We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each ...

We present an explicit sixth order compact finite difference scheme for fast high accuracy numerical solutions of the two dimensional convection diffusion equation with variable coefficients. The sixth order scheme is based on the well-known fourth order compact scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multisca...

Fourth-order combined compact finite difference scheme is given for solving the time fractional convection–diffusion–reaction equation with variable coefficients. We introduce the flux as a new variable and transform the original equation into a system of two equations. Compact difference is used as a high-order approximation for spatial derivatives of integer order in the coupled partial diffe...

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing SummationBy-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the...

Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study, we use a simple boundary value problem for a one-dimensional parabolic equation to discuss the selection of an approximation with respect to time. We conside...

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations: 1 2 σ2(x)u′′(x) + b(x)u′(x)− u(x) = −f(x), x ∈ R, where σ is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain c...

The multiscale multigrid method is presented in this article to solve the linear systems arising from a fourth order discretisation. We used a symbolic algebra packageMathematica to derive a family of finite difference approximations on a 27 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at selected grid points. A set of test problems are presente...

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick’s law. In this paper we combine the forward time centered space (FTCS) method, well k...

An implicit, time-dependent variable grid finite difference method based on the addition of an artificial diffusivity is introduced and analyzed for approximating the solution of a scalar conservation law in a single space variable. No relation between the grids at successive time steps is required for convergence. Two adaptive grid selection procedures are shown to be covered by the analysis. ...

This paper is concerned with the construction and developing of several nonstandard finite difference (NSFD) schemes in matrix form in order to obtain numerical solutions of epidemic models. In particular, we deal with a classical SIR epidemic model and a seasonal model associated with the evolution of the transmission of respiratory syncytial virus RSV in the human population. The first model ...