Jan Verwer

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.

Benchmarking stiff ODE solvers for atmospheric chemistry problems II: Rosenbrock solvers

Abstract In the numerical simulation of atmospheric transport-chemistry processes, a major task is the integration of the stiff systems of ordinary differential equations describing the chemical transformations. It is therefore of interest to systematically search for stiff solvers which can be identified as close to optimal for atmospheric applications. In this paper we continue our investigation from Sandu et al. (1996, CWI Report NM-R9603 and Report in Comput. Math., No. 85) and compare eight solvers on a set of seven box-models used in present day models. The focus is on Rosenbrock solvers. These turn out to be very well suited for our application when they are provided with highly efficient sparse matrix techniques to economize on the linear algebra. Two of the Rosenbrock solvers tested are from the literature, viz. rodas and Ros 4, and two are new and specially developed for air quality applications, viz. rodas 3 and ros 3.

Benchmarking stiff ode solvers for atmospheric chemistry problems-I. implicit vs explicit

Abstract In many applications of atmospheric transport-chemistry problems, a major task is the numerical integration of the stiff systems of ordinary differential equations describing the chemical transformations. This paper presents a comprehensive numerical comparison between five dedicated explicit and four implicit solvers for a set of seven benchmark problems from actual applications. The implicit solvers use sparse matrix techniques to economize on the numerical linear algebra overhead. As a result they are often more efficient than the dedicated explicit ones, particularly when approximately two or more figures of accuracy are required. In most test cases, sparse RODAs, a Rosenbrock solver, came out as most competitive in the 1% error region. Of the dedicated explicit solvers, TWOSTEP came out as best. When less than 1% accuracy is aimed at, this solver performs very efficiently for tropospheric gas-phase problems. However, like all other dedicated explicit solvers, it cannot efficiently deal with gas-liquid phase chemistry. The results presented may constitute a guide for atmospheric modelers to select a suitable integrator based on the type and dimension of their chemical mechanism and on the desired level of accuracy. Furthermore, we would like to consider this paper an open invitation for other groups to add new representative test problems to those described here and to benchmark their numerical algorithms in our standard computational environment.

Gauss-Seidel iteration for stiff odes from chemical kinetics

A simple Gauss–Seidel technique is proposed that exploits the special form of the chemical kinetics equations. Classical Aitken extrapolation is applied to accelerate convergence. The technique is meant for implementation in stiff solvers that are used in long range transport air pollution codes using operator splitting. Splitting necessarily gives rise to a great deal of integration restarts. Because the Gauss–Seidel iteration works matrix free, it has much less overhead than the modified Newton method. Start-up costs therefore can be kept low with this technique. Preliminary promising numerical results are presented for a prototype of a second order backward differentiation formula (BDF) solver applied to a stiff ordinary differential equation (ODE) from atmospheric chemistry. A favourable comparison with the general purpose BDF code DASSL is included. The matrix free technique may also be of interest for other chemically reacting fluid flow problems.

Explicit methods for stiff ODEs from atmospheric chemistry

The subject of research is the numerical integration of atmospheric chemical kinetics systems. The application lies in the study of air pollution, modelled by atmospheric chemistry-transport problems. This application puts high demands on the efficiency of the stiff solver. Three explicit methods are discussed and compared for a selected chemical kinetics system which is representative for the state of the art. The first and the second method are of the explicit QSSA type and the third is based on the two-step backward differentiation formula, combined with Gauss-Seidel iteration to approximately solve the implicitly defined solution. This also renders the method explicit. In the comparison, the two-step method comes out best.

A Comparison of Stiff ODE Solvers for Atmospheric Chemistry Problems

In the operator splitting solution of atmospheric transport-chemistry problems modeling air pollution, a major task is the numerical integration of the stiff systems of ordinary differential equations describing the chemical transformations. In this paper a numerical comparison is presented between two special purpose solvers developed for this task.

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.

Forced prey-predator oscillations

In this paper existence and stability of subharmonic solutions of the Volterra-Verhulst equations with a periodic coefficient are analyzed by the method of Urabe. The study supports the view that the observed 4- and 10-yr cycles of prey-predator systems are due to seasonal fluctuations.

The sparse-grid combination technique applied to time-dependent advection problems

In the numerical technique considered in this paper, time-stepping is performed on a set of semi-coarsened space grids. At given time levels the solutions on the different space grids are combined to obtain the asymptotic convergence of a single, fine uniform grid. We present error estimates for the two-dimensional, spatially constant-coefficient model problem and discuss numerical examples. A spatially variable-coefficient problem (Molenkamp–Crowley test) is used to assess the practical merits of the technique. The combination technique is shown to be more efficient than the single-grid approach, yet for the Molenkamp–Crowley test, standard Richardson extrapolation is still more efficient than the combination technique. However, parallelization is expected to significantly improve the combination techniques performance.

Solution of time-dependent advection-diffusion problems with the sparse-grid combination technique and a Rosenbrock solver

Abstract In the current paper the efficiency of the sparse-grid combination tech- nique applied to time-dependent advection-diffusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe±cient problem and a system of 2D non- linear Burgers equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers equations this gain in efficiency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned.