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A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative
Ercília Sousa, Can Li
TL;DR
The paper studies numerical solution of a one-dimensional fractional diffusion equation with a nonlocal Riemann-Liouville derivative. It develops a second-order derivative discretization and a weighted average finite difference method, proving unconditional von Neumann stability and convergence. Numerical tests agree with the theoretical results.
Problem
Nonlocal fractional derivatives complicate numerical methods for anomalous diffusion, motivating accurate discretization of the Riemann-Liouville derivative.
Method
The paper combines a second-order Riemann-Liouville derivative approximation with a weighted average finite difference scheme and von Neumann stability analysis.
Results
The scheme is unconditionally von Neumann stable for τ ∈[1/2, 1], with second-order accuracy at τ = 1/2 and numerical tests agreeing with theory.
Takeaways & Limitations
The method provides a convergent weighted numerical approach for the fractional diffusion equation based on the Riemann-Liouville derivative on an unbounded domain.
Takeaways & Limitations
The stability analysis treats d_j as locally constant, while convergence also assumes conditions ensuring the Riemann-Liouville integrals converge.
Abstract
from arXiv · showhide
A one dimensional fractional diffusion model with the Riemann-Liouville fractional derivative is studied. First, a second order discretization for this derivative is presented and then an unconditionally stable weighted average finite difference method is derived. The stability of this scheme is established by von Neumann analysis. Some numerical results are shown, which demonstrate the efficiency and convergence of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method.
1 Introduction
The paper addresses numerical challenges for anomalous diffusion modeled by a nonlocal Riemann-Liouville derivative, focusing on superdiffusion with 1 < α < 2. It introduces a second-order derivative approximation and a weighted finite difference scheme with established stability and convergence properties.
- Fractional space derivatives model anomalous diffusion, whose mean square displacement grows nonlinearly over time.
- Superdiffusion is studied for fractional derivative orders 1 < α < 2, with experimental evidence reported in prior work.
- Nonlocal fractional derivatives create challenges for numerical methods compared with local integer-order derivatives.
- The proposed Riemann-Liouville derivative approximation has second-order consistency on an unbounded domain and is represented by a series.
- The weighted average finite difference τ-scheme covers τ ∈[1/2, 1], includes Crank-Nicolson and backward Euler cases, and is unconditionally stable.
- The paper analyzes a one-dimensional equation on the real line with 1 < α ≤2, positive diffusion coefficient d(x), initial data, and boundary conditions.
2 The numerical method
The numerical method combines a linear-spline approximation of the Riemann-Liouville integral with weighted time discretization. The resulting spatial operator is assembled from series coefficients and integrated into a weighted average finite difference scheme.
- Approximation of the Riemann-Liouville derivative: The discretization uses uniform mesh points x_j = j∆x on the unbounded spatial grid.
- Approximation of the Riemann-Liouville derivative: The fractional operator is formed from coefficients q_j,k derived from neighboring a_j,k values and applied through a series representation.
- Approximation of the Riemann-Liouville derivative: The series representation converges absolutely for 1 < α < 2 and every bounded function u(x,t) at fixed time.
- Approximation of the Riemann-Liouville derivative: A linear spline approximates the integral representation of the Riemann-Liouville derivative to obtain a second-order spatial approximation.
- Weighted average finite difference methods: For α = 2, the fractional operator reduces to the central second-order difference operator.
3 Convergence of the numerical scheme
The paper establishes accuracy and stability for a weighted finite-difference scheme discretizing the fractional diffusion equation. The spatial approximation is second order, while the temporal order depends on the weighting parameter, and von Neumann analysis proves unconditional stability.
- 3.1 Consistency: The fractional derivative approximation is second-order accurate under u ∈ C(4)(IR) with u(4)(x) = 0 for x ≤ a.The associated spatial error term is O(∆x2).
