Mathematical modeling of a wave field by finite difference method

Abstract

The system of differential equations of hydrodynamics is reduced to the inhomogeneous wave equation, for which a discrete model is constructed. The computational domain is a rectangle covered with a two-dimensional uniform calculation grid. Discrete analogues of the wave equation for Dirichlet and Neumann boundary conditions were developed. The paper describes the choice of calculation grids. Analytical expressions describing the approximation error in spatial coordinate directions for the second derivative operator based on the second and fourth order of accuracy schemes were obtained and, in this range of error, the sizes of grids were estimated. The computational experiments demonstrated that, to keep the calculation error in the specified limit 0.1- 1%, it is necessary to perform calculations on grids with the number of nodes per half wavelength in the range from 9 to 30 when using the second-order of accuracy scheme, and from 4 to 6 when using the fourth-order of accuracy scheme. It is important to choose the values of the time step and the weight parameter. At the weight parameter σ =1/12, the relative error is significantly less than at σ =0 (corresponding to an explicit scheme) and at σ =1/4 (corresponding to a symmetrical setting of coefficients in a weight scheme). The optimal values of the weight parameter were calculated in the context of maintaining the propagation frequency of oscillatory processes. The analysis of the discrete model gave a condition for the stability of the difference scheme, and an expression describing the approximation error in a time variable, depending on time steps and spatial coordinate directions, was determined. It was established that the approximation of initial conditions makes a smaller contribution to the total error than the approximation of the equation for subsequent time layers. Based on the proposed algorithms and approaches, a software package was designed to simulate the propagation of vibrations in a two-dimensional computational domain. A number of computational experiments were carried out to study, for example, the propagation of acoustic waves from antennas with different directional characteristics and the scattering of waves on the obstacles of different types. To determine the farfields of an acoustic antenna, it is proposed to make the calculation window movable and to find its location in space. This significantly reduces the calculation time for the propagation of sound waves over long distances.