By using the algorithm of Nfimberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sq(d 1 ) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q >~ 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Bi...

Variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff methods of order 4 to 14, denoted by HBO(4-14)3, are constructed for solving non-stiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. These methods use y′ and y′′ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads...

Abstract In this work, we develop a class of high‐order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite‐Birkhoff‐Predictor‐Corrector (HBPC) and technique relaxation for numerical ordinary differential equations (ODEs). We explain algorithm in detail show results two‐ three‐derivative metho...

A one-step 4-stage Hermite-Birkhoff-Taylor method of order 12, denoted by HBT(12)4, is 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(9) as in Taylor methods combined with a 4-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the tr...

We show how to combine incidence matrices, which admit Hermite-Birkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the co...

The ordinary and delay differential equations (ODEs/DDEs) solver HB515DDE is based on the discrete hybrid variable-step variable-order 3-stage Hermite–Birkhoff ODE solver of (consistency) order 5 to 15. The current version of the solver can handle ODEs and DDEs with statedependent, non-vanishing, small, vanishing and asymptotically vanishing delays. Delayed values are computed using Hermite int...