16 research outputs found
Contributions to the numerical solution of heterogeneous fluid mechanics models
A high order projection hybrid finite volume – finite element method is developed to solve incompressible and compressible low Mach number flows. Furthermore, turbulent regimes are also considered thanks to the k–ε model. The unidimensional advection-diffusion-reaction equation is used to construct, analyze and assess high order finite volume schemes. Two families of methods are studied: Kolgan-type schemes and ADER methodology. A modification of the last one is proposed providing a new numerical method called Local ADER. The designed method is extended to solve the transport-diffusion stage of the three-dimensional projection method. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the schemes and to assess the performance of the method on several realistic test problems
A new thermodynamically compatible finite volume scheme for magnetohydrodynamics
In this paper we propose a novel thermodynamically compatible finite volume
scheme for the numerical solution of the equations of magnetohydrodynamics
(MHD) in one and two space dimensions. As shown by Godunov in 1972, the MHD
system can be written as overdetermined symmetric hyperbolic and
thermodynamically compatible (SHTC) system. More precisely, the MHD equations
are symmetric hyperbolic in the sense of Friedrichs and satisfy the first and
second principles of thermodynamics. In a more recent work on SHTC systems,
\cite{Rom1998}, the entropy density is a primary evolution variable, and total
energy conservation can be shown to be a \textit{consequence} that is obtained
after a judicious linear combination of all other evolution equations. The
objective of this paper is to mimic the SHTC framework also on the discrete
level by directly discretizing the \textit{entropy inequality}, instead of the
total energy conservation law, while total energy conservation is obtained via
an appropriate linear combination as a \textit{consequence} of the
thermodynamically compatible discretization of all other evolution equations.
As such, the proposed finite volume scheme satisfies a discrete cell entropy
inequality \textit{by construction} and can be proven to be nonlinearly stable
in the energy norm due to the discrete energy conservation. In multiple space
dimensions the divergence-free condition of the magnetic field is taken into
account via a new thermodynamically compatible generalized Lagrangian
multiplier (GLM) divergence cleaning approach. The fundamental properties of
the scheme proposed in this paper are mathematically rigorously proven. The new
method is applied to some standard MHD benchmark problems in one and two space
dimensions, obtaining good results in all cases
A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers
We present a novel staggered semi-implicit hybrid FV/FE method for the
numerical solution of the shallow water equations at all Froude numbers on
unstructured meshes. A semi-discretization in time of the conservative
Saint-Venant equations with bottom friction terms leads to its decomposition
into a first order hyperbolic subsystem containing the nonlinear convective
term and a second order wave equation for the pressure. For the spatial
discretization of the free surface elevation an unstructured mesh of triangular
simplex elements is considered, whereas a dual grid of the edge-type is
employed for the computation of the depth-averaged momentum vector. The first
stage of the proposed algorithm consists in the solution of the nonlinear
convective subsystem using an explicit Godunov-type FV method on the staggered
grid. Next, a classical continuous FE scheme provides the free surface
elevation at the vertex of the primal mesh. The semi-implicit strategy followed
circumvents the contribution of the surface wave celerity to the CFL-type time
step restriction making the proposed algorithm well-suited for low Froude
number flows. The conservative formulation of the governing equations also
allows the discretization of high Froude number flows with shock waves. As
such, the new hybrid FV/FE scheme is able to deal simultaneously with both,
subcritical as well as supercritical flows. Besides, the algorithm is well
balanced by construction. The accuracy of the overall methodology is studied
numerically and the C-property is proven theoretically and validated via
numerical experiments. The solution of several Riemann problems attests the
robustness of the new method to deal also with flows containing bores and
discontinuities. Finally, a 3D dam break problem over a dry bottom is studied
and our numerical results are successfully compared with numerical reference
solutions and experimental data
A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics
We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism.
