Rolf Jeltsch

Waveform relaxation with overlapping splittings

In this paper, an extension of the waveform relaxation algorithm for solving large systems of ordinary differential equations is presented. The waveform relaxation algorithm is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces. Allowing the subspaces to overlap, i.e., dropping the assumption of disjointedness, an extension of this algorithm is obtained. This new algorithm, the so-called multisplitting algorithm, is also well suited for parallel computation. As numerical examples demonstrate, this overlapping of the subsystems heavily reduces the computation time.

Stability and accuracy of time discretizations for initial value problems

SummaryThis paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.

Largest disk of stability of explicit Runge-Kutta methods

The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [−2m, 0] as a proper subset.

Essentially optimal explicit Runge–Kutta methods with application to hyperbolic–parabolic equations

Optimal explicit Runge–Kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time-step restrictions. This idea, so far used mostly for real-line spectra, is generalized to more general spectra in the form of a thin region. In thin regions the eigenvalues may extend away from the real axis into the imaginary plane. We give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases. Semi-discretizations of hyperbolic–parabolic equations are a relevant application which exhibit a thin region spectrum. As a model, linear, scalar advection–diffusion is investigated. The second-order-stabilized explicit Runge–Kutta methods derived from the stability polynomials are applied to advection–diffusion and compressible, viscous fluid dynamics in numerical experiments. Due to the stabilization the time step can be controlled solely from the hyperbolic CFL condition even in the presence of viscous fluxes.

Stability of quadrature rule methods for first kind volterra integral equations

Gladwin [4] proved that Newton-Gregory formulas of order larger than 2 produce unstable algorithms when applied to nonlinear Volterra integral equations of the first kind. It is shown that similar results are true for all interpolatory quadrature rules using equidistant nodes. Upper bounds for the error order of quadrature rules, which lead to stable methods are given. Some higher order stable methods are indicated.

Multistep methods using higher derivatives and damping at infinity

Linear multistep methods using higher derivatives are discussed. The order of damping at infinity which measures the stability behavior of a k-step method for large h is introduced. A-stable methods with positive damping order are most suitable for stiff problems. A method for computing the damping order is given. Necessary and sufficient conditions for A-stability, A(oe)-stability and stiff stability are presented. A new A-stable two-step method of order 4 with damping order 1 is found and numerical results are given.

Stiff Stability and Its Relation to

Necessary and sufficient conditions for an

A_0

A_0

- and

-stable multistep method to be stiffly stable or

A(0)

A(0)