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

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A variational multiscale stabilized finite element method for the solution of the Euler equations of nonhydrostatic stratified flows
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  A Variational Multiscale Stabilized Finite Element Method for the Solutionof the Euler Equations of Nonhydrostatic Stratified Flows Simone Marras a, ∗ , Margarida Moragues a , Mariano Vázquez a,b, ∗∗ , Oriol Jorba a , GuillaumeHouzeaux a a  Barcelona Supercomputing Center, Spain  b IIIA-CSIC. Barcelona, Spain  Abstract We present a compressible version of the Variational Multiscale Stabilization (VMS) method ap-plied 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 frame-work of nonhydrostatic atmospheric dynamics. This effort is justified by the previously observedgood performance of VMS and by the advantages that a compact Galerkin formulation offers onmassively parallel architectures —a paradigm for both Computational Fluid Dynamics (CFD) andNumerical Weather Prediction (NWP) practitioners. We also propose a simple technique to con-struct 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 theproximity of steep topography. To evaluate the performance of the method for stratified environ-ments, six standard 2D and two 3D test cases are selected. Of these, two admit a semi-analyticsolution, while the remaining six are non-steady and non-linear thermal problems with dominantbuoyancy effects that challenge the algorithm in terms of stability. Keywords:  Variational Multiscale Stabilization; Finite Element Method; Nonhydrostatic Flows;Euler Equations. ∗ simone.marras@bsc.es ∗∗ mariano.vazquez@bsc.es Preprint submitted to Journal of Computational Physics September 10, 2012  *ManuscriptClick here to view linked References  1. Introduction The past decade has seen an important growth in the development of faster and cheaper super-computers for high-performance computing (HPC). This trend is such that many research groupscan now enhance their models in terms of efficiency and accuracy through higher resolution, by lim-iting the time of computation. Among many fields of computational mechanics, this is even moreso in the context of atmospheric simulations, where the wall-clock time is a discriminant metricin the selection of the underlying algorithm. It has been widely proved that, to effectively see animprovement in computation on these architectures, the numerical algorithm at hand should havecertain characteristics. Of these, being local by construction, which implies the smallness of thecommunication foot-print, is maybe the most important. On these grounds, we propose a FiniteElement (FE) scheme with Variational Multiscale Stabilization (VMS) for the solution of the fullycompressible Euler equations of stratified flows, having it proved great parallelization efficiency onmassively parallel architectures [1]. The use of Finite Element schemes in atmospheric simulationsbegan four decades ago with [2] and [3] in the 60s, continued in the 70s (e.g.[4, 5]), and was followed by an extensive production of articles in the 80s and 90s with, e.g., [6, 7, 8], who set the foun- dations of the operational  Global Environmental Multiscale   (GEM) model [9, 10] of the Canadian Meteorological Center & Meteorological Research Branch (CMC-MRB). In the UK, [11] used finiteelements for the vertical discretization of a semi-Lagrangian transport scheme and introduced it inthe operational version of the European Centre for Medium-Range Weather Forecasts (ECMWF)global spectral model, with great improvement with respect to the finite difference (FD) versionof the code. At the same time, in [12] the British Met Office committed to a FE approach forthe treatment of the vertical atmosphere within their global and regional models. More Galerkin-type models appear since the beginning of 2000 in the domain of Geophysical fluid dynamics: in[13, 14, 15, 16, 17, 18], different variational formulations mostly based on Spectral Elements (SE) and Discontinuous Galerkin (DG) techniques are employed to solve the shallow water problem,hyperbolic systems on the sphere, or the Navier-Stokes and Euler equations in non-hydrostaticmesoscale and global modeling with unstructured grids. Similarly and with dynamic mesh adap-tivity in mind, Numerical Weather Prediction (NWP) Finite Volume (FV) softwares such as theJapanese  Nonhydrostatic Icosahedral Atmospheric Model   (NICAM) on global domains ([19, 20]) and the American  Operational Multiscale Environmental model with Grid Adaptivity   (OMEGA)2  [21] are further examples of new trends in NWP as alternatives to FD methods. More recently,examples of element-based models are the SE/DG  Nonhydrsotatic Unified Model for the Atmo-sphere   (NUMA), whose linear scalability up to 12,288 CPUs using MPI was shown in [22], and the SE  Community Earth System Model   (CESM) with scalability shown up to 160,000 CPUs in [23]. Müller et al. are working on adaptive grid refinement and built an adaptive solver for atmosphericproblems proposed in [24]. In 2010, [25] presented their results from an edge-based finite element solver built on a fully unstructured grid. In this context, the finite element algorithm proposed hereis, at our knowledge, the first continuous Galerkin method with VMS stabilization applied to strat-ified nonhydrostatic flows. No special numerical treatment or special assumption on the governingequations are considered for the