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A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations
WenYi Tian, Han Zhou, Weihua Deng
TL;DR
Space fractional diffusion equations require accurate numerical treatment of Riemann-Liouville fractional derivatives. The paper proposes weighted and shifted Grünwald operators and uses them in finite-difference schemes whose stability and convergence are established for one- and two-dimensional problems. The schemes achieve O(τ^2 + h^2) accuracy in the discrete L2 norm, while some alternative approximations are not unconditionally stable.
Problem
Accurately approximating Riemann-Liouville fractional derivatives is difficult, motivating higher-accuracy numerical solutions for space fractional diffusion equations.
Method
The paper combines shifted Grünwald-Letnikov formulae with weights to construct second-order approximations and finite-difference schemes for one- and two-dimensional space fractional diffusion equations.
Results
The schemes are unconditionally stable in the stated one- and two-dimensional cases and achieve O(τ^2 + h^2) accuracy in the discrete L2 norm.
Takeaways & Limitations
Weighted and shifted Grünwald operators provide theoretically analyzed and numerically verified discretizations for space fractional diffusion equations.
Takeaways & Limitations
The (p, q) = (0, −1) method is unstable for the relevant fractional-diffusion setting, and schemes based on (2.23) or (2.28) are not unconditionally stable for time-dependent fractional problems.
Abstract
from arXiv · showhide
A class of second order approximations, called the weighted and shifted Grünwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.
1 Introduction
The paper addresses the difficulty of accurately approximating Riemann-Liouville fractional derivatives by proposing weighted and shifted Grünwald operators for space fractional diffusion equations. It develops and analyzes finite-difference schemes for one- and two-dimensional problems.
- Standard Grünwald-Letnikov discretization provides only first-order accuracy for Riemann-Liouville fractional derivatives.
- Weighted and shifted Grünwald-Letnikov formulae combine distinct shifts and weights to achieve second and higher order accuracy.The weights depend on both the shifted numbers and the fractional-derivative order.
- The proposed finite-difference schemes solve one- and two-dimensional space fractional diffusion equations, with stability analysis for each case.
- The finite-difference solutions approximate the exact solutions with O(τ^2 + h^2) accuracy in the discrete L2 norm.Numerical experiments are used to verify the methods' efficiency and accuracy.
2 High Order Approximations for Riemann-Liouville Fractional Derivatives
The paper develops weighted and shifted Grünwald operators for higher-order approximation of Riemann-Liouville fractional derivatives, then analyzes their matrix properties and applicability on bounded domains. The second-order construction supports finite-difference schemes whose solutions achieve O(τ^2 + h^2) accuracy in the discrete L2 norm, while selected three-shift operators lack unconditional stability.
- Second-order approximations: The weighted and shifted Grünwald approach combines distinct shifted Grünwald-Letnikov formulas with weights to obtain second- and higher-order approximations.The construction is motivated by the first-order accuracy of the standard shifted Grünwald operator.
- Second-order approximations: For bounded nonperiodic intervals, shifts p and q should satisfy |p| ≤1 and |q| ≤1 so required grid nodes remain inside the domain.The paper also notes zero extension when homogeneous endpoint values permit it.
- Stability analysis: The choices (p, q) = (1, 0) and (1, −1) yield negative real parts for the difference-matrix eigenvalues over the stated fractional-order ranges.For (1, 0), the result covers 1 < α ≤2; for (1, −1), it covers 1 ≤α ≤2.
- Stability analysis: For 1 < α ≤2, the matrix A is negative definite, and c1A + c2A^T has eigenvalues with negative real parts when c1, c2 ≥0 and c1 + c2 ≠0.The proof uses the symmetric or Hermitian part, generating functions, and Toeplitz eigenvalue properties.
3 One Dimensional Space Fractional Diffusion Equation
The one-dimensional scheme applies Crank–Nicolson time discretization with second-order WSGD approximations for the Riemann–Liouville spatial derivatives. Its stability and convergence are established for the resulting finite-difference method.
- Discretization: The problem is discretized on a uniform spatial and temporal mesh using the Crank–Nicolson technique and second-order WSGD formulas.The matrix form uses (I − B)U^n+1 = (I + B)U^n + τF^n + H^n.
- Discretization: The WSGD spatial operator approximates the left and right Riemann–Liouville fractional derivatives with second-order accuracy for (p, q) = (1, 0) or (1, −1).The diffusion equation uses nonnegative constant coefficients K1 and K2 and assumes a unique, sufficiently smooth exact solution.
