This paper presents an overview of the acoustic wave equation and the common time-domain numerical solution strategies in closed environments. First, the wave equation is presented and its qualities analyzed. Common principles of numerical approximation of derivatives are then reviewed. Based on them, the finite difference (FD) and the finite element methods (FEM) for the solution of the wave e...

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential increasing (decreasing) is proposed. It is based on approximation of the solution to find in the form of superposition of certain polynomial-exponential basic fun...

in this paper, a chebyshev finite difference method has been proposed in order to solvenonlinear two-point boundary value problems for second order nonlinear differentialequations. a problem arising from chemical reactor theory is then considered. the approachconsists of reducing the problem to a set of algebraic equations. this method can be regardedas a non-uniform finite difference scheme. t...

An explicit, adaptive, hybrid finite element/finite difference method is proposed for the numerical solution of Maxwell’s equations in the time domain. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. Thus, we combine the flexibility of finite elements with the efficiency of finit...

In this study, a first-order, nonstationary stochastic model of transient flow is developed which is applicable to the entire domain of a bounded vadose zone in the presence of sink/source. We derive general equations governing the statistical moments of the flow quantities by perturbation expansions. Owing to the mathematical complexity of the equations, in general we need to solve them numeri...

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x′(t) = ax(t) + bx(t − 1) + cx(t + 1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation o...

In this paper, a solution-adaptive algorithm is presented for the simulation of incompressible viscous flows. The framework of this method consists of an adaptive local stencil refinement algorithm and 3-points central difference discretization. The adaptive local stencil refinement is designed in such a manner that 5-points symmetric stencil is guaranteed at each interior node, so that convent...