5,997 research outputs found
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows
In this article we present a novel staggered semi-implicit hybrid
finite-volume/finite-element (FV/FE) method for the resolution of weakly
compressible flows in two and three space dimensions. The pressure-based
methodology introduced in Berm\'udez et al. 2014 and Busto et al. 2018 for
viscous incompressible flows is extended here to solve the compressible
Navier-Stokes equations. Instead of considering the classical system including
the energy conservation equation, we replace it by the pressure evolution
equation written in non-conservative form. To ease the discretization of
complex spatial domains, face-type unstructured staggered meshes are
considered. A projection method allows the decoupling of the computation of the
density and linear momentum variables from the pressure. Then, an explicit
finite volume scheme is used for the resolution of the transport diffusion
equations on the dual mesh, whereas the pressure system is solved implicitly by
using continuous finite elements defined on the primal simplex mesh.
Consequently, the CFL stability condition depends only on the flow velocity,
avoiding the severe time restrictions that might be imposed by the sound
velocity in the weakly compressible regime. High order of accuracy in space and
time of the transport diffusion stage is attained using a local ADER (LADER)
methodology. Moreover, also the CVC Kolgan-type second order in space and first
order in time scheme is considered. To prevent spurious oscillations in the
presence of shocks, an ENO-based reconstruction, the minmod limiter or the
Barth-Jespersen limiter are employed. To show the validity and robustness of
our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are
analysed, showing a good agreement with available exact solutions and numerical
reference data from low Mach numbers, up to Mach numbers of the order of unity
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks
Recent work has introduced a simple numerical method for solving partial
differential equations (PDEs) with deep neural networks (DNNs). This paper
reviews and extends the method while applying it to analyze one of the most
fundamental features in numerical PDEs and nonlinear analysis: irregular
solutions. First, the Sod shock tube solution to compressible Euler equations
is discussed, analyzed, and then compared to conventional finite element and
finite volume methods. These methods are extended to consider performance
improvements and simultaneous parameter space exploration. Next, a shock
solution to compressible magnetohydrodynamics (MHD) is solved for, and used in
a scenario where experimental data is utilized to enhance a PDE system that is
\emph{a priori} insufficient to validate against the observed/experimental
data. This is accomplished by enriching the model PDE system with source terms
and using supervised training on synthetic experimental data. The resulting DNN
framework for PDEs seems to demonstrate almost fantastical ease of system
prototyping, natural integration of large data sets (be they synthetic or
experimental), all while simultaneously enabling single-pass exploration of the
entire parameter space.Comment: 21 pages, 14 figures, 3 table
High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
In this paper we present an efficient discretization method for the solution
of the unsteady incompressible Navier-Stokes equations based on a high order
(Hybrid) Discontinuous Galerkin formulation. The crucial component for the
efficiency of the discretization method is the disctinction between stiff
linear parts and less stiff non-linear parts with respect to their temporal and
spatial treatment. Exploiting the flexibility of operator-splitting time
integration schemes we combine two spatial discretizations which are tailored
for two simpler sub-problems: a corresponding hyperbolic transport problem and
an unsteady Stokes problem. For the hyperbolic transport problem a spatial
discretization with an Upwind Discontinuous Galerkin method and an explicit
treatment in the time integration scheme is rather natural and allows for an
efficient implementation. The treatment of the Stokes part involves the
solution of linear systems. In this case a discretization with Hybrid
Discontinuous Galerkin methods is better suited. We consider such a
discretization for the Stokes part with two important features:
H(div)-conforming finite elements to garantuee exactly divergence-free velocity
solutions and a projection operator which reduces the number of globally
coupled unknowns. We present the method, discuss implementational aspects and
demonstrate the performance on two and three dimensional benchmark problems.Comment: 21 pages, 3 figures, 4 tabl
A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics
In this paper we present a novel pressure-based semi-implicit finite volume
solver for the equations of compressible ideal, viscous and resistive
magnetohydrodynamics (MHD). The new method is conservative for mass, momentum
and total energy and in multiple space dimensions it is constructed in such a
way as to respect the divergence-free condition of the magnetic field exactly,
also in the presence of resistive effects. This is possible via the use of
multi-dimensional Riemann solvers on an appropriately staggered grid for the
time evolution of the magnetic field and a double curl formulation of the
resistive terms. The new semi-implicit method for the MHD equations proposed
here discretizes all terms related to the pressure in the momentum equation and
the total energy equation implicitly, making again use of a properly staggered
grid for pressure and velocity. The time step of the scheme is restricted by a
CFL condition based only on the fluid velocity and the Alfv\'en wave speed and
is not based on the speed of the magnetosonic waves. Our new method is
particularly well-suited for low Mach number flows and for the incompressible
limit of the MHD equations, for which it is well-known that explicit
density-based Godunov-type finite volume solvers become increasingly
inefficient and inaccurate due to the increasingly stringent CFL condition and
the wrong scaling of the numerical viscosity in the incompressible limit. We
show a relevant MHD test problem in the low Mach number regime where the new
semi-implicit algorithm is a factor of 50 faster than a traditional explicit
finite volume method, which is a very significant gain in terms of
computational efficiency. However, our numerical results confirm that our new
method performs well also for classical MHD test cases with strong shocks. In
this sense our new scheme is a true all Mach number flow solver.Comment: 26 pages, 12 figures,1 tabl
Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting
In this paper we present a novel arbitrary high order accurate discontinuous
Galerkin (DG) finite element method on space-time adaptive Cartesian meshes
(AMR) for hyperbolic conservation laws in multiple space dimensions, using a
high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}.
