Browsing Faculty of Science and Engineering by Publisher "De Gruyter"
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An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth dataIn this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Error estimates of highorder numerical methods for solving time fractional partial differential equationsError estimates of some highorder numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a highorder numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3 \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one and twodimensional cases are given to show that the numerical results are consistent with the theoretical results.

Some time stepping methods for fractional diffusion problems with nonsmooth dataWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Timestepping error bounds for fractional diffusion problems with nonsmooth initial data, Journal of Computational Physics, 293(2015), 201217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be selfadjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the RiemannLiouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Stability of a numerical method for a fractional telegraph equationIn this paper, we introduce a numerical method for solving the timespace fractional telegraph equations. The numerical method is based on a quadrature formula approach and a stability condition for the numerical method is obtained. Two numerical examples are given and the stability regions are plotted.

Two highorder time discretization schemes for subdiffusion problems with nonsmooth dataTwo new highorder time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders $O(k^{3 \alpha})$ and $O(k^{4 \alpha})$ with $0< \alpha <1$ can be restored for any fixed time $t$ for both smooth and nonsmooth data, respectively. The error estimates for these two new highorder schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.