A parameter-robust iterative method for Stokes–Darcy problems retaining local mass conservation

We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analysis, the performance of the iterative scheme is shown to be robust with respect to material and discretization parameters. By construction, the scheme is applicable to a wide range of locally conservative discretization schemes and we consider explicit examples in the framework of Mixed Finite Element methods. Finally, the theoretical results are confirmed with the use of numerical experiments.

Numerical Simulation for 2D Sedimentation Model by an Upwind Discontinuous Galerkin Procedure

We reformulate the mathematical model for the 2D sedimentation in an estuary as a coupled nonlinear differential system. Combining the mass-conservation character of the discontinuous Galerkin method and the jump-capturing property of Lesaint-Raviart upwind technique, we design an upwind discontinuous Galerkin finite element method, which obeys the local mass conservation and possesses good stability. Our theoretical analysis shows that there exists a unique solution to the numerical procedure and the discrete solution permits O(hk+Δt) convergence rate. Numerical experiments are conducted to verify our theoretical findings. This may provide a theoretical principle for better understanding of the mechanism and morphological characters of sedimentation at estuaries.

Inherently mass-conservative version of the semi-Lagrangian absolute vorticity (SL-AV) atmospheric model dynamical core

The semi-Lagrangian absolute vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at the Hydrometeorological Centre of Russia. The distinct feature of the SL-AV dynamical core is the semi-implicit, semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. A semi-implicit, semi-Lagrangian approach allows for long time steps but violates the global and local mass conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no a posteriori mass-fixing algorithm is applied. However, the global mass-fixing algorithms degrade the local mass conservation. The new inherently mass-conservative version of the SL-AV model dynamical core presented here ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume, semi-Lagrangian discretization for a continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). Numerical experiments show that the new version of the SL-AV dynamical core presented combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain-induced Rossby-wave test and baroclinic instability test for the mass-conservative dynamical core are found to be in agreement with the results available in the literature.

Inherently mass-conservative version of the semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model dynamical core

The semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at Hydrometeorological centre of Russia. The distinct feature of SL-AV dynamical core is the semi-implicit semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. Semi-implicit semi-Lagrangian approach allows for long time steps while violates the global and local mass-conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no aposteriori mass-fixing algorithms are applied. However, the global mass-fixing algorithms degrade the local mass conservation. The inherently mass-conservative version of SL-AV model dynamical core presented in the article ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume semi-Lagrangian discretization for continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). The numerical experiments show that the presented new version of SL-AV dynamical core combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain induced Rossby wave test and baroclinic instability test for mass-conservative dynamical core are found to be in agreement with the results available in literature.

Trajectory-Tracking Scheme in Lagrangian Form for Solving Linear Advection Problems: Interface Spatial Discretization

A previous Lagrangian linear advection scheme (trajectory-tracking scheme) is modified to achieve local mass conservation in this paper, which is more favorable to climate modeling. The discretized tracer parcels are volumes with interfaces instead of centroids. In 2D problems, the parcels are polygons and the interfaces are described by polygonal edges with a finite number. Because polygons will deform under the background wind field, a curvature-guard algorithm (CGA) is devised to retain accurate representation of the deformed interfaces among parcels. The tracer mass carried by parcels is mapped onto the regular latitude–longitude mesh by a first-order conservative remapping algorithm that can handle concave polygons. Several standard test cases have been carried out to show the effectiveness of the new scheme.