This contribution discusses two attitudes to artificial dissipation reduction in numerical schemes for solving initial boundary value problems for the Allen-Cahn equation with anisotropy incorporated into the diffusion operator. In the first case, a weighted first order finite difference scheme is used for spatial discretization of the anisotropic texture diffusion problem in 2D, designed for v...

We show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite diff...

Numerical computations of frequency domain field problems or elliptical partial differential equations may be based on differential equations or integral equations. The new concept of field computation presented in this paper is based on the postulate of the existence of linear equations of the discretized nodal values of the fields, different from the conventional equations, but leading to the...

The fast adaptive composite grid (FAC) method is an iterative method for solving discrete boundary value problems on composite grids. McCormick introduced the method in [10J and considered the convergence behaviour for discrete problems resulting from finite volume element discretiza-tion on composite grids. In this paper we consider discrete problems resulting from finite difference discretiza...

We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart–Thomas mixed finite element met...

We provide error analyses for explicit, implicit, and semi-implicit monotone finitedifference schemes on uniform meshes with nonlocal numerical fluxes. We are motivated by finite-difference discretizations of certain long-wave (Sobolev) regularizations of the conservation laws that explicitly add a dispersive term as well as a nonlinear dissipative term. We also develop certain relationships be...

We derive a family of sixth-order compact finite-difference schemes for the three-dimensional Poisson’s equation. As opposed to other research regarding higher-order compact difference schemes, our approach includes consideration of the discretization of the source function on a compact finite-difference stencil. The schemes derived approximate the solution to Poisson’s equation on a compact st...

Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point...