Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by m-dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical B-convergence theory to...

This paper is concerned with the numerical solution of implicit neutral functional diierential equations. Based on the continuous Runge{Kutta method (for ordinary diierential equations) and the collocation method (for functional equations), two general one-step methods are formulated and their uniform order of approximation are discussed. Numerical stability of a class of Runge{Kutta-Collocatio...

Implicit Runge-Kutta methods are considered which combine the single-implicitness or diagonal-implicitness property with a zero first row in the coefficient matrix. Acceptable stability for stiff problems is retained by requiring the last stage of a step to be identical to the output value. This requirement, which corresponds to the FSAL property for explicit Runge-Kutta methods, allows the met...

Spectral deferred correction is a flexible technique for constructing high-order, stiffly-stable time integrators using a low order method as a base scheme. Here we examine their use in conjunction with splitting methods to solve initial-boundary value problems for partial differential equations. We exploit their close connection with implicit Runge–Kutta methods to prove that up to the full ac...

In this paper a fully explicit, stabilized projection method called the Runge–Kutta–Chebyshev (RKC) projection method is presented for the solution of incompressible Navier–Stokes systems. This method preserves the extended stability property of the RKC method for solving ODEs, and it requires only one projection per step. An additional projection on the time derivative of the velocity is perfo...

We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations y′(t) = f(t, y(t)). The function f is smooth in y and we suppose that f and D1 yf are of bounded variation in t and that D2 yf is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte...

Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. Th...

A general class of one-step methods for index 2 differential-algebraic systems in Hessenberg form is studied. This family of methods, which we call partitioned Runge-Kutta methods, includes all one-step methods of Runge-Kutta type proposed in the literature for integrating such DAE systems, including the more recently proposed classes of half-explicit methods. A new family of super-convergent p...