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Semi-implicit formulations of the Navier-Stokes equations: applications to nonhydrostatic atmospheric modeling
- SIAM JOURNAL ON SCIENTIFIC COMPUTING. IN REVIEW
"... We present semi-implicit (IMEX) formulations of the compressible Navier-Stokes 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 semi-implicit (IMEX) formulations of the compressible Navier-Stokes 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 semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge-Kutta 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 semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including: inertia gravity waves, rising thermal bubbles (i.e., Rayleigh-Taylor instabilities), density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of non-reflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and non-reflecting boundary conditions. It is shown that the non-reflecting boundary conditions exert a strong impact on the accuracy and efficiency of the models.
A comparison of two shallow water models with non-conforming adaptive grids.
, 2007
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High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model
- Internat. J. Numer. Methods Fluids
"... SUMMARY We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders o ..."
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Cited by 7 (1 self)
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SUMMARY We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order 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 element-type area integrals, but also the finite volume-type 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 time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order 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 time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order 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 high-order 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 high-order (HO) time-integrators 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 low-order 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 Gauss-Lobatto spectral elements
- SIAM J. Sci. Comput
"... Abstract. The classical overlapping Schwarz algorithm is here extended to the triangular/tetra-hedral 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/tetra-hedral 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 par-titioning 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 Three-Dimensional Cubed-Sphere Grids: A Parallel Solution-Adaptive Simulation Framework
, 2012
"... An accurate, efficient and scalable parallel, cubed-sphere 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 well-known Euler and ideal magnetohydrodynamics (M ..."
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Cited by 1 (1 self)
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An accurate, efficient and scalable parallel, cubed-sphere 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 well-known Euler and ideal magnetohydrodynamics (MHD) equations. Our main contribution compared to existing cubed-sphere-based algorithms lies in the design of a cubed-sphere framework that is based on a genuine and consistent multi-block implementation, leading to flux calculations, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubed-sphere grid. This results in the first fully adaptive three-dimensional cubed-sphere 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 cubed-sphere grid, multi-dimensional k-exact 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 inter-block communication, flux calculation, adaptivity and parallelization. A second-order Godunov-type finite-volume scheme is employed to discretize the governing equations on the multi-block hexahedral mesh defining the cubed-sphere grid. An overview of the algorithm parallelization and of the flexible block-based
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 Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional 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 Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional 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 two-dimensional 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 stability-preserving third order Runge-Kutta 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 high-order 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 steady-state 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 time-integration method limits the model error. Thus, the time step ∆t is restricted by both, the CFL-condition 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 Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional 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 Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R3, are locally represented in terms of spherical triangu-lar coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional 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 stability-preserving third order Runge-Kutta 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 high-order 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.
unknown title
, 805
"... A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling ..."
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A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling
