Related papers: High Order Finite Difference Methods on Non-unifor…
Finite difference approximation, in addition to Taylor truncation errors, introduces numerical dispersion-and-dissipation errors into numerical solutions of partial differential equations. We analyze a class of finite difference schemes…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
The second order weighted and shifted Gr\"{u}nwald difference (WSGD) operators are developed in [Tian et al., arXiv:1201.5949] to solve space fractional partial differential equations. Along this direction, we further design a new family of…
A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each…
Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order…
The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are considered and the numerical method with its error analysis is presented in this paper. First, a $L2$ approximation is proved with first order accuracy to the…
We propose a new first-order-system least squares (FOSLS) finite-element discretization for singularly perturbed reaction-diffusion equations. Solutions to such problems feature layer phenomena, and are ubiquitous in many areas of applied…
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the…
For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The…
We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected…
High-order difference operators with the summation-by-parts (SBP) property can be used to build stable discretizations of hyperbolic conservation laws; however, most high-order SBP operators require a conforming, high-order mesh for the…
Discrete Fracture Network models are largely used for very large scale geological flow simulations. For this reason numerical methods require an investigation of tools for efficient parallel solutions on High Performance Computing systems.…
A method for performing high order mesh refinement multigrid computations is presented. The Full Approximation Scheme (FAS) multigrid technique is utilized for a sequence of nested patches of increasing resolution. Conservation forms are…
High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and…
When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical…
The parameter-uniform convergence of a fitted operator method for a singularly perturbed differential equation is normally available only for uniform meshes. Here we establish the parameter-uniform convergence of a fitted operator method on…