The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of twoand three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization...

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...

Conclusions This paper compares three numerical methods for reliability calculation of Markov, closed, fault-tolerant systems which give rise to continuous-time, timehomogeneous, finite-state, acyclic Markov chains. We consider a modified version of Jensen’s method (a probabilistic method, also known as uniformization or randomization), a new version of ACE (Acyclic Markov Chain Evaluator) algo...

The paper presents a sixth-order numerical algorithm for studying the completely integrable CamassHolm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We...

stability in the autonomous, scalar case was proven in 1963 in Johansson and Kreiss (9), see also (4), and for non-commuting, autonomous systems in 2012 (12), see also (17) for background. We prove herein weak instability in the nonautonomous case. The extension of stability for ODEs from autonomous to nonautonomous (with test problem y′ = λ (t)y) has a rich history. Dahlquist (3) proved that a...

Runge-Kutta methods are an important family of implicit and explicit iterative methods used for the approximation of solutions of ordinary differential equations. Explicit RungeKutta methods are unsuitable for the solution of stiff equations as their region of stability is small. Stiff equation is a differential equation for which certain numerical methods for solving the equation are numerical...

This research work presents a new evolutionary optimization algorithm, EVO-RUNGE-KUTTA in theoretical mathematics with applications in scientific computing. We illustrate the application of EVO-RUNGE-KUTTA, a two-phase optimization algorithm, to a problem of pure algebra, the study of the parameterization of an algebraic variety, an open problem in algebra. Results show the design and optimizat...

The paper presents an error estimate for RungeKutta direct discretizations of terminal optimal control problems for linear systems. The optimal control for such problems is typically discontinuous, and Lipschitz stability of the solution with respect to perturbations does not necessarily hold. The estimate (in terms of the optimal controls) is of first order if certain recently obtained suffici...

Real-time simulation often requires a computer to interact with external hardware in a dynamic loop. Data must be sampled from input signals and incorporated into the numerical integration algorithm to evaluate the derivatives of the state variables. Also, data from the integration must be available for external use with a minimum of delay. This paper will show that, because of these requiremen...

This paper is concerned with the study of the stability of RungeKutta-Pouzet methods for Volterra integro-differential equations with delays. We are interested in the comparison between the analytical and numerical stability regions. First, we focus on scalar equations with real coefficients. It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical soluti...