Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

[1] A novel stratospheric chemical data assimilation system has been developed and applied to Environmental Satellite Michelson Interferometer for Passive Atmospheric Sounding (ENVISAT/MIPAS) data, aiming to combine the sophistication of the four-dimensional variational (4D-var) technique with flow-dependent covariance modeling and also to improve numerical performance. The system is tailored for operational stratospheric chemistry state monitoring. The atmospheric model of the assimilation system includes a state-of-the-art stratospheric chemistry transport module along with its adjoint and the German weather service's global meteorological forecast model, providing meteorological parameters. Both models share the same grid and same advection time step, to ensure dynamic consistency without spatial and temporal interpolation errors. A notable numerical efficiency gain is obtained through an icosahedral grid. As a novel feature in stratospheric variational data assimilation a special focus was placed on an optimal spatial exploitation of satellite data by dynamic formulation of the forecast error covariance matrix, providing potential vorticity controlled anisotropic and inhomogeneous influence radii. In this first part of the study the design and numerical features of the data assimilation system is presented, along with analyses of two case studies and a posteriori validation. Assimilated data include retrievals of O 3 , CH 4 , N 2 O, NO 2 , HNO 3 , and water vapor. The analyses are compared with independent observations provided by Stratospheric Aerosol and Gas Experiment II (SAGE II) and Halogen Occultation Experiment (HALOE) retrievals. It was found that there are marked improvements for both analyses and assimilation based forecasts when compared with control model runs without any data ingestion.

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