A new family of semi-implicit Finite Volume / Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of non-linear hyperbolic partial differential equations on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is achieved using the semi-implicit IMEX Runge-Kutta schemes, and the novel schemes are proven to be asymptotic preserving and well-balanced. As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier-Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids

Large scale finite element solvers for the large eddy simulation of incompressible turbulent flows

Premi extraordinari doctorat UPC curs 2015-2016, àmbit d’Enginyeria CivilIn this thesis we have developed a path towards large scale Finite Element simulations of turbulent incompressible flows. We have assessed the performance of residual-based variational multiscale (VMS) methods for the large eddy simulation (LES) of turbulent incompressible flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. We show that VMS thought as an implicit LES model can be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we also consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we define a symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is also assessed in this thesis. Furthermore, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. Precisely, the symmetric projection stabilization approach is suitable for segregated Runge-Kutta time integration schemes. This combination, together with the use of block-preconditioning techniques, lead to elasticity-type and Laplacian-type problems that can be optimally preconditioned using the balancing domain decomposition by constraints preconditioners. The weak scalability of this formulation have been demonstrated in this document. Additionally, we also contemplate the weak imposition of the Dirichlet boundary conditions for wall-bounded turbulent flows. Four well known problems have been mainly considered for the numerical experiments: the decay of homogeneous isotropic turbulence, the Taylor-Green vortex problem, the turbulent flow in a channel and the turbulent flow around an airfoil.En aquesta tesi s'han desenvolupat diferents algoritmes per la simulació a gran escala de fluxos turbulents incompressibles mitjançant el mètode dels Elements Finits. En primer lloc s'ha avaluat el comportament dels mètodes de multiescala variacional (VMS) basats en el residu, per la simulació de grans vòrtexs (LES) de fluxos turbulents. S'han considerat diferents models VMS tenint en compte diferents aproximacions de les subescales, que inclouen tant subescales estàtiques o dinàmiques, una definicó lineal o nolineal, i diferents seleccions de l'espai de les subescales. S'ha demostrat que els mètodes VMS pensats com a models LES poden ser una alternativa als models basats en la física del problema. Aquest tipus de mètode normalment es combina amb l'ús de parelles de velocitat i pressió amb igual ordre d'interpolació. En aquest treball, també s'ha considerat un enfocament diferent, basat en l'ús d'elements inf-sup estables conjuntament amb estabilització del terme convectiu. Amb aquest objectiu, s'ha definit un mètode d'estabilització amb projecció simètrica del terme convectiu mitjançant una descomposició ortogonal de les subescales. En aquesta tesi també s'ha valorat la precisió i eficiència d'aquest mètode comparat amb mètodes basats en el residu fent servir interpolacions amb igual ordre per velocitats i pressions. A més, s'ha proposat un esquema d'integració en temps basat en els mètodes de Runge-Kutta que té dues propietats destacables. En primer lloc, el càlcul de la velocitat i la pressió es segrega al nivell de la integració temporal, sense la necessitat d'introduir tècniques de fraccionament del pas de temps. En segon lloc, els esquemes segregats de Runge-Kutta proposats, mantenen el mateix ordre de precisió tant per les velocitats com per les pressions. Precisament, els mètodes d'estabilització amb projecció simètrica són adequats per ser integrats en temps mitjançant esquemes segregats de Runge-Kutta. Aquesta combinació, juntament amb l'ús de tècniques de precondicionament en blocs, dóna lloc a problemes tipus elasticitat i Laplacià que poden ser òptimament precondicionats fent servir els anomenats \textit{balancing domain decomposition by constraints preconditioners}. La escalabilitat dèbil d'aquesta formulació s'ha demostrat en aquest document. Adicionalment, també s'ha contemplat la imposició de forma dèbil de les condicions de contorn de Dirichlet en problemes de fluxos turbulents delimitats per parets. En aquesta tesi principalment s'han considerat quatre problemes ben coneguts per fer els experiments numèrics: el decaïment de turbulència isotròpica i homogènia, el problema del vòrtex de Taylor-Green, el flux turbulent en un canal i el flux turbulent al voltant d'una ala.Award-winningPostprint (published version

A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward difference formulae (BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection-diffusion equations