High order schemes for the tempered fractional diffusion equations

1 Introduction

• Exponential tempering makes large Lévy-flight jumps less likely and gives finite moments, addressing models of bounded-domain processes with finite-moment observables.The untempered Lévy-flight model has a divergent second moment.Source: §1 · p. 1
• Tempered fractional calculus is treated as a space operator, unlike fractional substantial calculus, which is time-space coupled and discretized in time.The paper describes the two calculi as arising from different physical backgrounds.Source: §1 · p. 2
• Earlier work developed WSGD approximations for Riemann-Liouville derivatives, while research on tempered fractional diffusion schemes remained limited.Some existing tempered-equation schemes had second-order accuracy; stability and convergence were not given for one compared approach.Source: §1 · p. 2
• The paper fills the high-order scheme gap by weighting and shifting first-order Grunwald approximations into tempered-WSGD operators, with Crank-Nicolson time discretization.It discusses numerical stability and convergence and reports numerical experiments.Source: §1 · p. 2
• The paper introduces the calculus and discretizations, analyzes the diffusion schemes, verifies their effectiveness and convergence orders numerically, and concludes with remarks.Source: §1 · p. 2

2.1 Definitions and Fourier transforms of the tempered fractional calculus

• The section defines left and right Riemann-Liouville tempered fractional integrals and derivatives, and states their Fourier transforms for u and its n-times derivative in Lq(R), q ≥ 1.Source: §2.1 · pp. 3–4

2.2 Discretizations of the tempered fractional calculus

• Because standard Grunwald discretization can yield unstable time-dependent fractional PDE schemes, shifting is necessary for tempered derivatives as well.The shifted approximation has a superconvergent point and O(h^2) accuracy.Source: §2.2 · p. 5
• Tempered-WSGD operators improve approximation accuracy by modifying Grunwald weights, without adding computational cost.The operators combine weighting and shifting of the Grunwald approximations.Source: §2.2 · p. 6
• The left and right tempered-WSGD operators provide O(h^ell) approximations uniformly for x in R.The stated results apply for any integer m ≥ ell.Source: §2.2 · pp. 6–7
• For bounded non-periodic problems, choose |p_j| ≤ 1 to keep approximation nodes within the interval; the paper focuses on second-order schemes for computational purposes.The passage notes that arbitrarily high-order approximations are possible, but focuses on second order with three weights.Source: §2.2 · p. 10
• For Riemann–Liouville calculus (λ=0), Lemma 5’s restrictions on γ1, γ2, and γ3 can be relaxed when using the generating-function method [34] to prove stability.Figure 2.1 illustrates the bounds on γ1, γ2, and γ3.Source: §2.2 · p. 12
• The p1=1, p2=0, p3=−1 choice yields third-order tempered-WSGD operators, but falls outside Lemma 5’s parameter domains and seems unstable for tempered fractional diffusion.The next section selects parameter choices intended to be stable for time-dependent tempered fractional PDEs.Source: §2.2 · pp. 12–13

3 Numerical schemes for the tempered fractional diffusion equation

• The model solves for solute concentration on a bounded spatial interval, with a source term and endpoint values specified by boundary conditions.The nonnegative weighting factors l and r sum to one and usually control dispersion bias.Source: §3 · p. 13

3.1 Crank-Nicolson-tempered-WSGD schemes

• Second-order tempered-WSGD approximations discretize the spatial derivatives, while Crank–Nicolson discretizes time.The scheme is formulated on a uniform spatial and temporal grid.Source: §3.1 · pp. 13–14

3.2 Stability and convergence

• The Crank–Nicolson-tempered-WSGD scheme is unconditionally stable, with the analysis showing its iteration matrix has spectral radius below one.The stated stability result applies for λ ≥ 0 and 1 < α < 2.Source: §3.2 · p. 15
• The local truncation error is O(τ^3 + τh^2), and the stated error bound is proportional to τ^2 + h^2.The analysis assumes a unique, sufficiently smooth solution.Source: §3 · p. 13, §3.2 · p. 16

4 Numerical results

• For computations on the finite interval [0, 1], the numerical errors and orders of accuracy reported in Tables 1–6 confirm the desired second-order accuracy.Source: §4 · p. 17
• Eq. (4.1) is solved by the Crank-Nicolson-tempered-WSGD scheme (3.5), and numerical results with second-order accuracy are obtained, with results collected in Tables 7–9.Source: §4 · p. 19
• Tables 10–12 report numerical errors and convergence behaviors for the Crank-Nicolson-tempered-WSGD schemes (3.5), confirming Theorem 10; they evaluate Example 3 at t = 1.The tables use fixed stepsizes τ = h, λ = 1.0, α = 1.2, different weights, and parameters (γ1, γ2, γ3) selected in set Sα.Source: §4 · p. 20, Table 10 · p. 20, Table 11 · p. 20

• Exponential tempering makes very large Lévy-flight jumps unlikely, motivating tempered fractional diffusion models for processes in bounded physical domains.Source: §5 · p. 21
• Unlike fractional substantial calculus, the tempered fractional derivative used here is a space operator without time coupling.Source: §5 · p. 21
• The paper derives high-order discretizations for left and right Riemann–Liouville tempered fractional derivatives and integrals, with a superconvergent point for first-order discretizations.Source: §5 · p. 21
• The stability domains are derived analytically and illustrated, while second-order schemes for tempered fractional diffusion have stability and convergence proved and numerically verified.Source: §5 · p. 21