This work overcomes the difficulty of the previous matched interface and boundary (MIB) method in dealing with interfaces with non-constant curvatures for optical waveguide analysis. This difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved MIB method. Instead, novel jump conditions are derived along global Cartesian directions for the transver...

We present a new method, based on averaging, to simulate certain systems with multiple time scales efficiently and demonstrate its utility in the context of the shallow-water equations. We first develop the method in a simple linear setting and analytically prove its stability. This is followed by an extension to the full equations and a presentation of a computational model for it. In this pre...

Conventional nite diierence eikonal solvers produce only the rst arrival time. However suitable solvers (of suuciently high order of accuracy) may be extended via Fermat's principle to yield a simple algorithm which computes all traveltimes to each subsurface point, with cost on the same order as that of a rst arrival solver.

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. It combines the best features of structured and unstructured grid methods to attain computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finit...

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...