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Compact Finite Difference Approximations for Space Fractional Diffusion Equations
Han Zhou, WenYi Tian, Weihua Deng
TL;DR
The paper addresses high-accuracy discretization of one- and two-dimensional space fractional diffusion equations. It constructs compact WSGD operators and analyzes the resulting schemes' stability and error. The schemes achieve third-order spatial and second-order temporal convergence, with numerical experiments confirming the theoretical results.
Problem
Existing shifted Grünwald methods provide limited spatial accuracy, while a prior WSGD operator was not unconditionally stable for time-dependent space fractional diffusion equations.
Method
The paper constructs compact WSGD operators from WSGD discretizations and uses them to build one- and two-dimensional compact finite difference schemes.
Results
The schemes are theoretically stable and converge with third-order accuracy in space and second-order accuracy in time, confirmed by numerical experiments.
Takeaways & Limitations
Compact WSGD discretizations provide effective high-order numerical approximations for the stated one- and two-dimensional space fractional diffusion problems.
Takeaways & Limitations
The compact approximations use shift choices constrained by bounded-domain nodes, and unconditional stability for one choice is established only over specified fractional-order ranges.
Abstract
from arXiv · showhide
Based on the weighted and shifted Grünwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the one and two dimensional space fractional diffusion equations. The detailed numerical stability and error analysis are theoretically performed. We theoretically prove and numerically verify that the provided numerical schemes have the convergent orders 3 in space and 2 in time.
1 Introduction
The paper targets high-accuracy numerical methods for space fractional diffusion equations, extending WSGD discretizations to compact schemes. It establishes stability and convergence theoretically and verifies the results numerically.
- 1 Introduction: Anomalous diffusion is characterized by mean square displacement scaling as ⟨x2(t)⟩−⟨x(t)⟩2 ∼ t^α, with α distinguishing subdiffusion, normal diffusion, and superdiffusion.The cases are 0 < α < 1, α = 1, and α > 1, respectively.
- 1 Introduction: Space fractional diffusion equations describe superdiffusion by replacing the classical second-order derivative with a Riemann-Liouville fractional derivative of order 1 < α < 2.
- 1 Introduction: Earlier shifted Grünwald discretizations generally provide first-order spatial accuracy, while WSGD operators achieve second or higher order accuracy.
- 1 Introduction: The paper constructs compact WSGD operators and uses them to develop compact finite difference schemes for one- and two-dimensional space fractional diffusion equations.
- 1 Introduction: The schemes receive theoretical stability and error analysis, with numerical experiments verifying the predicted convergence behavior.
2 Compact WSGD Operators for the Riemann-Liouville Fractional Derivatives
This section constructs third-order compact WSGD approximations for Riemann-Liouville fractional derivatives and establishes matrix properties used in stability analysis. The construction recovers familiar compact finite differences at integer derivative orders.
- Construction: The compact WSGD operators are derived from WSGD operators and Taylor expansions of shifted Grünwald finite difference formulas.
- Construction: The Riemann-Liouville derivatives are defined as left and right fractional derivatives on a bounded interval, with boundary conditions incorporated into their approximations.
- Construction: Third-order compact WSGD approximations are constructed for the fractional operators under the stated smoothness assumptions.
- Construction: For bounded nonperiodic problems, shifts must keep required nodes inside the interval; the choices (p, q) = (1, 0) and (1, −1) are used for compact approximations.
- Properties: At α = 2, the operators recover centered and compact second-derivative differences, while α = 1 recovers centered and compact first-derivative schemes.
- Stability foundations: The matrix analysis establishes positive definiteness for compact matrices and negative-real-part eigenvalues for the WSGD discretization matrices in the specified α ranges.
- Stability foundations: These matrix properties, together with Lyapunov stability criteria, provide the basis for analyzing the compact difference schemes.
3 One Dimensional Space Fractional Diffusion Equation
The one-dimensional scheme discretizes a two-sided space fractional diffusion equation using compact WSGD operators and Crank–Nicolson time stepping. The analysis establishes unconditional stability under specified parameter ranges and a third-order spatial, second-order temporal error estimate.
- Problem formulation: The problem uses left and right Riemann–Liouville fractional derivatives of order 1 < α ≤ 2 on a bounded interval with prescribed initial and boundary data.
- Compact difference scheme: The discretization applies compact fractional operators on a uniform spatial-time grid and uses Crank–Nicolson for time discretization.The resulting scheme is also written in matrix form for implementation and analysis.
