NUMERICAL INTEGRATORS FOR FOURTH ORDER INITIAL AND BOUNDARY VALUE PROBLEMS

Abstract

In this paper, a continuous k-step linear multistep method (LMM) is developed and used to generate new finite difference methods (NFDMs), which are assembled and applied as simultaneous numerical integrators to solve fourth order initial and boundary value problems without reducing them to an equivalent first order system. The NFDMs are analyzed for convergence via consistency and zero-stable by conveniently expressing them as block methods. The initial value problems (IVPs) are solved without the need for either predic- tors or starting values from other methods, while the boundary value problems (BVPs) are solved by assembling the NFDMs into a single block matrix equa- tion. We illustrate our process using a specific example for k = 4. Numerical examples are given to show the efficiency of the methods.