Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã = Sã + g(t)w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W ₁–W ₂ and S = S ₁ + S ₂ and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã = W ^–1 Sã + g(t)W ^–1 w are somewhat faster but less convenient to implement since the matrix W ^–1 S is no longer sparse. The pseudospectral methods lead to the system = D + g(t)w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).     

Strong Stability Preserving Properties of Runge–Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation u _t =Lu where L is a linear operator. We present optimal SSP Runge–Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge–Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.     

A Computer System Supporting Administrative Decision Making under the Conditions of Mixed Information for Systems of Ecological Monitoring of the Atmosphere

A mathematical model and a block diagram of a system for making administrative decisions in distributed monitoring systems under the conditions of mixed information are developed by methods of systems analysis. The structure of applied decision-making systems for supporting information systems of analysis and prediction of atmospheric air pollution is proposed.