The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundar...

The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a fu...

A combined compact difference scheme is proposed for linear second-order partial differential equations with mixed derivative. The scheme is based on a nine-point stencil at interior with sixth-order accurate local truncation error. Fourier analysis is used to analyze the spectral resolution of the proposed scheme. Numerical tests demonstrate at least sixth-order convergence rate with Dirichlet...

The parametric cubic curve is a well-developed subject in the field of computer-aided design. Yet there are still some aspects of its nature that need to be more fully investigated. In this paper, formulations of composite PC curves are presented in an order of increasing rigorousness and order of accuracy. A unique PC curve formulation is developed for the practical use in circle approximation...

In standard equispaced finite difference (FD) formulas, symmetries can make the order of accuracy relatively high compared to the number of nodes in the FD stencil. With scattered nodes, such symmetries are no longer available. The generalization of compact FD formulas that we propose for scattered nodes and radial basis functions (RBFs) achieves the goal of still keeping the number of stencil ...

This paper presents a conservative front-tracking method for shocks and contact discontinuities that is second-order accurate. It is based on a volume-of-fluid method that treats the moving front with concepts similar to those of an embedded-boundary method. Special care is taken in the centering of the data to ensure the right order of accuracy at the front, and a redistribution step guarantee...

A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection–diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing de...

Difference schemes are treated as operator equations in real linear normed space [l, 23. Twoand three-layer schemes are discussed in real Hilbert space (more precisely, in unitary space, since no use is made of completeness). Stable schemes are classified. Schemes are written in a canonical form that enables stability operators, or what will be termed regularizers, to be introduced. Sufficient ...

A novel spectral-element approach for the eikonal equation is presented based on the work of Tucker et al. This approach is readily implemented in standard finiteelement flow solvers developed for the Navier-Stokes equations. Results are presented for distance function (level set) for three planar configurations: circular cylinder, NACA 0012 airfoil, and T106 airfoil. Results demonstrate that t...

The paper proposes a novel multi-area state estimator based on wide area measurements where only boundary buses are considered in the coordination level. The power injection measurements are not used during the coordination level reducing the number of states and therefore size of the problem. Instead, a set of pseudo-measurements are included whenever a power injection measurement is available...