In this work, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both...

New software based on explicit Runge-Kutta formulas have been developed to replace well-established, widely-used codes written by the authors (RKF45 and its successors in the SLATEC Library and the NAG Fortran 77 Library Runge-Kutta codes). The new software has greater functionality than its predecessors. Also, it is more eecient, more robust and better documented.

This paper presents a family of Runge{Kutta type integration schemes of arbitrarily high order for diierential equations evolving on manifolds. We prove that any classical Runge{Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher–order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order–optimal for A–stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical inv...

In this paper, we analyze second-order Runge–Kutta approximations to a nonlinear optimal control problem with control constraints. If the optimal control has a derivative of bounded variation and a coercivity condition holds, we show that for a special class of Runge–Kutta schemes, the error in the discrete approximating control is O(h2) where h is the mesh spacing.

The non-linear wave equation is taken as a model problem for the investigation. Different multisymplectic reformulations of the equation are discussed. Multi-symplectic Runge–Kutta methods and multi-symplectic partitioned Runge–Kutta methods are explored based on these different reformulations. Some popular and efficient multi-symplectic schemes are collected and constructed. Stability analyses...

We prove that to every rational function R(z) satisfying R(−z)R(z) = 1, there exists a symplectic Runge-Kutta method with R(z) as stability function. Moreover, we give a surprising relation between the poles of R(z) and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.