Weighted Residual Methods

Classification of Numerical Solutions of Differential Equations

Two main families of approximate methods could be identified in the literature. The discrete coordinate methods and the distributed coordinate methods.

Discrete coordinate methods depend on solving the differential relations at pre-specified points in the domain. When those points are determined, the differential equation may be approximately presented in the form of a difference equation. The difference equation presents a relation, based of the differential equation, between the values of the dependent variables at different values of the independents variables. When those equations are solved, the values of the dependent variables are determined at those points giving an approximation of the distribution of the solution. Examples of the discrete coordinate methods are finite difference methods and the Runge-Kutta methods. Discrete coordinate methods are widely used in fluid dynamics and in the solution of initial value problems.

The other family of approximate methods is the distributed coordinate methods. These methods, generally, are based on approximating the solution of the differential equation using a summation of functions that satisfy some or all the boundary conditions. Each of the proposed functions is multiplied by a coefficient, generalized coordinate, that is then evaluated by a certain technique that identifies different methods from one another. After the solution of the problem, you will obtain a function that represents, approximately, the solution of the problem at any point in the domain.

Stationary functional methods are part of the distributed coordinate methods family. These methods depend on minimizing/maximizing the value of a functional that describes a certain property of the solution, for example, the total energy of the system. Using the stationary functional approach, the finite element model of a problem may be obtained. It is usually much easier to present the relations of different variables using a functional, especially when the relations are complex as in the case of fluid structure interaction problems or structure dynamics involving control mechanisms.

The weighted residual methods, on the other hand, work directly on the differential equations. As the approximate solution is introduced, the differential equation is no more balanced. Thus, a residue, a form of error, is introduced to the differential equation. The different weighted residual methods handle the residue in different ways to obtain the values of the generalized coordinates that satisfy a certain criterion.