A finite difference method for an anomalous sub-diffusion equation, theory and applications

Mustapha, Kassem; AlMutawa, Jaafar

doi:10.1007/s11075-012-9547-0

Cited by 41 publications

(18 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(For other numerical methods of (3), see [2,8,9,20,23,24,27,28,35] and related references therein.) When f ≡ 0 (that is, homogeneous case), the two representations (1a) and (3) are different ways of writing the same equation, as they are equivalent under reasonable assumptions on the initial data.…”

Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

2015

Self Cite

View full text Add to dashboard Cite

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T ], the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L ∞ 0, T ; L 2 (Ω)norm and to ∇u in the L ∞ 0, T ; L 2 (Ω) -norm are proven to converge with the rate h k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

show abstract

Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

2015

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finite difference methods that can work with variable timesteps are scarce. Some examples are the matrix approach on non-equidistant grids by Podlubny et al [9], a generalized Crank-Nicolson method by Mustapha et al [10,11], and a non-uniform L1 time discretization [12,13,14]. The finite difference method we employ in this paper is an unconditionally stable implicit method discussed in Ref.…”

Section: Introductionmentioning

confidence: 99%

Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

Yuste

Quintana-Murillo

2015

Numer Algor

View full text Add to dashboard Cite

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

show abstract

“…Several authors have proposed a variety of numerical methods for this problem. For finite difference (FD) methods with convergence rates of order O(h 2 ) in space, where h is the maximum meshsize, see, for example, [4,5,19,20,31,46,47,50,51]. In [11], FD schemes were considered which are first-order accurate in time but O(h 4 )-accurate in space provided u is sufficiently smooth including at t = 0.…”

Section: Introductionmentioning

confidence: 99%

A hybridizable discontinuous Galerkin method for fractional diffusion problems

Cockburn

Mustapha

2014

Numer. Math.

Self Cite

View full text Add to dashboard Cite

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order −α with −1 < α < 0. For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time t ∈ [0, T ] the approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω , the approximations to u in the L ∞ 0, T ; L 2 (Ω ) -norm and to ∇u in the L ∞ 0, T ; L 2 (Ω ) -norm are proven to converge with the rate h k+1 , where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate of log(T h −2/(α+1) ) h k+2 .

show abstract

A finite difference method for an anomalous sub-diffusion equation, theory and applications

Cited by 41 publications

References 28 publications

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations

A hybridizable discontinuous Galerkin method for fractional diffusion problems

Contact Info

Product

Resources

About