Basic Issues and Concepts of Numerical Integration

This chapter presents a brief insight into the theory of numerical integration methods for ordinary differential equations. In particular, it gives precise definitions and explains all basic issues and notions of numerical integration. These include the concepts of stepping formulas, their consistency, convergence and stability. A special emphasis is paid to implementation aspects of implicit numerical schemes, including the accuracy and stability of the fixed-point iteration as well as those of various Newton-type iterations. The important families of reversible and Hamiltonian problems are also covered, here. This chapter pays a particular attention to raising the accuracy of one-step methods via the Richardson extrapolation technique and elaborates all technical details of contemporary local and global error control mechanisms used for achieving a desirable accuracy of numerical integration in automatic mode. The presented theoretical analysis of stepping methods under consideration, which are all summarized in the form of pseudo-codes situated in Appendix of this chapter, is always supported with illustrative calculations performed in MATLAB.