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WSLD operators: A class of fourth order difference approximations for space Riemann-Liouville derivative
Minghua Chen, Weihua Deng
TL;DR
Fractional PDEs are difficult to solve analytically because fractional operators are nonlocal, while higher-order discretizations can improve accuracy at nearly the same computational cost. This paper develops Lubich-based high-order schemes for space fractional derivatives and verifies fourth-order spatial convergence in one- and two-dimensional variable-coefficient problems.
Problem
Nonlocal fractional operators make analytical solutions of fractional PDEs challenging or sometimes impossible, motivating efficient numerical schemes with improved accuracy.
Method
The paper derives high-order space-fractional discretization schemes from Lubich operators and examines their full discretization for one- and two-dimensional variable-coefficient problems.
Results
The numerical results confirm global truncation errors O(τ^2 + h^4) in one dimension and O(τ^2 + (∆x)^4 + (∆y)^4) in two dimensions.
Takeaways & Limitations
The derived schemes provide high-order discretizations for space fractional derivatives, with fourth-order spatial convergence verified in both one- and two-dimensional tests.
Takeaways & Limitations
The analyzed diffusion problems assume derivative orders 1 < α,β < 2 and nonnegative variable coefficients; the two-dimensional analysis further assumes coefficient dependencies specified by d±(x,y) and e±(x,y).
Abstract
from arXiv · showhide
Because of the nonlocal properties of fractional operators, higher order schemes play more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first-order schemes but greatly improve the accuracy. Nowadays, there are already two types of second order discretization schemes for space fractional derivatives: the first type is given and discussed in [Sousa & Li, arXiv:1109.2345; Chen & Deng, arXiv:1304.3788; Chen et al., Appl. Numer. Math., 70, 22-41]; and the second type is a class of schemes presented in [Tian et al., arXiv:1201.5949]. The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference (WSLD) operators, for space fractional derivatives. Then we use the derived schemes to solve the space fractional diffusion equation with variable coefficients in one-dimensional and two-dimensional cases. And the unconditional stability and the convergence with the global truncation error $\mathcal{O}(τ^2+h^4)$ are theoretically proved and numerically verified.
1. Introduction
Fractional PDEs motivate efficient high-order discretization because analytical solutions are difficult or less useful, while existing space-fractional schemes leave room for fourth-order approaches. The paper derives WSLD operators and applies them to variable-coefficient fractional diffusion problems.
- Nonlocal fractional operators make analytical solutions difficult or sometimes impossible, motivating numerical methods for fractional PDEs.
- High-order finite-difference schemes can retain nearly the same computational cost as first-order schemes while improving accuracy.The relevant matrix-vector multiplication cost is O(NlogN) for the Toeplitz-like systems.
- Two types of second-order discretization schemes for space fractional derivatives had already been developed before this work.One combines centered second-derivative differences with piecewise linear fractional-integral approximation; another is associated with Tian et al.
- The paper derives fourth-order weighted and shifted Lubich difference operators for space fractional derivatives.The operators are based on Lubich’s fractional linear multistep framework and its generating functions.
- Direct use of second-order Lubich formulas can be unstable for time-dependent problems with derivative order α in (1,2), whereas the proposed weighted and shifted formulas target fourth-order accuracy.
2. Derivation of a class of fourth order discretizations for space fractional operators
The paper derives second-, third-, and fourth-order approximation operators for left and right Riemann–Liouville fractional derivatives, including bounded-domain matrix forms. It then analyzes parameter choices and establishes negative-real-part eigenvalue properties for schemes used in space fractional problems.
- Approximation operators: The derived operators approximate left and right Riemann–Liouville fractional derivatives with second-, third-, and fourth-order truncation errors.The results are stated under regularity assumptions involving fractional derivatives and Fourier transforms in L1(R).
- Bounded-domain implementation: The schemes extend to bounded domains by zero-extending the functions outside the computational interval and using uniform spatial mesh points.The resulting operators can be rewritten in matrix form using the interior grid values.
- Stability analysis: Direct use of the second-order Lubich formula for α in (1,2) is unstable for time-dependent problems, motivating shifted or weighted constructions.The instability is linked to the spectral behavior of the associated matrix.
- Stability analysis: Parameter choices are analyzed through Toeplitz-like matrix generating functions and eigenvalue properties.The analysis uses positive-definiteness results for symmetric or Hermitian parts of matrices.
- Stability analysis: For selected parameter ranges, the matrices associated with the derived operators are negative definite or have eigenvalues with negative real parts.The paper identifies this spectral property as the reason the corresponding schemes work for space fractional derivatives.
- Approximation operators: Fourth-order left and right approximation operators are used to discretize the corresponding Riemann–Liouville derivatives.The operators are incorporated into the space-fractional diffusion discretization.
4. Convergence and Stability Analysis
The analysis establishes unconditional stability and fourth-order spatial, second-order temporal convergence for the one- and two-dimensional schemes under the stated coefficient relations and fractional-order ranges.
- Convergence: The schemes are fourth-order convergent in space and second-order convergent in time.
- 2D operator properties: ℜ(λ(Ax)) < 0 and ℜ(λ(Ay)) < 0 for the two-dimensional operators under D− = καD+ and E− = κβE+.
- 1D stability: The one-dimensional scheme is unconditionally stable for α ∈ (1,2) when D− = καD+.
- 1D stability: The one-dimensional stability proof uses a spectral radius below one for (I − A)−1(I + A).
- 2D stability: The two-dimensional scheme is unconditionally stable for 1 < α,β < 2 under D− = καD+ and E− = κβE+.
5. Numerical results
Numerical experiments evaluate maximum errors and convergence for one- and two-dimensional variable-coefficient fractional diffusion equations, confirming the theoretically predicted rates.
- Experimental setup: The experiments use the l∞ norm to measure numerical errors.
- One-dimensional results: The one-dimensional test uses variable coefficients d+(x) = x^α and d−(x) = 2x^α with a specified exact solution and zero boundary conditions.
- One-dimensional results: At t = 1 with τ = h^2, the one-dimensional results confirm global truncation error O(τ^2 + h^4).
- Two-dimensional results: The two-dimensional test uses variable coefficients in both spatial directions and zero boundary conditions.
- Two-dimensional results: The two-dimensional scheme confirms global truncation error O(τ^2 + (∆x)^4 + (∆y)^4).
6. Conclusions
The paper develops a Lubich-based approach for constructing high-order space-fractional derivative discretizations, including effective second-order difference operators.
- Lubich’s operators provide a new basis for deriving high-order discretization schemes for space fractional derivatives.