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Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation
Zhibo Wang, Seakweng Vong
TL;DR
Earlier schemes for these fractional equations generally provide temporal accuracy below two, motivating more accurate discretizations. The paper uses weighted and shifted Grünwald operators to construct compact finite difference schemes, establishing temporal order two and spatial order four. Numerical experiments support the theoretical convergence analysis, while stated limitations concern a nonnegativity property and an example without an exact solution.
Problem
Previous schemes for the modified anomalous fractional sub-diffusion and fractional diffusion-wave equations have temporal accuracy that depends on the fractional derivative order and is usually less than two.
Method
The paper develops compact finite difference schemes based on weighted and shifted Grünwald difference operators for both fractional equations.
Results
The proposed schemes have accuracy order τ^2 + h^4, with temporal step size τ and spatial step size h, and numerical convergence matches the theoretical order.
Takeaways & Limitations
The schemes provide second-order temporal and fourth-order spatial accuracy for the studied fractional sub-diffusion and diffusion-wave equations.
Takeaways & Limitations
The paper cannot prove that h(α, x) is nonnegative, and one numerical example uses a reference numerical solution because an exact solution is not readily available.
Abstract
from arXiv · showhide
In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied. Schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Based on the idea of weighted and shifted Grunwald difference operator, we establish schemes with temporal and spatial accuracy order equal to two and four respectively.
1 Introduction
The paper targets accurate numerical approximations for fractional sub-diffusion and diffusion-wave equations, whose prior schemes usually have temporal accuracy below two. It develops weighted-and-shifted-Grünwald-based high-order compact schemes for both equations.
- Motivation: Fractional sub-diffusion and diffusion-wave equations are studied as numerical-approximation problems for physically relevant fractional models.Fractional derivatives of order between zero and one model anomalous diffusion, while orders between one and two support electromagnetic, acoustic, and mechanical response modeling.
- Related work: Previous numerical methods for the modified anomalous fractional sub-diffusion and diffusion-wave equations have been developed by many authors.
- Related work: The main numerical challenge is discretizing the fractional derivatives accurately within finite difference schemes.Fractional derivatives are defined through integrals, motivating approximation through interpolating polynomials and related discretizations.
- Related work: Weighted and shifted Grünwald difference operators provide an alternative basis for constructing accurate finite difference schemes.Earlier work also analyzed forward-Euler and weighted averaged schemes based on Grünwald–Letnikov approximations.
- Contribution: The paper establishes compact schemes with temporal and spatial accuracy orders two and four, respectively.The schemes address both the fractional diffusion-wave equation and the modified anomalous fractional sub-diffusion equation.
2 Preliminaries
The preliminaries define the fractional derivatives and integral used in the paper, then introduce shifted and weighted-shifted Grünwald operators. Coefficient conditions are used to target second-order accuracy before constructing high-order compact schemes.
- Fractional operators: The section defines the Caputo derivative for order γ ∈(1, 2), the Riemann–Liouville derivative for α ∈(0, 1), and the associated fractional integral.These definitions establish the fractional operators used in the subsequent analysis.
- Shifted Grünwald operators: A shifted Grünwald approximation is introduced for the Riemann–Liouville fractional derivative.The approximation uses coefficients g(α)_k for k ≥ 0.
- Shifted Grünwald operators: A corresponding shifted operator is introduced for the Riemann–Liouville fractional integral, with coefficients ω(α)_k for k ≥ 0.The construction is motivated by related work cited in the paper.
- Accuracy construction: The coefficients μ1 and μ2 are constrained by a system to achieve second-order accuracy, preparing the high-order compact schemes developed next.A preceding lemma and its Fourier-transform argument support the operator analysis.
- Weighted operators: Weighted and shifted Grünwald difference operators are defined using integer shifts p and q with p ≠ q.The preliminaries also state Fourier-transform conditions and use Fourier analysis in the derivation.
3 A compact scheme for the fractional sub-diffusion equation
The section develops a compact finite-difference scheme for the modified anomalous fractional sub-diffusion equation using weighted and shifted fractional-derivative approximations. The scheme is analyzed for solvability, stability, and convergence, with a compact spatial discretization designed to raise accuracy.
