Journal of Computational and Applied Mathematics
Two FORTRAN packages for assessing initial value methods
ACM Transactions on Mathematical Software (TOMS)
Numerical comparisons of some explicit Runge-Kutta pairs of orders 4 through 8
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
High Phase-Lag-Order Runge--Kutta and Nyström Pairs
SIAM Journal on Scientific Computing
Pracniques: further remarks on reducing truncation errors
Communications of the ACM
N-body simulations: The performance of some integrators
ACM Transactions on Mathematical Software (TOMS)
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Explicit Runge---Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.