High-resolution Wave Propagation Method for Stratified Flows

The implementation of the multidimensional f-waves Riemann solver for the time-dependent, three-dimensional, nonhydrostatic, meso- and microscale atmospheric flows is described in detail. The Riemann solver employs flux-based wave decomposition (f-waves) for the calculation of Godunov fluxes in which the flux differences are written directly as the linear combination of the right eigenvectors of the hyperbolic system. The scheme incorporates the source term due to gravity without introducing discretization errors which is an important property in the context of atmospheric flows. The resulting flow solver is conservative, accurate, stable, and well-balanced. The implementation of the solver is evaluated using benchmark test cases for atmospheric dynamics

A family of well-balanced WENO and TENO schemes for atmospheric flows

We herein present a novel methodology to construct very high order well-balanced schemes for the computation of the Euler equations with gravitational source term, with application to numerical weather prediction (NWP). The proposed method is based on augmented Riemann solvers, which allow preserving the exact equilibrium between fluxes and source terms at cell interfaces. In particular, the augmented HLL solver (HLLS) is considered. Different spatial reconstruction methods can be used to ensure a high order of accuracy in space (e.g. WENO, TENO, linear reconstruction), being the TENO reconstruction the preferred method in this work. To the knowledge of the authors, the TENO method has not been applied to NWP before, although it has been extensively used by the computational fluid dynamics community in recent years. Therefore, we offer a thorough assessment of the TENO method to evidence its suitability for NWP considering some benchmark cases which involve inertia and gravity waves as well as convective processes. The TENO method offers an enhanced behavior when dealing with turbulent flows and underresolved solutions, where the traditional WENO scheme proves to be more diffusive. The proposed methodology, based on the HLLS solver in combination with a very high-order discretization, allows carrying out the simulation of meso- and micro-scale atmospheric flows in an implicit Large Eddy Simulation manner. Due to the HLLS solver, the isothermal, adiabatic and constant Brunt-Väisälä frequency hydrostatic equilibrium states are preserved with machine accuracy

Local preconditioning and variational multiscale stabilization for Euler compressible steady flow

This paper introduces a preconditioned variational multiscale stabilization (P-VMS) method for compressible flows. In this introductory paper we focus on inviscid flow and steady state problems. The Euler equations are solved on fully unstructured grids and discretized using the finite element method. The P-VMS method can be decomposed in three parts. First, a local preconditioner is applied to the continuous equations to reduce the stiffness while covering a wide range of Mach numbers. Then, the resulting preconditioned system is discretized in space using finite elements and stabilized with a variational multiscale stabilization method adapted for the preconditioned equations. In this paper, the solution is advanced in time using a fully explicit time discretization, although P-VMS is general and can be applied to fully implicit solvers. The proposed method is assessed by comparing convergence and accuracy of the solutions between the non-preconditioned and preconditioned cases, in particular for van Leer-Lee-Roe’s and Choi-Merkle’s preconditioners, in some selected examples covering a large range of Mach numbers.Peer ReviewedPostprint (author's final draft

A variational multiscale stabilized finite element method for the solution of the Euler equations of nonhydrostatic stratified flows

We present a compressible version of the variational multiscale stabilization (VMS) method applied to the finite element (FE) solution of the Euler equations for nonhydrostatic stratified flows. This paper is meant to verify how the algorithm performs when solving problems in the framework of nonhydrostatic atmospheric dynamics. This effort is justified by the previously observed good performance of VMS and by the advantages that a compact Galerkin formulation offers on massively parallel architectures - a paradigm for both computational fluid dynamics (CFD) and numerical weather prediction (NWP) practitioners. We also propose a simple technique to construct a well-balanced approximation of the dominant hydrostatics that, if not properly discretized, may cause unacceptable vertical oscillations. This is a relevant problem in NWP, especially in the proximity of steep topography. To evaluate the performance of the method for stratified environments, six standard 2D and two 3D test cases are selected. Of these, two admit a semi-analytic solution, while the remaining six are non-steady and non-linear thermal problems with dominant buoyancy effects that challenge the algorithm in terms of stability. © 2012 Elsevier Inc..M.M. was supported by the Catalan autonomous government through an FIE scholarship. Part of this work was supported through Grants CGL2008-02818 and CSD00C-06-08924 of the Spanish Ministry of Science and TechnologyPeer Reviewe

