Advanced Numerical Integration Based on Runge–Kutta Formulas

This chapter presents a brief insight into the theory of Runge–Kutta methods for ordinary differential equations. In particular, it gives precise definitions and explains all basic issues and notions in such numerical integration tools. These include the concept of Runge–Kutta formula, its consistency, convergence and stability. The important families of Runge–Kutta schemes for reversible and Hamiltonian problems are also covered, here. A special emphasis is paid to implementation aspects of four subclasses in the implicit Runge–Kutta method realm, which allow cheap and efficient Newton-type iterations to be applied. This chapter pays its particular attention to variable-stepsize implementation of stepping formulas under consideration and elaborates all technical details of contemporary local and global error control mechanisms used for achieving a desirable accuracy of adaptive numerical integration in automatic mode. The presented theoretical analysis of Runge–Kutta schemes is accompanied by detailed pseudo-codes, which clarify algorithmic particulars of such numerical integration procedures and are situated in Appendix of this chapter. Finally, a proper theoretical analysis and exhaustive numerical examination of five ODE solvers used commonly in nonlinear Kalman filtering is fulfilled in MATLAB.