One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10

Abstract

A one-step 9-stage Hermite-Birkhoff-Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order differential equations of the form y = f (x, y), y(x 0 ) = y 0 . The method uses y and higher derivatives y (2) to y (4) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor-and Runge-Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than Dormand-Prince DP(8,7)13M. The stepsize is controlled by means of y (2) and y (4) . HBT(10)9 is superior to DP(8,7)13M and Taylor method of order 10 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. These numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

“…The one-step 9-stage Hermite-Birkhoff-Taylor (HBT) method HBT(10)9 of order 10 constructed in [20] for solving ODEs is expanded into the DAE solver HBT(10)9DAE by the addition of a scheme that generates first-order derivatives at off-step points besides Pryce scheme that generates high order derivatives at step points. The stepsize is controlled by a local error estimator.…”

“…The order conditions, the Vandermonde-type formulation and the region of absolute stability of HBT(10)9DAE are as for HBT(10)9 and are found in [20]. For DAEs, we need only add a step control predictor P 10 .…”

“…It is to be noted that, as in Runge-Kutta methods, y n+c i are evaluated by numerical formulae which satisfy the order conditions listed in [20,Appendix A]. Thus, the values x 1,n+c i , x 2,n+c i , x 1,n+c i , x 2,n+c i , x 1,n+c i , x 2,n+c i , λ n+c i at off-step points should not be projected onto the constraints manifold.…”

“…It is seen that HBT(10)9DAE uses fewer derivatives than T10DAE and requires fewer steps, uses less CPU time, and has higher accuracy than DP(8,7)DAE. Section 2 reviews HBT(10)9 for ODEs constructed in [20]. Section 3 presents Pryce structural analysis for HBT(10)9DAE.…”