Eckart W. Gekeler

On the order conditions of Runge-Kutta methods with higher derivatives

SummaryRunge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy′=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.

Galerkin-Runge-Kutta methods and hyperbolic initial boundary value problems

Inhomogeneous but time-homogeneous linear hyperbolic initial boundary value problems are solved using Galerkin procedures for the space discretization and Runge-Kutta methods for the time discretization. The space discretized system is not transformed a-priori in a linear system of first order. For the difference of the Ritz projection of the exact solution and the numerical approximation error estimates are derived under the assumption that the applied Runge-Kutta methods have a non-empty interval of absolute stability. It is shown that this class of schemes is not empty in the present case of second order systems, too.ZusammenfassungDie vorliegende Arbeit befaßt sich mit der numerischen Lösung inhomogener linearer hyperbolischer Anfangsrandwert-probleme, die zeitlich homogen sind. Das analytische Problem wird zunächst mit Hilfe von Galerkinverfahren in den Raumveränderlichen diskretisiert. Dann wird das resultierende semidiskrete System ohne vorherige Transformation durch Runge-Kutta-Verfahren approximiert. Für die Differenz zwischen der Ritzprojektion der exakten Lösung und dem numerischen Ergebnis werden Fehlerabschätzungen hergeleitet unter der Voraussetzung, daß das verwendete Runge-Kutta-Verfahren ein nichtleeres Intervall absoluter Stabilität besitzt. Ein Beispiel zeigt, daß solche Verfahren auch für Systeme zweiter Ordnung existieren.

Galerkin-obrechkoff methods and hyperbolic initial boundary value problems with damping

The numerical solution of linear inhomogeneous but time-homogeneous hyperbolic initial value problems with damping is considered. A special class of high order Galerkin-Obrechkoff methods is investigated, and L2 error bounds are derived.

A-priori error estimates of Galerkin backward differentiation methods in time-inhomogeneous parabolic problems

SummaryBackward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.

On the convergence of the von Neumann difference approximation to hyperbolic initial boundary value problems

AbstractThe general implicit finite-difference approximation of second order devised by von Neumann is applied to the initial boundary value problem for a somewhat generalized wave equation. By means of eigenvalue expansion it is shown that the method is uniform convergent of order

On the numerical solution of double-periodic elliptic eigenvalue problems

\log \left( {\Delta x^{ - 1} } \right)O\left( {\Delta t^2 + \Delta x^2 } \right)

On the stability of backward differentiation methods

Δt, Δx mesh widths). Moreover, the convergence on the linet=T reveals to be proportional toT.

Abriss der Tensorrechnung

Trigonometric collocation methods are used to approximate simple Hopf bifurcation problems, wy’ + f(y,μ) = 0, with 2π-periodic solutions. The discrete system is solved by a constructive Ljapunov-Schmidt reduction originally due to Keller and Langford. In this method no component of the solution is fixed in advance therefore codes for the Fast Fourier Transform can be applied in the iteration which is of the form