Finite difference approximations for a fractional advection diffusion problem

TLDR: This work derives explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation and presents the order of accuracy of the schemes, and proves they are stable under certain conditions.

About: This article is published in Journal of Computational Physics.The article was published on 2009-06-01 and is currently open access. It has received 147 citations till now. The article focuses on the topics: Anomalous diffusion & Fractional calculus.

Figures

Figure 8. Solution of (1) for V = 0.5, D = 0.2 at t = 1 and for β = −1 (− · −), β = 1 (−): (a) α = 1.8; (b) α = 1.6.

Figure 10. Solution of (1) for V = 0.5, D = 0.2 at t = 1 and for α = 1.8 (- −); α = 1.6 (− · −); α = 1.4 (· · ·); α = 1.2 (−−): (a) β = −1; (b) β = 1.

Figure 9. Solution of (1) for V = 0.5, D = 0.2 at t = 1 and for β = −1 (− · −), β = 1 (−): (a) α = 1.4; (b) α = 1.2.

Figure 3. Stability regions obtained with condition (23) for the LaxWendroff scheme (12): (a) α = 1.1 – The curves are plotted for β = −1,−0.6 (left to right). For −0.6 ≤ β ≤ 1 the stability regions are the same as for β = −0.6; (b) α = 1.2 – The curves are plotted for β = −1,−0.7 (left to right). For −0.7 ≤ β ≤ 1 the stability regions are the same as for β = −0.7.

Figure 1. Stability regions obtained with condition (23) for β = 0 and α = 2, 1.8, 1.6, 1.4, 1.2 (left to right). (a) Upwind scheme (11): these regions are defined by ν + 2α−1µ α ≤ 1 also given in Theorem 3; For different values of β the stability regions are the same. (b) Lax-Wendroff scheme (17): these regions are defined by ν2 +2α−1µ α ≤ 1 also given in Theorem 4. For values of β such as −0.5 ≤ β ≤ 1 the stability regions are the same.

Figure 5. Stability regions obtained with condition (23) for the central scheme (17), with β = −1,−0.5, 0, 0.5, 1 (bottom to top): (a) α = 1.2; (b) α = 1.4.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Finite difference approximations for a fractional advection diffusion problem" ?

The authors consider a one dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. The authors present the order of accuracy of the schemes and in order to show its convergence they prove they are stable under certain conditions. In the end the authors present a test problem. The authors derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. 

Q2. What is the stability region for the upwind and Lax-Wendroff schemes?

for β = −1 the stability regions are smaller than for the rest of the β values, that is, as β increases the regions become bigger. 

Q3. What is the stability region of the Lax-Wendroff scheme?

Additionally the stability region of the Lax-Wendroff scheme is more adequate than the stability region of the central scheme since for small diffusive parameters µα and larger Courant numbers ν, the latter can be unstable. 

Q4. What is the form of the matrix iteration M?

The matrix iteration M has the formM = A+ 12 µαB, (19)where A and B are matrices of dimension (2N−1)×(2N−1) and A is related with the advection discretisations and B with the diffusion discretisations. 

Q5. What are some numerical schemes for fractional differential equations?

Some numerical schemes have been developed for diffusion problems, [7], [8], [9], [10], [11], [12] and for advection diffusion problems [13], [14]. 

Q6. What is the amplitude of the p-th harmonic?

The von Neumann analysis assumes that any finite mesh function, such as, the numerical solution Unj will be decomposed into a Fourier series asUnj = N ∑p=−N κnpe iξp(j∆x), j = −N, . . . , N,where κnp is the amplitude of the p-th harmonic and ξp = pπ/N∆x. 

Q7. What is the simplest way to explain the finite difference schemes?

To derive a finite difference scheme the authors suppose there are approximations Un := {Unj } to the values U(xj, tn) at the mesh pointsxj = j∆x, j = −N, . . . ,−2,−1, 0, 1, 2, . . . , N and tn = n∆t, n ≥ 0,where ∆x denotes the uniform space step and ∆t the uniform time step. 

Q8. What is the Riemann-Letnikov formula for fractional derivatives?

If the authors define a fractional operator, ∇αβ , such as,2∇αβu = (1 + β) ∂αu ∂xα + (1 − β) ∂ αu ∂(−x)α , (2)equation (1) can be written in a simple form∂u ∂t + V ∂u ∂x = D∇αβu. 

Q9. What are the conditions derived from the von Neumann analysis?

The conditions derived here, using the property (23), allow to conclude that some of the analytical necessary conditions obtained with the von Neumann analysis are necessary and sufficient conditions for stability. 

Q10. What is the order of accuracy of the Lax-Wendroff scheme?

For small D, from (15), it followsT nj =(∂u∂t)nj+O(∆t2)+V ( ∂u∂x)nj+O(∆x2)+O(∆x2)−D(∇αβu)nj +O(∆x),and the Lax-Wendroff scheme has an order of accuracy close to O(∆t2) + O(∆x2) + O(∆x).4.2.