- Stability analysis: The scheme is unconditionally stable for (p, q) = (1, 0) when 1 < α ≤ 2, and for (p, q) = (1, −1) over the stated restricted range of α.The proof uses positive definiteness of Cα, negative definiteness of Dα + Dα^T, and a spectral-radius bound.
- Convergence analysis: The error estimate is O(τ^2 + h^3), establishing second-order accuracy in time and third-order accuracy in space for the fractional scheme.The estimate follows from the discrete energy argument and a discrete Gronwall lemma.
- Classical limiting cases: When α = 1 or 2 under (p, q) = (1, 0), and α = 2 under (p, q) = (1, −1), the compact schemes recover classical finite-difference cases with O(τ^2 + h^4) error.
4 Two Dimensional Space Fractional Diffusion Equation
The paper discretizes a two-dimensional two-sided space fractional diffusion equation using compact finite difference and splitting schemes. Stability is established under specified shifted-Grünwald parameter cases, while the schemes achieve third-order spatial convergence and second-order temporal accuracy.
- Model and discretization: The two-dimensional model uses Riemann–Liouville fractional derivatives of orders 1 < α, β ≤ 2 with prescribed initial and boundary conditions.The solution is assumed unique and sufficiently smooth.
- Model and discretization: The domain is discretized on a uniform rectangular grid with spatial stepsizes h_x and h_y and time stepsize τ.Grid points are defined by x_i = a + ih_x, y_j = c + jh_y, and t_n = nτ.
- Compact schemes: The Crank–Nicolson time discretization and compact spatial operators produce a compact finite difference scheme with local truncation error O(τ^2 + h^4).The construction uses finite difference operators and Taylor expansions before factorization.
- Compact schemes: Several splitting alternatives are derived, including compact Peaceman–Rachford, Douglas, D’yakonov, and locally one-dimensional schemes.Intermediate variables are introduced for the alternating-direction constructions.
- Stability: For (p, q) = (1, 0), the difference schemes are unconditionally stable for 1 < α, β ≤ 2; for (p, q) = (1, −1), stability holds under an additional lower bound on α and β.The stability proof uses matrix definiteness, Kronecker-product properties, eigenvalue bounds, and spectral-radius arguments.
- Convergence: The error analysis establishes discrete L2-norm convergence for the schemes, with a bound of order O(τ^2 + h^3) under the stated parameter conditions.The result compares the exact solution with the numerical solution and assumes a positive constant in the error estimate.
5 Numerical Experiments
Numerical experiments test the compact schemes for one- and two-dimensional fractional diffusion problems using maximum and L2 errors. The results examine stability thresholds, convergence behavior across fractional orders, and splitting schemes with time extrapolation.
- Experimental setup: The numerical section evaluates the effectiveness and convergence orders of the proposed schemes in one- and two-dimensional cases.The experiments use maximum and L2 error measures.
- Two-dimensional experiments: One time extrapolation raises the temporal accuracy to third order while substantially reducing computational time in the reported computations.The extrapolation combines solutions computed with time stepsizes τ and τ/2.
- One-dimensional experiments: For the one-dimensional problem, the (p, q) = (1, −1) scheme is neither stable nor convergent when α < 1 + √6/2 ≈ 1.59.This threshold agrees with the theoretical stability result.
- One-dimensional experiments: Figure 1 reports maximum- and L2-error convergence rates at t = 1 with N = 128, showing rates that fall from 4 to 3 near α = 1 and increase from 1.9 toward 2 as α increases.The comparison is made for different fractional orders α.
- Two-dimensional experiments: The two-dimensional experiments report maximum and L2 errors and convergence rates for compact difference splitting schemes at t = 1 with τ = h.The tested schemes include compact LOD, Peaceman–Richardson, Douglas, and D’yakonov schemes.
6 Conclusion
The paper extends second-order WSGD operators to third-order compact WSGD operators and uses them to construct schemes for one- and two-dimensional space fractional diffusion equations, with stability and convergence analysis.
- The CWSGD operators extend WSGD operators from second-order to third-order accuracy for fractional-derivative approximation.
- Compact finite difference schemes based on CWSGD operators are established for one- and two-dimensional space fractional diffusion equations.
- Theoretical stability and convergence analyses are presented for the resulting compact difference schemes.
- Numerical results illustrate the effectiveness of the compact difference approximation and confirm the schemes’ convergence orders.