- Problem setting: The analysis assumes zero initial data, while nonzero initial data are handled by introducing v(x, t) = u(x, t) − ψ(x).The boundary conditions prescribe u(0, t) = ϕ1(t) and u(L, t) = ϕ2(t).
- Scheme construction: The problem is discretized with an weighted Crank-Nicolson type time discretization and a compact spatial scheme based on a sixth-derivative regularity lemma.The construction uses shifted Grunwald approximations for fractional operators and a compact spatial formula.
- Solvability: At each time level, the difference scheme forms a linear tridiagonal system with a strictly diagonally dominant coefficient matrix, giving a unique solution.This establishes well-posedness of the discrete problem at every time level.
- Stability and convergence: The convergence analysis uses an error equation, quadratic-form positivity, and a discrete Gronwall inequality.The positivity argument is connected to a symmetric Toeplitz matrix and the Grenander-Szegő theorem.
- Analysis: A proof step notes that the function h(α, x) cannot be shown nonnegative by differentiating with respect to α as in earlier work.The section instead establishes nonnegativity through monotonicity in x and the Grenander-Szegő theorem.
- Stability and convergence: The compact scheme is shown to be stable for 0 < α, β < 1.The stability statement is obtained using techniques similar to those used in the convergence proof.
4 A Compact scheme for the fractional diffusion-wave equation
The fractional diffusion-wave problem is reformulated and discretized using a compact finite difference scheme, whose solvability, convergence, and stability are established through supporting lemmas.
- Problem formulation: The original problem is equivalently transformed into one involving a Riemann-Liouville fractional integral with α = γ − 1 ∈(0, 1).This reformulation is introduced before constructing the finite difference scheme.
- Scheme construction: The compact scheme for the reformulated problem is introduced after selecting parameters that yield µ1 = 1 − α.The parameter choice follows from the cited lemma used in constructing the approximation.
- Solvability: The resulting difference scheme has a unique solution because its linear system has the required structural properties.For the earlier scheme, the coefficient matrix is described as strictly diagonally dominant and tridiagonal; the later scheme is stated to be uniquely solvable for the same reasons.
- Convergence and stability: Convergence is established through a supporting lemma based on properties of the associated Toeplitz sequence and generating function.The proof uses positivity and related generating-function arguments, with details stated to parallel the earlier theorem.
- Convergence and stability: The proposed compact scheme is also shown to be stable by following the proof strategy used for the preceding scheme.The stability statement is given as a remark extending the earlier argument to this formulation.
5 Numerical experiments
Numerical experiments test the proposed schemes on modified anomalous fractional sub-diffusion and fractional diffusion-wave problems. The reported convergence results agree with the theoretical accuracy and remain effective when an exact solution is unavailable.
- Experimental setup: The experiments compute maximum-norm and 2-norm errors between exact and numerical solutions, using MATLAB with L = T = 1.These tests assess the proposed finite difference schemes across several examples.
- Modified anomalous fractional sub-diffusion: Example 5.1 compares exact and numerical solutions for α = 0.2, β = 0.7, and h = τ = 1.The example also varies parameters and reports temporal and spatial convergence orders in Tables 1–4.
- Modified anomalous fractional sub-diffusion: The convergence order of the numerical results matches the theoretical order in the modified anomalous fractional sub-diffusion tests.Tables report temporal convergence with h = 1 and spatial convergence with τ = 1 for Example 5.1.
- Modified anomalous fractional sub-diffusion: For Example 5.2, the exact solution is u(x, t) = t^2 sin(2πx), and both temporal and spatial convergence tables confirm the theoretical result.The temporal study uses h = 1, while the spatial study uses τ = 1; maximum-norm and 2-norm rates coincide for the listed digits.
- Fractional diffusion-wave equations: Tables 7 and 8 justify the accuracy of the Section 4 scheme for the fractional diffusion-wave Example 5.3.Example 5.3 has exact solution u(x, t) = e^x t^(α+2), with α = γ − 1.
- Fractional diffusion-wave equations: When the exact solution cannot be found readily in Example 5.4, a fine-grid numerical solution is used as the ‘true’ solution, and the scheme still works properly.The reference solution uses M = 400 and N = 4000; numerical results for sufficiently small τ justify the theoretical analysis.