Three-stage Hermite-Birkhoff-Taylor ODE solver with a C++ program

Abstract

One-step 3-stage Hermite-Birkhoff-Taylor methods, denoted by HBT( p)3, are constructed for solving nonstiff systems of first-order differential equations of the form y' = f( x, y), y(x0) = y0. The method uses derivatives y' to y(p--2) as in Taylor methods and is combined with a 3-stage Runge-Kutta method of order 3. 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 then reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method yields impressive results with regards to intervals of absolute stability. A detailed formulation of variable step size (VS) fixed order HBT( p)3 is presented, as well as the formulation of variable-step variable-order (VSVO) HBT(p)3. Several problems often used to test high order ODE solvers on the basis the number of steps, CPU time, maximum global error, and maximum global energy error are considered. The results stress that both VS and VSVO HBT(p)3 methods are superior to Dormand-Prince DP (8,7)13M and Taylor method of order p, denoted by T( p). To obtain results at high precision, high order VS and VSVO HBT( p)3 methods have been implemented in multiple precision. These numerical results clearly show the benefit of formulating a method by adding high order derivatives to Runge-Kutta method.