Marco Restelli

A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier-Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling. We study three different forms of the governing equations using seven test cases. Three test cases involve flow over mountains which require the implementation of non-reflecting boundary conditions, while one test requires viscous terms (density current). Including viscous stresses into finite difference, finite element, or spectral element models poses no additional challenges; however, including these terms to either finite volume or discontinuous Galerkin models requires the introduction of additional machinery because these methods were originally designed for first-order operators. We use the local discontinuous Galerkin method to overcome this obstacle. The seven test cases show that all of our models yield good results. The main conclusion is that equation set 1 (non-conservation form) does not perform as well as sets 2 and 3 (conservation forms). For the density current (viscous), the SE and DG models using set 3 (mass and total energy) give less dissipative results than the other equation sets; based on these results we recommend set 3 for the development of future multiscale research codes. In addition, the fact that set 3 conserves both mass and energy up to machine precision motives us to pursue this equation set for the development of future mesoscale models. For the bubble and mountain tests, the DG models performed better. Based on these results and due to its conservation properties we recommend the DG method. In the worst case scenario, the DG models are 50% slower than the non-conservative SE models. In the best case scenario, the DG models are just as efficient as the conservative SE models.

A Hybridizable Discontinuous Galerkin Method for Steady-State Convection-Diffusion-Reaction Problems

In this article, we propose a novel discontinuous Galerkin method for convection-diffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; this renders it efficiently implementable and competitive with the main existing methods for these problems. The second is that, when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable, optimal convergence properties are obtained for both variables; this is in sharp contrast with all other discontinuous methods for this problem. The third is that the method exhibits superconvergence properties of the approximation to the scalar variable; this allows us to postprocess the approximation in an element-by-element fashion to obtain another approximation to the scalar variable which converges faster than the original one. In this paper, we focus on the efficient implementation of the method and on the validation of its computational performance. With this aim, e...

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Modeling

A discontinuous Galerkin (DG) finite element formulation is proposed for the solution of the compressible Navier-Stokes equations for a vertically stratified fluid, which are of interest in mesoscale nonhydrostatic atmospheric modeling. The resulting scheme naturally ensures conservation of mass, momentum, and energy. A semi-implicit time-integration approach is adopted to improve the efficiency of the scheme with respect to the explicit Runge-Kutta time integration strategies usually employed in the context of DG formulations. A method is also presented to reformulate the resulting linear system as a pseudo-Helmholtz problem. In doing this, we obtain a DG discretization closely related to those proposed for the solution of elliptic problems, and we show how to take advantage of the numerical integration rules (required in all DG methods for the area and flux integrals) to increase the efficiency of the solution algorithm. The resulting numerical formulation is then validated on a collection of classical two-dimensional test cases, including density driven flows and mountain wave simulations. The performance analysis shows that the semi-implicit method is, indeed, superior to explicit methods and that the pseudo-Helmholtz formulation yields further efficiency improvements.

Semi-Implicit Formulations of the Navier-Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

We present semi-implicit (implicit-explicit) 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, density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting 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 nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models.

A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows

A new, conservative semi-Lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form. The approach combines the accuracy and conservation properties of the Discontinuous Galerkin (DG) method with the computational efficiency and robustness of Semi-Lagrangian (SL) techniques. Unconditional stability in the von Neumann sense is proved for the proposed discretization in the one-dimensional case. A monotonization technique is then introduced, based on the Flux Corrected Transport approach. This yields a multi-dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard DG methods. The accuracy and stability of the method are further demonstrated by two-dimensional tracer advection tests in the case of incompressible flows. The comparison with results obtained by standard SL and DG methods highlights several advantages of the new technique.

A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations

A semi-implicit and semi-Lagrangian discontinuous Galerkin method for the shallow water equations is proposed, for applications to geophysical scale flows. A non conservative formulation of the advection equation is employed, in order to achieve a more treatable form of the linear system to be solved at each time step. The method is equipped with a simple p-adaptivity criterion, that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution. Numerical results show that the method captures well the main features of gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear test cases with analytic solutions. The accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree.

A locally p-adaptive approach for Large Eddy Simulation of compressible flows in a DG framework

Abstract We investigate the possibility of reducing the computational burden of LES models by employing local polynomial degree adaptivity in the framework of a high-order DG method. A novel degree adaptation technique especially featured to be effective for LES applications is proposed and its effectiveness is compared to that of other criteria already employed in the literature. The resulting locally adaptive approach allows to achieve significant reductions in computational cost of representative LES computations.

Finite element discretization of a Stokes-like model arising in plasma physics

Abstract We consider a time-dependent diffusion–reaction model for two vector unknowns, satisfying a divergence-free constraint, and the associated scalar Lagrange multiplier. The motivation for studying such a model is provided by a plasma physics problem arising in the modeling of nuclear fusion devices (Braginskii equations), where the two vector unknowns represent ion and electron velocities, the scalar unknown is the electrostatic potential and the divergence-free constraint reflects the physical assumption of quasi-neutrality. We first recast the problem in a form reminiscent of the standard Stokes problem, which allows us to recognize the importance of using a compatible discretization for the vector and scalar unknowns, then propose and analyze a stable finite element formulation. Following this, we address some peculiar geometrical aspects of the model, showing how they can be naturally dealt with within our formulation, and finally discuss a solution procedure for the resulting linear system based on the classical Uzawa algorithm. Some numerical experiments complete the paper.

Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau--Lifshitz--Gilbert equations, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. These extra elaborations are needed due to the fact that any crude time integration either does not keep the unit length at the nodes or does not satisfy an energy law. We will carry out a linear Euler--like time-stepping method...

Analysis of a Hybrid RANS/LES Model Using RANS Reconstruction

The Hybrid RANS/LES method proposed is based on the hybrid filter approach introduced by Germano in 2004. In this work, instead of using two explicit models respectively for LES and RANS, we use only a model for LES and we reconstruct the RANS field exploiting the properties of hybrid filter. The reconstruction is obtained from the resolved velocity and the LES subgrid stress tensor. The model has been implemented using a variational approach in a DG-FEM framework and tested for the turbulent channel flow test case. Different configurations for hybrid terms have been compared with DNS data and with pure LES results.