low Mach regime typical of atmospheric dynamics. Nevertheless,this algorithm can treat a widespread range of flow regimes including very low Mach number flows.When convection dominates, because of the centered nature of a straight finite element dis-cretization instabilities form until the solution eventually explodes. These instabilities can betreated by means of different stabilization procedures. The  Streamline Upwind Petrov-Galerkin  (SUPG) [26, 27], the  Galerkin/Least-Squares   (GLS) [28], Galerkin methods with bubble functions [29, 30, 31], or sub-grid projection methods [32] are some of the most used stabilization techniques for finite elements. Due to its high efficiency, robustness, and validity at all Mach regimes, in thispaper we focus on the Variational Multiscale approach pioneered by [33, 34]. To our knowledge, VMS methods for compressible flows today only appear in [35, 36, 37, 38] for Computational Fluid Dynamics (CFD) simulations where compressibility effects are large (i.e.  M   ≥  0 . 36, where  M  indicates the flow Mach number). In the framework of atmospheric modeling, VMS was recentlyapplied by [39] to solve the advection-diffusion equation using high-order spectral elements. Here we use a simplified version of the method presented in [38].The main advantages of the method can be summarized as follows. The algorithm is halo-free,which is one of the biggest advantages for efficient parallelization regardless of the order of accuracyof the scheme [16]. The only information that is needed by each element is that of the shared nodes between neighboring elements. This makes the method fully local and highly parallelizable.Although quadrilateral elements are used, the algorithm is fully unstructured and does not relyon specific characteristics of the grid. Furthermore, no dimensional splitting is used in that boththe horizontal and vertical directions are not distinguished in the discretization process. In other3  words, the Euler equations are solved by the same numerical method in all space dimensions. Thegreat advantage of this is that the code becomes totally free from the geometry of the grid. Thisapproach is classical in CFD and was first applied in NWP by [21, 40, 41] using finite volumes, finite differences, and high order spectral elements, respectively.The paper is organized as follows. Section 2 describes the set of equations used throughout.The numerical method of solution is described in Section 3 and is followed by a description of well-balanced solutions in Section 4. In Sections 5 and 6 the algorithm is tested against standard benchmarks for atmospheric simulation in two and three dimensions respectively. Conclusions arereported in Section 7. 2. Governing equations Let  d  = 3 be the space dimension. Given a bounded domain Ω ⊂ R d and a time interval (0 ,t f  ), t f   ∈ R + , we write the Euler equations of atmospheric flows [42] with no Coriolis effects as: ∂  q ∂t  + L ( q ) =  f  ( q ) ,  (1)where  q  is the vector of the unknowns,  L  is the differential operator, and  f   is the source vector.We have: q  =  ρU V W θ  ,  L ( q ) =  ∇· U ∇·  U U ρ  +  p e x  ∇·  U V ρ  +  p e y  ∇·  U W ρ  +  p e z  U ρ  ·∇ θ  ,  f  ( q ) =  000 − ρg 0  , where  ∇  =   ∂ ∂x ,  ∂ ∂y ,  ∂ ∂z  , and  e x  = (1 , 0 , 0) T ,  e y  = (0 , 1 , 0) T , and  e z  = (0 , 0 , 1) T are the unitaryvectors in the space directions  x ,  y , and  z , respectively. Superscript T indicates the transposeoperator. Density  ρ , momentum  U  = ( U,V,W  ) T , and potential temperature  θ  are functions of space  x  = ( x,y,z ) and time  t . We also define the velocity  u  =  U /ρ  = ( u,v,w ). The acceleration4  of gravity, of modulus  g  , acts along the vertical direction. It is understood that in two dimensions,the  y -direction disappears. System (1) is closed by the state law for pressure  p  =  p 0  Rρθ p 0  γ  ,  (2)where  γ   =  c  p /c v , being  c  p  and  c v  the coefficients of specific heat at constant pressure and volume,respectively,  R  =  c  p  − c v  is the constant of perfect gases, and  p 0  = 10 5 Pa  is the surface pres-sure. The problem consists in finding  q ( x ,t) such that Eq. (1) with proper boundary and initialconditions is verified  ∀ ( x ,t )  ∈  Ω × (0 ,t f  ). 3. Numerical formulation Set (1) is discretized in space using Finite Elements stabilized by the Variational Multiscalemethod and in time using an explicit Finite Difference scheme. 3.1. Finite Elements: Weak Form  Given a polyhedral approximation Ω h of Ω, let P  h = { K  i } i =1 ,...,n el  be its finite element partition,where  K  i ⊂  Ω h are  n el  conforming quadrilaterals of characteristic length  h i . In this paper,  h i corresponds to the shortest edge of the element. We consider the trial and basis functions space W  h associated with  P  h as the space generated by the Lagrange polynomials of order one. Let {  p k } k =1 ,...,n nodes  be the nodes of the grid and  ψ hk  the Lagrange polynomial associated with node  p k ,then  { ψ hk } k =1 ,...,n nodes  is a basis for  W  h . We project Eq. (1) onto  W  h by the  L 2  scalar product,where  L 2  is the space of square-integrable real-valued functions. We hence obtain the weak (orvariational) form of Eq. (1):   Ω h ψ h ∂  q ∂t d Ω h +   Ω h ψ h L ( q ) d Ω h =   Ω h ψ h f  ( q ) d Ω h ∀ ψ h ∈  W  h .  (3)We define  q h as the projection of   q  onto  W  h and expands it as q h ( x ,t ) = n nodes  k =1 ψ hk ( x ) q hk ( t ) ,  (4)5
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