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Semiimplicit formulations of the NavierStokes equations: applications to nonhydrostatic atmospheric modeling
 SIAM JOURNAL ON SCIENTIFIC COMPUTING. IN REVIEW
"... We present semiimplicit (IMEX) formulations of the compressible NavierStokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur ..."
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We present semiimplicit (IMEX) formulations of the compressible NavierStokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semiimplicit timeintegration method for these equations. Finally, we compare all five equations and compare the semiimplicit formulations of these equations both using the Schur and No Schur forms against an explicit RungeKutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semiimplicit formulations are faster than the explicit RungeKutta method for all the tests studied especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semiimplicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semiimplicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including: inertia gravity waves, rising thermal bubbles (i.e., RayleighTaylor instabilities), density current (i.e., KelvinHelmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semiimplicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: noflux (reflecting) and nonreflecting boundary conditions. It is shown that the nonreflecting boundary conditions exert a strong impact on the accuracy and efficiency of the models.
A comparison of two shallow water models with nonconforming adaptive grids.
, 2007
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Highorder semiimplicit timeintegrators for a triangular discontinuous Galerkin oceanic shallow water model
 Internat. J. Numer. Methods Fluids
"... SUMMARY We extend the explicit in time highorder triangular discontinuous Galerkin (DG) method to semiimplicit (SI) and then apply the algorithm to the twodimensional oceanic shallow water equations; we implement highorder SI timeintegrators using the backward difference formulas from orders o ..."
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SUMMARY We extend the explicit in time highorder triangular discontinuous Galerkin (DG) method to semiimplicit (SI) and then apply the algorithm to the twodimensional oceanic shallow water equations; we implement highorder SI timeintegrators using the backward difference formulas from orders one to six. The reason for changing the timeintegration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for highorder DG methods. Changing the timeintegration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite elementtype area integrals, but also the finite volumetype boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI timeintegrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: highorder accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of timeexplicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new highorder SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new highorder SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use highorder (HO) timeintegrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with loworder time discretizations. Published in
Geometric error of finite volume schemes for conservation laws on evolving surfaces. arXiv preprint arXiv:1301.1287
, 2013
"... Abstract. This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of R3. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schem ..."
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Abstract. This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of R3. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal. 1.
Overlapping Schwarz methods for Fekete and GaussLobatto spectral elements
 SIAM J. Sci. Comput
"... Abstract. The classical overlapping Schwarz algorithm is here extended to the triangular/tetrahedral spectral element (TSEM) discretization of elliptic problems. This discretization, based on Fekete nodes, is a generalization to nontensorial elements of the tensorial Gauss–Lobatto–Legendre quadrila ..."
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Abstract. The classical overlapping Schwarz algorithm is here extended to the triangular/tetrahedral spectral element (TSEM) discretization of elliptic problems. This discretization, based on Fekete nodes, is a generalization to nontensorial elements of the tensorial Gauss–Lobatto–Legendre quadrilateral spectral elements (QSEM). The overlapping Schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains, solving local problems on these subdomains, and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh. The overlap size is generous, i.e., one element wide, in the TSEM case, while it is minimal or variable in the QSEM case. The results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains N and the spectral degree p in case of generous overlap; otherwise it depends inversely on the overlap size. The proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains.
Hyperbolic Conservation Laws on ThreeDimensional CubedSphere Grids: A Parallel SolutionAdaptive Simulation Framework
, 2012
"... An accurate, efficient and scalable parallel, cubedsphere grid numerical framework is described for solution of hyperbolic conservation laws in domains between two concentric spheres. The particular conservation laws considered in this work are the wellknown Euler and ideal magnetohydrodynamics (M ..."
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An accurate, efficient and scalable parallel, cubedsphere grid numerical framework is described for solution of hyperbolic conservation laws in domains between two concentric spheres. The particular conservation laws considered in this work are the wellknown Euler and ideal magnetohydrodynamics (MHD) equations. Our main contribution compared to existing cubedspherebased algorithms lies in the design of a cubedsphere framework that is based on a genuine and consistent multiblock implementation, leading to flux calculations, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubedsphere grid. This results in the first fully adaptive threedimensional cubedsphere grid framework, with excellent parallel scalability on thousands of computing cores. Crucial elements of the proposed approach are: unstructured connectivity of the six grid root blocks that correspond to the six sectors of the cubedsphere grid, multidimensional kexact reconstruction that automatically takes into account information from neighbouring cells isotropically and is able to automatically handle varying stencil size, and adaptive division of the solution blocks into smaller blocks of varying spatial resolution that are all treated exactly equally for interblock communication, flux calculation, adaptivity and parallelization. A secondorder Godunovtype finitevolume scheme is employed to discretize the governing equations on the multiblock hexahedral mesh defining the cubedsphere grid. An overview of the algorithm parallelization and of the flexible blockbased
A Discontiuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates
"... Abstract A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a RungeKutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a twodimensional surface in R 3 , are locally repr ..."
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Abstract A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a RungeKutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a twodimensional surface in R 3 , are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a twodimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation using a Rusanov numerical flux. A strong stabilitypreserving third order RungeKutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires highorder quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, exept in a preprocessing step. The validation of the atmospheric model has been done considering steadystate and unsteady analytical solutions of the nonlinear shallow water equations. Experimental convergence was observed and the order of convergence k + 1 was achieved. Furthermore, the article presents a numerical experiment, for which the third order timeintegration method limits the model error. Thus, the time step ∆t is restricted by both, the CFLcondition and accuracy demands. As a second step of validation, the model could reproduce a known barotropic instability caused by a small initial perturbation of a geostrophic balanced jet stream. Conservation of mass was shown up to machine precision and energy conservation converges with decreasing grid resolution and increasing polynomial order k.
A Discontiuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates
"... A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a RungeKutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a twodimensional surface in R3, are locally represented in t ..."
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A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a RungeKutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a twodimensional surface in R3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a twodimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation using a Rusanov numerical flux. A strong stabilitypreserving third order RungeKutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires highorder quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, exept in a preprocessing step.
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, 805
"... A mixed discontinuous/continuous finite element pair for shallowwater ocean modelling ..."
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A mixed discontinuous/continuous finite element pair for shallowwater ocean modelling