Numerical methods for initial value problems

We will consider the discretization of initial value problems for first order o.d.e´s,

We will first briefly review some basic stability properties of this problem, assuming that f satisfies either the Lipschitz condition or the one-sided Lipschitz condition. Then, the simplest scheme for ( ), namely the Euler method, will be discussed. We will then focus on the two most popular classes of methods for initial value problems, namely the Runge-Kutta methods and multistep methods but will also briefly discuss Galerkin type schemes. Various stability as well as consistency properties of the schemes will be analyzed and advantages and drawbacks of each class will be discussed.