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A new difference scheme for the time fractional diffusion equation
A. A. Alikhanov
TL;DR
Time fractional diffusion equations are costly to approximate because each time layer depends on all previous layers, while high-order time approximation is difficult for singular fractional derivatives. The paper constructs a higher-order Caputo difference analog and variable-coefficient schemes, proving stability and L2-norm convergence at the approximation-error rate. Numerical tests confirm the theoretical results.
Problem
Time fractional diffusion equations require all previous time layers for numerical approximation, while singular fractional derivatives make high-order time accuracy difficult.
Method
The paper constructs a Caputo difference analog of order O(τ 3−α) and develops second- and fourth-order spatial, second-order temporal schemes for variable-coefficient diffusion equations.
Results
The schemes are proved stable and convergent in the mesh L2-norm at the rate equal to the approximation-error order, with numerical tests confirming the theoretical results.
Takeaways & Limitations
The paper provides stability conditions and higher-order difference schemes for time fractional diffusion equations, with the method extendable to other time fractional problems and boundary conditions.
Takeaways & Limitations
The L1 method does not achieve second-order approximation at all points of a mesh, even when a locally refined non-uniform mesh improves its accuracy.
Abstract
from arXiv · showhide
In this paper we construct a new difference analog of the Caputo fractional derivative (called the $L2$-$1_σ$ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and forth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $L_2$ - norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.
1. Introduction
The paper addresses the computational difficulty of time fractional diffusion equations by developing a higher-order Caputo derivative approximation and associated stable schemes.
- Motivation: Time fractional diffusion equations require information from all previous time layers, making numerical algorithms time-consuming even in one dimension.The complexity increases substantially for two- and three-dimensional problems.
- Related methods: The L1 method has local truncation error O(τ 2−α) on a uniform mesh.Mesh refinement near the current time can improve approximation, but does not produce second-order accuracy at every mesh point.
- Related methods: High-order time approximation is difficult because fractional derivatives are singular.Existing compact and spectral approaches primarily improve spatial accuracy.
- Contribution: The paper constructs a Caputo derivative analog with approximation order O(τ 3−α) for α ∈(0, 1).This operator underlies schemes with second- and fourth-order spatial accuracy and second-order temporal accuracy for variable-coefficient diffusion equations.
- Contribution: Stability and convergence are proved in the mesh L2-norm using energy inequalities, and numerical tests support the results.The schemes target time fractional diffusion equations with variable coefficients.
2. Family of difference schemes. Stability and convergence
The paper studies general difference schemes on non-uniform time meshes, establishes stability criteria and a priori estimates, and proves L2-norm convergence at the approximation-error rate.
- Family of difference schemes: Families of difference schemes on non-uniform time meshes are analyzed through a mesh L2-norm stability criterion.The criterion is intended to simplify stability checks for a wide class of schemes.
- Family of difference schemes: The schemes use a discrete Caputo derivative and a spatial difference operator whose negative preserves positive definiteness.Mesh functions approximate the variable coefficients and source term.
- Stability: The schemes are proved unconditionally stable under the stated conditions and satisfy an a priori estimate.The proof uses inner-product and energy-inequality arguments.
- Convergence: A scheme with approximation order O(N^-r1 + M^-r2) converges in the mesh L2-norm at rate O(N^-r1 + M^-r2).The result follows by applying the stability estimate to the error problem.
- Convergence: The convergence proof introduces the error z = y−u and derives an estimate for the resulting error problem.This connects consistency, stability, and convergence for the difference scheme.
3. A new L2 −1σ fractional numerical differentiation formula
The paper constructs and analyzes the L2-1σ formula, a new Caputo fractional-derivative difference analog with approximation order O(τ 3−α), and demonstrates its numerical accuracy on a test problem.
- 3. A new L2 −1σ fractional numerical differentiation formula: The L2-1σ formula is introduced as a new difference analog of the Caputo fractional derivative with approximation order O(τ 3−α).It is defined for 0 < α < 1 and is named in formula (27).
- 3. A new L2 −1σ fractional numerical differentiation formula: The construction uses a uniform temporal mesh with σ = 1 −α/2 and evaluates the Caputo derivative at the shifted points t_j+σ.The derivation applies to u(t) ∈ C3[0, T] and 0 < α < 1.
