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...

In this paper, we consider reinitializing level functions through equation /tþ sgnð/Þðkr/k 1Þ 1⁄4 0 [16]. The method of Russo and Smereka [11] is taken in the spatial discretization of the equation. The spatial discretization is, simply speaking, the second order ENO finite difference with subcell resolution near the interface. Our main interest is on the temporal discretization of the equation...

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...

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EP-TRK) methods for rst-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the eecient sequential Runge-Kutta method dopri5.

Randomness may exist in the initial value or in the differential operator or both. In [1,2], the authors discussed the general order conditions and a global convergence proof is given for stochastic Runge-Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. In [3,4], the authors discussed the random Euler method and the conditions for the mean square...

A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special form y(t) = f(t, y(t)). The method is based on the exact integration of some given functions in order to solve the problem exactly when the solution is the linear combination of these functions. If this is not the case, the algebraic order (order of accuracy) of the m...

Three new Runge–Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of...

The e ciency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the e ciency of higher-order Runge-Kutta schemes in comparison with the popular Backward Di erencing Formulations. For this comparison an unsteady two-dimensional laminar ow problem is chosen, i.e. ow around a circular cylinder at Re=1200. It is conclude...