As atmospheric models are pushed towards non-hydrostatic resolutions, there is a growing need for new numerical discretizations that are accurate, robust and effective at these scales. In this paper we describe a new arbitrary-order staggered nodal finite-element method (SNFEM) vertical discretization motivated by flux reconstruction methods. The SNFEM formulation generalizes traditional second...

Two new formulations of a symmetric WENO method for the direct numerical simulation of compressible turbulence are presented. The schemes are designed to maximize order of accuracy and bandwidth, while minimizing dissipation. The formulations and the corresponding coefficients are introduced. Numerical solutions to canonical flow problems are used to determine the dissipation and bandwidth prop...

Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and second order hyperbolic, partial differential equations are considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundary without reducing th...

We present a numerical method for computing the signed distance to a piecewise-smooth surface defined as the zero set of a function. It is based on a marching method by Kim [Kim01] and a hybrid discretization of firstand second-order discretizations of the signed distance function equation. If the solution is smooth at a point and at all of the points in the domain of dependence of that point, ...

In this paper we combine nonstandard finite-difference (NSFD) schemes and Richardson’s extrapolation method to obtain numerical solutions of two biological systems. The first biological system deals with the dynamics of phytoplankton–nutrient interaction under nutrient recycling and the second one deals with the modeling of whooping cough in the human population. Since both models requires posi...

Construction of signed distance to a given interface is a topic of special interest to level set methods. There are currently, however, few algorithms that can efficiently produce highly accurate solutions. We introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations. We provide operati...

We present a direct, adaptive solver for the Poisson equation which can achieve any prescribed order of accuracy. It is based on a domain decomposition approach using local spectral approximation, as well as potential theory and the fast multipole method. In two space dimensions, the algorithm requires O(NK) work where N is the number of discretization points and K is the desired order of accur...

We describe a 2-D finite difference algorithm for inverting the Poisson equation on an irregularly shaped domain, with mixed boundary conditions, with the domain embedded in a rectangular Cartesian grid. We give both linear and quadratic boundary treatments and derive 1D error expressions for both cases. The linear approach uses a 5-point formulation and is first-order accurate while the quadra...

In this work, a new formulation for central schemes based on staggered grids is proposed. It is based on a novel approach, in which first a time discretization is carried out, followed by the space discretization. The schemes obtained in this fashion have a simpler structure than previous central schemes. For high order schemes, this simplification results in higher computational efficiency. In...