A variational multiscale stabilized finite element method for the solution of the Euler equations of nonhydrostatic stratified flows

We present a compressible version of the variational multiscale stabilization (VMS) method applied to the finite element (FE) solution of the Euler equations for nonhydrostatic stratified flows. This paper is meant to verify how the algorithm performs when solving problems in the framework of nonhydrostatic atmospheric dynamics. This effort is justified by the previously observed good performance of VMS and by the advantages that a compact Galerkin formulation offers on massively parallel architectures – a paradigm for both computational fluid dynamics (CFD) and numerical weather prediction (NWP) practitioners. We also propose a simple technique to construct a well-balanced approximation of the dominant hydrostatics that, if not properly discretized, may cause unacceptable vertical oscillations. This is a relevant problem in NWP, especially in the proximity of steep topography. To evaluate the performance of the method for stratified environments, six standard 2D and two 3D test cases are selected. Of these, two admit a semi-analytic solution, while the remaining six are non-steady and non-linear thermal problems with dominant buoyancy effects that challenge the algorithm in terms of stability.Peer Reviewe

A variational multiscale stabilized finite element method for the solution of the Euler equations of nonhydrostatic stratified flows

We present a compressible version of the variational multiscale stabilization (VMS) method applied to the finite element (FE) solution of the Euler equations for nonhydrostatic stratified flows. This paper is meant to verify how the algorithm performs when solving problems in the framework of nonhydrostatic atmospheric dynamics. This effort is justified by the previously observed good performance of VMS and by the advantages that a compact Galerkin formulation offers on massively parallel architectures – a paradigm for both computational fluid dynamics (CFD) and numerical weather prediction (NWP) practitioners. We also propose a simple technique to construct a well-balanced approximation of the dominant hydrostatics that, if not properly discretized, may cause unacceptable vertical oscillations. This is a relevant problem in NWP, especially in the proximity of steep topography. To evaluate the performance of the method for stratified environments, six standard 2D and two 3D test cases are selected. Of these, two admit a semi-analytic solution, while the remaining six are non-steady and non-linear thermal problems with dominant buoyancy effects that challenge the algorithm in terms of stability.Peer Reviewe

Variational multiscale stabilization of finite and spectral elements for dry and moist atmospheric problems

