Beyond conventional Runge-Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models (Q879410)

The paper deals with the numerical integration of index 3 differential algebraic equations (DAEs) of index 3 in Lagrangian and Hamiltonian form. The author introduces the new class of \((s, s)\)-Gauss-Lobatto specialized partitioned additive Runge-Kutta schemes. Applied to Hamiltonian DAEs it is shown that these methods preserve the contraints, the symmetry and the symplecticness of the system. Discretzing the Lagrangian form these schemes maintain the variational structure of the problem. Moreover, in both cases optimal order of convergence is shown. In the second part of the paper a new approach for solving ordinary differential equations (ODEs) in a local area of the phase space is presented. The main idea is to solve a local linearization of the problem analytically and the remaining nonlinear part numerically on the basis of symmetric Runge-Kutta schemes. Finally, some numerical computations confirm the theoretical results and display the sharpness of the convergence results.