Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation

A reaction-diffusion problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time $t=0$, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

The fractional derivative D α t denotes the Caputo fractional derivative defined by The derivative definition is not local (unlike classical derivatives).

Example (part 1)
Example.Consider the fractional heat equation where the Mittag-Leffler function .
M-L function is fractional analogue of the exponential function:

Graph of solution to Example
Plot of surface v (x, t) and its cross-section at x = π/2 when α = 0.3.An initial layer in v at t = 0 is evident.
-uses separation of variables to prove existence and uniqueness of a classical solution to this problem -i.e., a function u whose derivatives exist and satisfy the PDE and the initial-boundary conditions pointwise -under some extra hypotheses on the data Here and subsequently, C denotes a generic constant that depends only on the data α, p, r , f , φ, l, T .
These bounds are sharp: they agree with the behaviour of our earlier example You can't assume too much regularity!
Consider the time-fractional heat equation You can't assume too much regularity!
Consider the time-fractional heat equation You can't assume too much regularity!
Consider the time-fractional heat equation

Outline
The PDE and the behaviour of its solution Finite difference method on a uniform mesh Finite difference method on a graded mesh
Computed approximation to the solution at each mesh point (x n , t m ) is denoted by u m n .
u xx is discretised using a standard approximation: The Caputo fractional derivative ∂t ds is approximated by the so-called L1 approximation Here The scheme Thus we approximate the IBVP by the discrete problem This discretisation is standard; it is considered for example in F.Liu, P.Zhuang & K.Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput.Math.Appl., 64 (2012), 2990-3007.

Properties of discrete system
At each time level, Must solve a tridiagonal linear system; matrix is an M-matrix so scheme satisfies a discrete maximum principle.
Have to use computed solutions at all previous time levels Previous numerical analysis: a criticism -In our discussion of convergence, we consider only the discrete L ∞ norm-There exist papers (e.g., Liu, Zhang & Burrage 2012) that consider problems and discretisations like ours, and prove O(h 2 + τ 2−α ) convergence of the numerical method, under the hypothesis that the solution u of the original problem is in C 4,2 ( Q) -which is satisfied only for very special data!
We are interested in proving a convergence result under the realistic hypothesis that u ∈ C 4,0 ( Q) with Previous numerical analysis: a criticism -In our discussion of convergence, we consider only the discrete L ∞ norm-There exist papers (e.g., Liu, Zhang & Burrage 2012) that consider problems and discretisations like ours, and prove O(h 2 + τ 2−α ) convergence of the numerical method, under the hypothesis that the solution u of the original problem is in C 4,2 ( Q) -which is satisfied only for very special data!
We are interested in proving a convergence result under the realistic hypothesis that u ∈ C 4,0 ( Q) with

Numerical evidence
Numerical experiments with our simple but typical first Example show that for our numerical method one obtains O(h 2 + τ α ) convergence, not the O(h 2 + τ 2−α ) that occurs only for unrealistically smooth solutions.
Temporal truncation error: one can show (a bit long and messy) that Also need to sharpen stability estimate of Liu, Zhang & Burrage 2012.

Theorem
For m = 1, 2, . . ., M the solution u m n of the scheme satisfies max Numerical experiments show that this bound is sharp.

Outline
The PDE and the behaviour of its solution Finite difference method on a uniform mesh Finite difference method on a graded mesh

Mesh graded in time
Let M and N be positive integers.Set x n := nh for n = 0, 1, . . ., N with h := l/N, t m := T (m/M) r for m = 0, 1, . . ., M with mesh grading r ≥ 1 chosen by the user.
Computed approximation to the solution at each mesh point (x n , t m ) is denoted by u m n .