On advanced variational formulation of the method of lines and its application to the wave propagation problems

Abstract

The method of lines is a simple and reliable method of numerical analysis of parabolic and hyperbolic problems of mathematical physics. By means of this method mixed initial-boundary problems described by partial differential equations are transformed into systems of ordinary differential equations with initial conditions. This reduction is obtained by means of application of particular finite difference schemes to the spatial derivatives. Many of the wave propagation problems describing by the hyperbolic equations could be formulated in terms of the variational principles. In the present paper we demonstrate how to derive the systems of ordinary differential equations of the method of lines directly from the Lagrangians of the corresponding variational formulations of the wave propagation problems. The discussed method has several obvious advantages in comparison with the traditional methods of deriving of the initial problems of systems for ordinary differential equations from the initial-boundary problems for partial differential equations. First, in Lagrangians we need to use a finite difference representation of the spatial derivatives of lower order than in equations. For example, in the wave equations describing longitudinal vibration of bars, torsional vibration of rods, etc., we need to represent the second partial derivative of displacements by its finite difference but in the corresponding Lagrangian we need to use the finite difference representation of the first partial derivative of displacements. In the equations of longitudinal vibration of the Rayleigh-Bishop bar as well as in the equations of lateral vibration of the Euler-Bernoulli beam we need to use a finite difference representation of the fourth order partial derivatives of displacements, but in the corresponding Lagrangians we need a finite difference representation of the second order partial derivatives of displacements. The second advantage of the variational approach to the method of lines is connected with number of terms in equations and the corresponding Lagrangians. As a rule, number of terms in Lagrangians is substantially less than in the corresponding equations. For example, in the equation of vibration of the Rayleigh-Bishop bar with variable parameters there are eight terms including spatial partial derivatives of displacement of the first, second, third and fourth order and first and second partial derivatives of combinations of geometrical and physical parameters. In the corresponding Lagrangian there are four terms including first and second spatial partial derivatives of displacement and derivatives of combinations of geometrical and physical parameters are absent. Despite the obvious advantages of the variational formulation of the method of lines there are some limitations of its practical application which are also discussed in the paper.