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The FORTRAN program RKC is intended for the time integration of parabolic partial differential equations discretized by the method of lines. It is based on a family of Runge-Kutta-Chebyshev formulas with a stability bound that is quadratic in the number of stages. Remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size. Moreover, they make it possible to evaluate the explicit formulas in just a few vectors of storage. These characteristics of the program make it especially attractive for problems in several spatial variables. RKC is compared to the BDF solver VODPK on two test problems in three spatial variables.

This paper examines a class of explicit finite-difference advection schemes derived along the method of lines. An important application field is large-scale atmospheric transport. The paper therefore focuses on the demand of positivity. For the spatial discretization, attention is confined to conservative schemes using five points per direction. The fourth-order central scheme and the family of K-schemes, comprising the second-order central, the second-order upwind, and the third-order upwind biased, are studied. Positivity is enforced through flux limiting. It is concluded that the limited third-order upwind discretization is the best candidate from the four examined. For the time integration attention is confined to a number of explicit Runge-Kutta methods of orders two up to four. With regard to the demand of positivity, these integration methods turn out to behave almost equally and no best method could be identified. '~' 1995 Academic Press, Inc. l. INTRODUCTIONThe subject of this paper is the numerical solution of the partial differential equation for linear advection of a scalar quantity w in an arbitrary velocity field u, given byLinear advection is an important (classical) problem in computational fluid dynamics and has been the subject of numerous investigations. The central theme is how to approximate the advection term \7 · (uw), such that the resulting errors in both phase and amplitude are minimized and the computational cost is still affordable. An important application we have in mind concerns atmospheric transport of chemical species. Then w represents a concentration or density and u a wind field. In addition to the usual accuracy and efficiency requirements, here the main consideration is that the transported concentrations must remain positive, because in actual applications also chemical reactions are modeled for which positivity is a prerequisite for avoiding non-physical chemical instabilities. We emphasize *The research reported helongs to the projects EUSMOG and CIRK which are carried out in cooperation with the Air Laboratory of the RIVM-Tbe Dutch National Institute of Public Health and Environmental Protection. The RIVM is acknowledged for financial support. 35that the demand of positivity is important and that it severely restricts the choice of method, as it is essentially equivalent to the demand of avoiding numerical under-and overshoots in regions of strong variation.The research objective of this paper is to ex.amine a class of positive, finite-difference advection schemes which we consider promising for atmospheric transport applications and to select from this class the best possible candidate. We hereby follow the method-of-lines approach which means that the spatial discretization and temporal integration are considered separately.For the spatial discretization we confine ourselves to stencils using five points per (spatial) direction. We consider this a good starting point since a 5-point stencil is computationally attractive for the following reasons. First, a 5-point stenc...

Abstract. 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 J1tcobian 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.

Dedicated to Peter van der Houwen for his numerous contributions in the field of numerical integration of differential equations Summary. The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length f3 proportional to s 2 . The method can be applied with s arbitrarily large, which is an attractive feature due to the proportionality of f3 with s 2 . The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of lst and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for any s and independent of the stiffness of the problem.Numerical examples are given to illustrate the theoretical results.