phase transformations in materials, and other time-dependent phenomena in material science.

Can the fast time-stepping L1 scheme on graded temporal meshes be combined with other spatial discretization techniques, such as spectral methods or isogeometric analysis, to further improve the computational efficiency?

Yes, the fast time-stepping L1 scheme on graded temporal meshes can be combined with other spatial discretization techniques, such as spectral methods or isogeometric analysis, to further enhance computational efficiency. By integrating the fast time-stepping L1 scheme with spectral methods, which are known for their high accuracy and efficiency in handling differential equations, the overall numerical scheme can benefit from the strengths of both approaches. Similarly, combining the fast time-stepping L1 scheme with isogeometric analysis, which utilizes non-uniform rational B-splines to represent the geometry and solution fields, can lead to more accurate and efficient simulations. Isogeometric analysis provides a seamless integration of CAD and analysis, allowing for more precise representation of complex geometries and smoother solutions. The combination of the fast time-stepping L1 scheme with spectral methods or isogeometric analysis can lead to improved accuracy, reduced computational cost, and enhanced efficiency in solving time-space fractional diffusion problems and other differential equations in various fields.