- 3. A new L2 −1σ fractional numerical differentiation formula: Quadratic interpolation Π2,su(t) over neighboring mesh points provides the approximation used to derive the fractional difference operator.The interpolation uses (t_s−1, u(t_s−1)), (t_s, u(t_s)), and (t_s+1, u(t_s+1)).
- 3.1. Basic properties of the new: The derivation expresses the approximation error through remainder terms and establishes coefficient inequalities for the new operator.The coefficient analysis includes κ_s and cases covering interior and boundary indices.
- 3.1. Basic properties of the new: The basic-properties analysis proves the inequalities needed for the L2-1σ formula using auxiliary functions, integration, and the inequality (1 + t)^γ < 1 + γt.The stated inequality applies for t > 0 and 0 < γ < 1.
- 3.2. Test example: A test example computes the α-order Caputo derivative of f(t) = t^4+α at t = 1 using the new formula.The exact solution is used for comparison.
- 3.2. Test example: Computations with temporal stepsizes M = 10 through 5120 compare L2-1σ against the L1-2 formula for α = 0.9, 0.5, and 0.1.Table 1 reports computational errors and numerical convergence orders at t_M−1+σ = 1.
4. A second order difference scheme for the time fractional diffusion equation
The paper constructs a second-order scheme in both space and time for the time-fractional diffusion equation, proves unconditional stability and convergence, and verifies the predicted rates numerically.
- The constructed difference scheme has approximation order O(h2 + τ 2) for the time-fractional diffusion equation.
- Stability and convergence: The scheme is unconditionally stable and satisfies an a priori estimate.
- Stability and convergence: For sufficiently smooth solutions and input data, the numerical solution converges in the mesh L2-norm at rate O(h2 + τ 2).
- Numerical results: The test problem uses exact solution t3 + 3t2 + 1 with k(x,t) = 2 − sin(xt), q(x,t) = 1 − cos(xt), l = 1, and T = 1.
- Numerical results: When h = τ, Table 2 reports decreasing maximum error and convergence order O(h2) = O(τ 2) as the grid is refined.
- Numerical results: With h = 1/1000, Table 3 reports decreasing maximum error and temporal convergence order O(τ 2) as the number of time steps increases.
5. A higher order difference scheme for the time fractional diffusion equation
The section develops a higher-order scheme for variable-coefficient time-fractional diffusion equations, proves its stability and convergence, and tests its predicted accuracy numerically.
- Scheme construction: The constructed scheme has approximation order O(h4 + τ 2) for the case k = k(t) and q = q(t).The scheme uses the operator Hh and is analyzed in the mesh L2-norm.
- Stability and convergence: The scheme is unconditionally stable and satisfies an a priori estimate.The stability proof uses inner-product transformations and the equivalence of the norms ∥Hhy∥0 and ∥y∥0.
- Stability and convergence: Under sufficient smoothness, the numerical solution converges in the mesh L2-norm at the approximation-error rate.The convergence result assumes u(x,t) ∈ C6,3x,t and a constant independent of τ and h.
- Numerical results: The test calculations report second-order temporal convergence for α = 0.75, 0.85, and 0.95.Table 4 reports L2-norm, maximum-norm errors, and temporal convergence order.
- Numerical results: The numerical tests report fourth-order spatial accuracy and the expected rates O(h4) and O(τ 2) under the stated grid refinements.The spatial test fixes τ = 1/20000, while the combined refinement uses h2 = τ.
6. Conclusion
The conclusion establishes stable, convergent second- and fourth-order spatial schemes with second-order temporal accuracy for variable-coefficient time-fractional diffusion equations. Numerical tests confirm the theoretical results, and the method is stated to extend to other time-fractional equations and boundary conditions.
- Conclusion: The paper studies a family of difference schemes for general time-fractional diffusion equations and obtains sufficient conditions for unconditional stability.The stability criterion is formulated in the mesh L2-norm.
- Conclusion: The constructed schemes achieve second- and fourth-order approximation in space and second-order approximation in time for variable coefficients.Their stability and convergence are proved in the mesh L2-norm at the approximation-error rate.
- Conclusion: The method can be extended to other time-fractional partial differential equations with other boundary conditions.
- Conclusion: Numerical tests completely confirm the obtained theoretical results.The calculations are carried out in MATLAB.