Variable-step variable-order 3-stage Hermite-Birkhoff ODE solver of order 5 to 15 with a C++ program

Abstract

Variable-step variable-order 3-stage Hermite-Birkhoff (HB) methods HB( p)3 of order p = 5 to 15 are constructed for solving nonstiff differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems of HB type. Fast algorithms are developed for solving these systems in O(p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The order and stepsize of these methods are controlled by four local error estimators. These methods, when programmed in Matlab, are superior to Matlab's ode113 in solving several problems often used to test higher order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. On the other hand, HB(5-15)3 are programmed in object-oriented C++ and the Dormand-Prince 13-stage nested Runge-Kutta pair DP(8,7)13M are programmed in C. DP(8,7) is found to use less CPU time, have smaller maximum global error but require a larger number of function evaluations than HB(5-15)3. However, for expensive equations, such as the Cubicwave, HB(5-15)3 is superior. In the C++ program, array and matrix are considered to be new objects. Algorithms, testing programs and new objects are structured separately as header files.