Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but also having optimal order of convergence. The new class of ( s , s ) -Gauss–Lobatto specialized partitioned additive Runge–Kutta (SPARK) methods uses greatly the structure of the DAEs and possesses all desired properties. In the second part we propose a unified approach for the solution of ordinary differential equations (ODEs) mixing analytical solutions and numerical approximations. The basic idea is to consider local models which can be solved efficiently, for example analytically, and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction. In order to preserve also symmetry we define the new class of symmetrized Runge–Kutta methods with local model (SRKLM).