A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density
A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics
Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGWe introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism. A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.Agencia Estatal de Investigación | Ref. PID2021-122625OB-I0
An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations
We present a novel second-order semi-implicit hybrid finite volume / finite
element (FV/FE) scheme for the numerical solution of the incompressible and
weakly compressible Navier-Stokes equations on moving unstructured meshes using
an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a
suitable splitting of the governing PDE into subsystems and employs staggered
grids, where the pressure is defined on the primal simplex mesh, while the
velocity and the remaining flow quantities are defined on an edge-based
staggered dual mesh. The key idea of the scheme is to discretize the nonlinear
convective and viscous terms using an explicit FV scheme that employs the
space-time divergence form of the governing equations on moving space-time
control volumes. For the convective terms, an ALE extension of the Ducros flux
on moving meshes is introduced, which is kinetic energy preserving and stable
in the energy norm when adding suitable numerical dissipation terms. Finally,
the pressure equation of the Navier-Stokes system is solved on the new mesh
configuration using a continuous FE method, with Lagrange
elements.
The ALE hybrid FV/FE method is applied to several incompressible test
problems ranging from non-hydrostatic free surface flows over a rising bubble
to flows over an oscillating cylinder and an oscillating ellipse. Via the
simulation of a circular explosion problem on a moving mesh, we show that the
scheme applied to the weakly compressible Navier-Stokes equations is able to
capture weak shock waves, rarefactions and moving contact discontinuities. We
show that our method is particularly efficient for the simulation of weakly
compressible flows in the low Mach number limit, compared to a fully explicit
ALE schem
On thermodynamically compatible finite volume schemes for continuum mechanics
In this paper we present a new family of semi-discrete and fully-discrete
finite volume schemes for overdetermined, hyperbolic and thermodynamically
compatible PDE systems. In the following we will denote these methods as HTC
schemes. In particular, we consider the Euler equations of compressible
gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model
of continuum mechanics, which, at the aid of suitable relaxation source terms,
is able to describe nonlinear elasto-plastic solids at large deformations as
well as viscous fluids as two special cases of a more general first order
hyperbolic model of continuum mechanics. The main novelty of the schemes
presented in this paper lies in the fact that we solve the \textit{entropy
inequality} as a primary evolution equation rather than the usual total energy
conservation law. Instead, total energy conservation is achieved as a mere
consequence of a thermodynamically compatible discretization of all the other
equations. For this, we first construct a discrete framework for the
compressible Euler equations that mimics the continuous framework of Godunov's
seminal paper \textit{An interesting class of quasilinear systems} of 1961
\textit{exactly} at the discrete level. All other terms in the governing
equations of the more general GPR model, including non-conservative products,
are judiciously discretized in order to achieve discrete thermodynamic
compatibility, with the exact conservation of total energy density as a direct
consequence of all the other equations. As a result, the HTC schemes proposed
in this paper are provably marginally stable in the energy norm and satisfy a
discrete entropy inequality by construction. We show some computational results
obtained with HTC schemes in one and two space dimensions, considering both the
fluid limit as well as the solid limit of the governing partial differential
equations
A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies
We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN)
model for the description of dispersive water waves. Contrarily to the
classical Boussinesq-type models, it contains only first order derivatives,
thus allowing to overcome the numerical difficulties and the severe time step
restrictions arising from higher order terms. The proposed model reduces to the
original SGN model when an artificial sound speed tends to infinity. Moreover,
it is endowed with an energy conservation law from which the energy
conservation law associated with the original SGN model is retrieved when the
artificial sound speed goes to infinity. The governing partial differential
equations are then solved at the aid of high order ADER discontinuous Galerkin
finite element schemes. The new model has been successfully validated against
numerical and experimental results, for both flat and non-flat bottom. For
bottom topographies with large variations, the new model proposed in this paper
provides more accurate results with respect to the hyperbolic reformulation of
the SGN model with the mild bottom approximation recently proposed in "C.
Escalante, M. Dumbser and M.J. Castro. An efficient hyperbolic relaxation
system for dispersive non-hydrostatic water waves and its solution with high
order discontinuous Galerkin schemes, Journal of Computational Physics 2018"