In this thesis the finite and spectral element methods (FEM and SEM, respectively) applied to problems in atmospheric simulations are explored through the common thread of Variational Multiscale Stabilization (VMS). This effort is justified by three main reasons. (i) the recognized need for new solvers that can efficiently execute on massively parallel architectures ¿a spreading framework in most fields of computational physics in which numerical weather prediction (NWP) occupies a prominent position. Element-based methods (e.g. FEM, SEM, discontinuous Galerkin) have important advantages in parallel code development; (ii) the inherent flexibility of these methods with respect to the geometry of the grid makes them a great candidate for dynamically adaptive atmospheric codes; and (iii) the localized diffusion provided by VMS represents an improvement in the accurate solution of multi-physics problems where artificial diffusion may fail. Its application to atmospheric simulations is a novel approach within a field of research that is still open. First, FEM and VMS are described and derived for the solution of stratified low Mach number flows in the context of dry atmospheric dynamics. The validity of the method to simulate stratified flows is assessed using standard two- and three-dimensional benchmarks accepted by NWP practitioners. The problems include thermal and gravity driven simulations. It will be shown that stability is retained in the regimes of interest and a numerical comparison against results from the the literature will be discussed. Second, the ability of VMS to stabilize the FEM solution of advection-dominated problems (i.e. Euler and transport equations) is taken further by the implementation of VMS as a stabilizing tool for high-order spectral elements with advection-diffusion problems. To the author¿s knowledge, this is an original contribution to the literature of high order spectral elements involved with transport in the atmosphere. The problem of monotonicity-preserving high order methods is addressed by combining VMS-stabilized SEM with a discontinuity capturing technique. This is an alternative to classical filters to treat the Gibbs oscillations that characterize high-order schemes. To conclude, a microphysics scheme is implemented within the finite element Euler solver, as a first step toward realistic atmospheric simulations. Kessler microphysics is used to simulate the formation of warm, precipitating clouds. This last part combines the solution of the Euler equations for stratified flows with the solution of a system of transport equations for three classes of water: water vapor, cloud water, and rain. The method is verified using idealized two- and three-dimensional storm simulations.En esta tesis los métodos de elementos finitos y espectrales (FEM - finite element method y SEM- spectral element method, respectivamente), aplicados a los problemas de simulaciones atmosféricas, se exploran a través del método de estabilización conocidocomo Variational Multiscale Stabilization (VMS). Tres razones fundamentales justifican este esfuerzo: (i) la necesidad de tener nuevos métodos de solución de las ecuaciones diferenciales a las derivadas parciales usando máquinas paralelas de gran escala –un entorno en expansión en muchos campos de la mecánica computacional, dentro de la cual la predicción numérica de la dinámica atmosférica (NWP-numerical weather prediction)representa una aplicación importante. Métodos del tipo basado en elementos(por ejemplo, FEM, SEM, Galerkin discontinuo) presentan grandes ventajas en el desarrollo de códigos paralelos; (ii) la flexibilidad intrínseca de tales métodos respecto a lageometría de la malla computacional hace que esos métodos sean los candidatos ideales para códigos atmosféricos con mallas adaptativas; y (iii) la difusión localizada que VMSintroduce representa una mejora en las soluciones de problemas con física compleja en los cuales la difusión artificial clásica no funcionaría. La aplicación de FEM o SEM con VMS a problemas de simulaciones atmosféricas es una estrategia innovadora en un campo de investigación abierto. En primera instancia, FEM y VMS vienen descritos y derivados para la solución de flujos estratificados a bajo número de Mach en el contexto de la dinámica atmosférica. La validez del método para simular flujos estratificados es verificada por medio de test estándar aceptado por la comunidad dentro del campo deNWP. Los test incluyen simulaciones de flujos térmicos con efectos de gravedad. Se demostrará que la estabilidad del método numérico se preserva dentro de los regímenesde interés y se discutirá una comparación numérica de los resultados frente a aquellos hallados en la literatura. En segunda instancia, la capacidad de VMS para estabilizarmétodos FEM en problemas de advección dominante (i.e. ecuaciones de Euler y ecuaciones de transporte) se implementa además en la solución a elementos espectrales de alto orden en problemas de advección-difusión. Hasta donde el autor sabe, esta es una contribución original a la literatura de métodos basados en elementos espectrales en problemas de transporte atmosférico. El problema de monotonicidad con métodos de alto orden es tratado mediante la combinación de SEM+VMS con una técnica de shockcapturing para un mejor tratamiento de las discontinuidades. Esta es una alternativa a los filtros que normalmente se aplican a SEM para eilminar las oscilaciones de Gibbsque caracterizan las soluciones de alto orden. Como último punto, se implementa un esquema de humedad acoplado con el núcleo en elementos finitos; este es un primer paso hacia simulaciones atmosféricas más realistas. La microfísica de Kessler se emplea para simular la formación de nubes y tormentas cálidas (warm clouds: no permite la formación de hielo). Esta última parte combina la solución de las ecuaciones de Eulerpara atmósferas estratificadas con la solución de un sistema de ecuaciones de transporte de tres estados de agua: vapor, nubes y lluvia. La calidad del método es verificadautilizando simulaciones de tormenta en dos y tres dimensiones