Willem Hundsdorfer

A Second-Order Rosenbrock Method Applied to Photochemical Dispersion Problems

A second-order, L-stable Rosenbrock method from the field of stiff ordinary differential equations is studied for application to atmospheric dispersion problems describing photochemistry, advective, and turbulent diffusive transport. Partial differential equation problems of this type occur in the field of air pollution modeling. The focal point of the paper is to examine the Rosenbrock method for reliable and efficient use as an atmospheric chemical kinetics box-model solver within Strang-type operator splitting. In addition, two W-method versions of the Rosenbrock method are discussed. These versions use an inexact Jacobian matrix and are meant to provide alternatives for Strang-splitting. Another alternative for Strang-splitting is a technique based on so-called source-splitting. This technique is briefly discussed.

The multiscale nature of streamers

Streamers are a generic mode of electric breakdown of large gas volumes. They play a role in the initial stages of sparks and lightning, in technical corona reactors and in high altitude sprite discharges above thunderclouds. Streamers are characterized by a self-generated field enhancement at the head of the growing discharge channel. We briefly review recent streamer experiments and sprite observations. Then we sketch our recent work on computations of growing and branching streamers, we discuss concepts and solutions of analytical model reductions and we review different branching concepts and outline a hierarchy of model reductions.

On the stability of implicit-explicit linear multistep methods

In many applications, large systems of ordinary differential equations (ODEs) have to be solved numerically that have both stiff and nonstiff parts. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson Leap-Frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms.

Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations

SummaryWe address the question of convergence of fully discrete Runge-Kutta approximations. We prove that, under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time will be strictly smaller than the conventional one, a phenomenon called order reduction. This phenomenon is extensively discussed, both analytically and numerically. As distinct from earlier contributions we here treat explicit Runge-Kutta schemes. Although our results are valid for both parabolic and hyperbolic problems, the examples we present are therefore taken from the hyperbolic field, as it is in this area that explicit discretizations are most appealing.

Monotonicity-Preserving Linear Multistep Methods

In this paper we provide an analysis of monotonicity properties for linear multistep methods. These monotonicity properties include positivity and the diminishing of total variation. We also pay particular attention to related boundedness properties such as the total variation bounded (TVB) property. In the analysis the multistep methods are considered in combination with suit- able starting procedures. This allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results.

An adaptive grid refinement strategy for the simulation of negative streamers

The evolution of negative streamers during electric breakdown of a non-attaching gas can be described by a two-fluid model for electrons and positive ions. It consists of continuity equations for the charged particles including drift, diffusion and reaction in the local electric field, coupled to the Poisson equation for the electric potential. The model generates field enhancement and steep propagating ionization fronts at the tip of growing ionized filaments. An adaptive grid refinement method for the simulation of these structures is presented. It uses finite volume spatial discretizations and explicit time stepping, which allows the decoupling of the grids for the continuity equations from those for the Poisson equation. Standard refinement methods in which the refinement criterion is based on local error monitors fail due to the pulled character of the streamer front that propagates into a linearly unstable state. We present a refinement method which deals with all these features. Tests on one-dimensional streamer fronts as well as on three-dimensional streamers with cylindrical symmetry (hence effectively 2D for numerical purposes) are carried out successfully. Results on fine grids are presented, they show that such an adaptive grid method is needed to capture the streamer characteristics well. This refinement strategy enables us to adequately compute negative streamers in pure gases in the parameter regime where a physical instability appears: branching streamers.

Interaction of Streamer Discharges in Air and Other Oxygen-Nitrogen Mixtures

The interaction of streamers in nitrogen-oxygen mixtures such as air is studied. First, an efficient method for fully three-dimensional streamer simulations in multiprocessor machines is introduced. With its help, we find two competing mechanisms how two adjacent streamers can interact: through electrostatic repulsion and through attraction due to nonlocal photoionization. The nonintuitive effects of pressure and of the nitrogen-oxygen ratio are discussed. As photoionization is experimentally difficult to access, we finally suggest to measure it indirectly through streamer interactions.

Accuracy and stability of splitting with stabilizing corrections

This paper contains a convergence analysis for the method of Stabilizing Corrections, which is an internally consistent splitting scheme for initial-boundary value problems. To obtain more accuracy and a better treatment of explicit terms several extensions are regarded and analyzed. The relevance of the theoretical results is tested for convection-diffusion-reaction equations.

Branching of negative streamers in free flight

We have recently shown that a negative streamer in a sufficiently high homogeneous field can branch spontaneously due to a Laplacian instability, rather than approach a stationary mode of propagation with fixed radius. In our previous simulations, the streamer started from a wide initial ionization seed on the cathode. We here demonstrate, in improved simulations, that a streamer emerging from a single electron branches in the same way. In fact, though the evolving streamer is much more narrow, it branches after an even shorter propagation distance.

A note on splitting errors for advection-reaction equations

Abstract In this note we consider proper ways to combine numerical schemes for advective transport and nonlinear chemistry. Obvious combinations are obtained with splitting in a so-called fractional step approach. We shall discuss for this approach correct implementations of source terms and inflow boundary conditions. Further we consider the use of multistep methods with explicit treatment of the advection terms and implicit chemistry.