Five and Six Stage Runge-Kutta Type Formulas of Orders Numerically Five and Six

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Abstract

New five-and six-stage Runge-Kutta type formulas of orders numerically five and six are proposed. It is well known that five and six stage explicit Runge-Kutta methods cannot achieve five and six order of accuracy. However fifth- and sixth-order formulas are obtained when the distance between some pairs of abscissas tends to zero. Such formulas are called limiting formulas and involve derivatives. In previous papers we presented numerically fifth- and sixth-order formulas which require only five and six evaluations of the function. However the parameters of these formulas are numbers of many decimal digits. Parameters of our new formulas are simple though our formulas can achieve the same accuracy as the limiting formulas. The values of derivatives in the limiting formulas do not require full significant digits carried in the computation. So we approximate derivatives by numerical differentiation and choose one of free parameters of the limiting formula to minimize the error caused by approximation. As a result the approximation errors in five- and six-stage formulas are reduced to O(h^4r^<-q/2>) and O(h^3r^<-q/2>) respectively with step size h in q-digits of r decimal system. Thus the approximation errors are not a significant part of the total error throughout the computation.

New five-and six-stage Runge-Kutta type formulas of orders numerically five and six are proposed. It is well known that five and six stage explicit Runge-Kutta methods cannot achieve five and six order of accuracy. However, fifth- and sixth-order formulas are obtained when the distance between some pairs of abscissas tends to zero. Such formulas are called limiting formulas and involve derivatives. In previous papers we presented numerically fifth- and sixth-order formulas which require only five and six evaluations of the function. However, the parameters of these formulas are numbers of many decimal digits. Parameters of our new formulas are simple, though our formulas can achieve the same accuracy as the limiting formulas. The values of derivatives in the limiting formulas do not require full significant digits carried in the computation. So, we approximate derivatives by numerical differentiation and choose one of free parameters of the limiting formula to minimize the error caused by approximation. As a result, the approximation errors in five- and six-stage formulas are reduced to O(h^4r^<-q/2>) and O(h^3r^<-q/2>) respectively with step size h in q-digits of r decimal system. Thus, the approximation errors are not a significant part of the total error throughout the computation.

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