- 3.1 Consistency: The weighted numerical method has truncation error O(∆x2) + O(∆tmτ), with mτ = 1 for τ ∈(1/2, 1] and mτ = 2 for τ = 1/2.Thus τ = 1/2 yields second-order accuracy in both time and space.
- 3.2 Fourier decomposition of the error: The stability proof represents the error as Fourier modes and evaluates their amplification factor through the coefficients qj,j−m.Stability follows when the relevant real part is nonpositive, ensuring that the mode amplitude does not grow.
- 3.2 Fourier decomposition of the error: The coefficient series defining the fractional operator converges absolutely for bounded functions.The coefficient properties include decay of qj,j−m as m tends to infinity and a structured sign pattern for the coefficients.
- 3.2 Fourier decomposition of the error: The weighted τ-scheme is unconditionally von Neumann stable for every τ ∈[1/2, 1].The analysis uses Fourier modes and requires the amplification factor to satisfy |κ(θ)| ≤ 1 for all phase angles θ.
4 Matricial form
The method is formulated in matrix form on a computational domain, incorporating boundary values and the fractional operator. For α > 1.5545, unconditional stability also follows from Gershgorin’s theorem.
- The computational domain [a, b] uses a mesh defined by x_j = a + j∆x.
- When u(a, t) = 0, the bounded-domain problem is equivalent to a whole-real-line problem with solution zero for x ≤ a.
- The numerical method is expressed in matrix form using boundary-value vector b_n, diagonal matrix µ_α, and matrix Q associated with the fractional operator.
- Matrix Q has a structured sparsity pattern, including Q_j,k = 0 for k > j + 1.
- For α > 1.5545, q_j,j−1 ≥ 0 enables an alternative proof of unconditional stability using Gershgorin’s theorem.
5 Numerical implementation
The numerical experiments assess accuracy, convergence, method comparisons, and simulated anomalous-diffusion behavior. Across the tests, τ = 1/2 gives the most accurate results and second-order convergence, while the physical simulations show long-tail diffusion that diminishes as α approaches 2.
- Accuracy and convergence: The experiments first verify the numerical method’s accuracy and convergence against exact solutions.The error is defined using the l∞ norm between approximate and exact solution vectors.
- Accuracy and convergence: τ = 1/2 produces the most accurate results, with second-order convergence observed for all tested α values.This behavior is reported for the first problem across the tested anomalous-diffusion orders.
- Method comparison: The weighted scheme with τ = 1/2 is more accurate than the compared shifted Grünwald-Letnikov Crank–Nicolson methods, before and after Richardson extrapolation.The comparison uses the same space and time steps and evaluates the largest absolute error at t = 1.
- Accuracy and convergence: For τ ≠ 1/2, the expected convergence rate is O(∆t + ∆x2), and choosing ∆t = ∆x2 yields second-order convergence.This rate is confirmed in Table 6 for the stated test configuration.
- Physical behavior: The physical simulation uses τ = 1/2 to track diffusion on [0, 4] with Gaussian initial data, unit diffusion coefficient, and zero boundary values.Figure 3 shows the evolution of u(x, t) for different α values and times.
- Physical behavior: The anomalous-diffusion parameter α controls the extent of long-tail diffusion, whose non-Fickian behavior gradually disappears as α → 2.The reported trend is consistent with cited experimental results.
6 Conclusions
The paper derives and analyzes a weighted numerical method for a fractional diffusion equation with an unbounded-domain Riemann–Liouville derivative. It establishes second-order accuracy at τ = 1/2, first-order accuracy for larger τ, and agreement between test problems and the theoretical convergence and stability results.
- The paper derives a weighted numerical method for the fractional diffusion equation based on the Riemann–Liouville derivative in an unbounded domain.
- The method is second-order accurate for τ = 1/2 and first-order accurate for τ ∈(1/2, 1] because of time discretization.
- Theoretical convergence is established through consistency and von Neumann stability, and test problems agree with these results.