- Stability: The finite-difference scheme is unconditionally stable because the relevant matrix eigenvalues have negative real parts, yielding spectral radius below one.The argument uses the negative eigenvalues of the symmetric matrix component and the resulting bound |(1 + λ)/(1 − λ)| < 1.
- Stability: The θ-weighted WSGD scheme is unconditionally stable when 1/2 ≤ θ ≤ 1.This extends the stability result beyond the Crank–Nicolson choice through the eigenvalue transformation of the iteration matrix.
- Convergence: The numerical solution converges to the exact solution in the discrete L2 norm with error bounded by a constant multiple of τ^2 + h^2.The convergence proof combines an error equation, negative definiteness of A and A^T, and the discrete Gronwall inequality.
4 Two Dimensional Space Fractional Diffusion Equation
The paper formulates a two-dimensional space fractional diffusion problem with Riemann-Liouville operators and develops Crank-Nicolson and alternating-direction schemes based on WSGD discretization. Stability and convergence are established for the resulting schemes.
- Problem formulation: The two-dimensional problem uses left- and right-sided Riemann-Liouville fractional operators of orders 1 < α, β ≤2 with diffusion coefficients and boundary conditions tied to their nonzero directions.The domain is Ω = (a, b) × (c, d), with homogeneous boundary values imposed on sides associated with nonzero directional coefficients.
- Discretization: The domain is discretized on a uniform spatial and temporal mesh, and the WSGD formula is used to construct a Crank-Nicolson finite difference approximation.The mesh uses steps h_x, h_y, and τ; the analysis also sets h_x = h_y = h for simplicity.
- Solution methods: The two-dimensional implementation includes LOD, Peaceman-Rachford ADI, Douglas ADI, and D’yakonov ADI techniques for efficiently solving the factored scheme.The ADI methods are introduced as solution techniques for the finite difference system.
- Stability analysis: The difference scheme (4.6) is unconditionally stable for 1 < α, β ≤2.The conclusion follows from the spectral radius of the directional evolution matrices being less than 1.
- Stability analysis: The stability proof represents grid functions as vectors and uses Kronecker-product matrix identities, commuting directional matrices, and spectral-radius bounds.The real parts of the directional eigenvalues are negative, making the relevant matrix factors contractive.
5 Numerical Examples
Numerical examples test the CN-WSGD method on one-dimensional constant- and variable-coefficient problems and compare four schemes on a two-dimensional problem. The experiments use exact solutions and report errors and convergence rates.
- One Dimensional Case: Examples 2–4 use exact one-dimensional solutions, including u(x, t) = e^-t x^(1+α) and u(x, t) = e^-t x^3(1 − x)^3.Example 4 extends the testing to a variable-coefficients problem with the same stated exact solution as Example 3.
- One Dimensional Case: Tables 2–4 report maximum and L2 errors with convergence rates at t = 1 for different α under the CN-WSGD scheme and τ = h.The tables cover Examples 2, 3, and 4, respectively.
- Two Dimensional Case: Example 5 tests a two-dimensional fractional diffusion problem on Ω = (0, 1)^2 with fractional orders 1.8 and 1.2 and homogeneous boundary conditions.Its exact solution is u(x, y, t) = e^-t x^3(1 − x)^3 y^3(1 − y)^3.
- Two Dimensional Case: Table 5 compares maximum and L2 errors and convergence rates at t = 1 with τ = h for the LOD, PR-ADI, Douglas-ADI, and D’yakonov-ADI schemes.The comparison uses N = N_x = N_y and shifted numbers p and q.
- Two Dimensional Case: The three ADI schemes obtain more accurate solutions than the LOD scheme and are equivalent in the two-dimensional case according to the numerical results.This conclusion is drawn from the maximum and L2 error comparisons in Table 5.
6 Conclusion
The paper presents WSGD operators as second-order approximations for fractional derivatives and applies them to one- and two-dimensional space fractional diffusion equations. Their effectiveness is supported by theoretical analysis and numerical verification.
- Conclusion: WSGD operators provide novel second-order approximations for Riemann-Liouville fractional derivatives and suggest routes toward higher-order and compact discretizations.The paper names the operators weighted and shifted Grünwald difference operators.
- Conclusion: The discretizations are applied to one- and two-dimensional space fractional diffusion equations using several numerical schemes whose effectiveness is theoretically proved and numerically verified.The conclusion covers both the analytical results and the numerical experiments.