Radau Ode Solvers with Hermite–Birkhoff Predictors

We construct and test new explicit general linear Hermite–Birkhoff–Radau (HBR) methods of orders 4, 5, 8, 9 and 10 with 1, 2, 3, 4, and 4 off-step points,respectively. It is observed that the order of HBRr methods with off-step points is + (= + 1) where is the order of the predictor used to obtain the solution value at the first off-step point, for = 3, 5, 7, 8 and order + + 1 for = 9. The stability region increases as the order increases. A local error estimator is used to control the step size. In practice, a set of methods with constrained step size can be precalculated to reduce overhead for roughly 10% increase in function evaluations. The methods are tested with 25 DETEST problems with constrained, and in some case, unconstrained variable step sizes and comparison is made with ODE23 and ODE45 of the M ODE suite, and Dormand–Prince DP(5,4)7M and DP(8,7)13M, respectively. Generally, the HBR methods have lower global errors and use fewer function evaluations