Solución de la ecuación de onda como un problema de valores iniciales usando diferencias finitas

Abstract

The solution of the wave equation is presented as the paradigm of the solution of initial value problems with boundary conditions using the finite differences approximation. First, it is developed an elementary solution and a direct discretization in order to introduce the method. Second, the wave equation is solved as a system of first order, the hyperbolicity properties of the resulting system of equations is studied, the characteristic variables and characteristic speeds of the system are calculated and boundary conditions are imposed in terms of the characteristic variables. In this case the method of lines is used as the evolution scheme. Special attention is devoted to the fact that numerical calculations require a criterion to be valid. In the case of the approximation using finite differences of a partial differential equation, the convergence to a correct solution in the continuum limit is presented as such criterion. Finally, it is expected that this manuscript serves as a guide to solve correctly other initial value problems with boundary conditions.