THE CONNECTION BETWEEN THE FINITE DIFFERENCE LIKE METHODS AND THE METHODS BASED ON INITIAL VALUE PROBLEMS FOR ODE

Abstract

This chapter highlights the connection between the finite difference like methods and the methods based on initial value problems forordinary differential equations (ODE). The formulation of the systems of equations stemming from different versions of finite difference methods or finite element methods, as presented in the theory, is not the entire story of the numerical solution of boundary value problems for ODE. An essential part of the process is the solution of the derived systems of equations. There are many ways to solve these systems, a variety of direct and iterative methods. Using the finite difference method, the initial value problem for equations can be solved by a second order method without automatic step selection. The use of the deferred corrections leads to the solution of the same equations using an extrapolation procedure based on second order schemes. Another point is that the second order method stemming from the elimination method needs a relatively very small number of operations. Iterative procedures are not used when solving boundary value problems for ODEs.