Comparison of finite-difference schemes for the first order wave equation problem

Abstract

A large number of problems in physics and technology lead to boundary value or initial boundary value problems for linear and nonlinear partial differential equations. Moreover, the number of problems that have an analytical solution is limited. These are problems in canonical domains such as, for example, a rectangle, circle, or ball, and usually for equations with constant coefficients. In practice, it is often necessary to solve problems in very complex areas and for equations with variable coefficients, often nonlinear. This leads to the need to look for approximate solutions using various numerical methods. A fairly effective method for the numerical solution of problems in mathematical physics is the finite difference method or the grid method, which makes it possible to reduce the approximate solution of partial differential equations to the solution of systems of algebraic equations. The article studied the most popular numerical methods of the first, second and third order of accuracy. All of these circuits have been compared with exact solutions.