Practical Runge–Kutta Processes

The development of embedded Runge–Kutta and Runge–Kutta–Nystrom formulae subject to various criteria is reviewed. An important criterion concerns the cost of achieving a particular global error in the numerical solution. By consideration of local truncation errors in the two formulae of an embedded pair, it is possible to produce a good process. Another criterion involves the provision of continuous solutions. Such a requirement can be at odds with the previous one of basic cost-effectiveness. However, it seems important to provide dense output without excessive cost in new function evaluations. Special RK/RKN formulae are preferable for practical global error estimation using the Zadunaisky pseudo-problem or related technique of solving for the error estimate. Two-term error estimation can be achieved and the pseudo-problem can be based on dense output values.