The Art of Writing a Runge-Kutta Code, Part I

Publisher Summary This chapter discusses Runge–Kutta methods of moderate order and their implementation as a computer code. There is a widely held belief that writing a Runge-Kutta code is not much more than a transcription of the basic formula into a high level computer language. One should restrict the attention to methods that are of order four but because of local extrapolation, this may also involve the formulas of orders three and five. A four-stage fourth order formula of Kutta is known as the classical scheme and has been extremely popular because of its simple coefficients. This principle involves taking two half steps with the basic formula and combining the result with that from one whole step of the basic formula so as to estimate the error of the more accurate result. The most important single aspect of a formula is its accuracy for it is the main factor affecting efficiency. One should not try to solve truly stiff problems with explicit Runge–Kutta formulas; however, some stiffness is not unusual. The essence of the matter is that the step size is limited for the reasons of stability in the presence of stiffness rather than for the reasons of accuracy that is more commonly the case.