Notoriously, the original DG method produces strong oscillations in the
presence of discontinuous solutions and several types of limiters have been
introduced over the years to cope with this problem. Following the innovative
idea recently proposed in \cite{Dumbser2014}, the discrete solution within the
troubled cells is \textit{recomputed} by scattering the DG polynomial at the
previous time step onto a suitable number of sub-cells along each direction.
Relying on the robustness of classical finite volume WENO schemes, the sub-cell
averages are recomputed and then gathered back into the DG polynomials over the
main grid. In this paper this approach is implemented for the first time within
a space-time adaptive AMR framework in two and three space dimensions, after
assuring the proper averaging and projection between sub-cells that belong to
different levels of refinement. The combination of the sub-cell resolution with
the advantages of AMR allows for an unprecedented ability in resolving even the
finest details in the dynamics of the fluid. The spectacular resolution
properties of the new scheme have been shown through a wide number of test
cases performed in two and in three space dimensions, both for the Euler
equations of compressible gas dynamics and for the magnetohydrodynamics (MHD)
equations.Comment: Computers and Fluids 118 (2015) 204-22
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
In this article we present a new class of high order accurate
Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for
solving nonlinear hyperbolic systems of conservation laws on moving two
dimensional unstructured triangular meshes. A WENO reconstruction algorithm is
used to achieve high order accuracy in space and a high order one-step time
discretization is achieved by using the local space-time Galerkin predictor.
For that purpose, a new element--local weak formulation of the governing PDE is
adopted on moving space--time elements. The space-time basis and test functions
are obtained considering Lagrange interpolation polynomials passing through a
predefined set of nodes. Moreover, a polynomial mapping defined by the same
local space-time basis functions as the weak solution of the PDE is used to map
the moving physical space-time element onto a space-time reference element. To
maintain algorithmic simplicity, the final ALE one-step finite volume scheme
uses moving triangular meshes with straight edges. This is possible in the ALE
framework, which allows a local mesh velocity that is different from the local
fluid velocity. We present numerical convergence rates for the schemes
presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler
equations of compressible gas dynamics.Comment: Accepted by "Communications in Computational Physics
Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids
In this paper two new families of arbitrary high order accurate spectral DG
finite element methods are derived on staggered Cartesian grids for the
solution of the inc.NS equations in two and three space dimensions. Pressure
and velocity are expressed in the form of piecewise polynomials along different
meshes. While the pressure is defined on the control volumes of the main grid,
the velocity components are defined on a spatially staggered mesh. In the first
family, h.o. of accuracy is achieved only in space, while a simple
semi-implicit time discretization is derived for the pressure gradient in the
momentum equation. The resulting linear system for the pressure is symmetric
and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D)
and can be solved very efficiently by means of a classical matrix-free
conjugate gradient method. The use of a preconditioner was not necessary. This
is a rather unique feature among existing implicit DG schemes for the NS
equations. In order to avoid a stability restriction due to the viscous terms,
the latter are discretized implicitly. The second family of staggered DG
schemes achieves h.o. of accuracy also in time by expressing the numerical
solution in terms of piecewise space-time polynomials. In order to circumvent
the low order of accuracy of the adopted fractional stepping, a simple
iterative Picard procedure is introduced. In this manner, the symmetry and
positive definiteness of the pressure system are not compromised. The resulting
algorithm is stable, computationally very efficient, and at the same time
arbitrary h.o. accurate in both space and time. The new numerical method has
been thoroughly validated for approximation polynomials of degree up to N=11,
using a large set of non-trivial test problems in two and three space
dimensions, for which either analytical, numerical or experimental reference
solutions exist.Comment: 46 pages, 15 figures, 4 table
High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE)
finite volume scheme on unstructured triangular meshes that is high order
accurate in space and time and that also allows for time-accurate local time
stepping (LTS). The new scheme uses the following basic ingredients: a high
order WENO reconstruction in space on unstructured meshes, an element-local
high-order accurate space-time Galerkin predictor that performs the time
evolution of the reconstructed polynomials within each element, the computation
of numerical ALE fluxes at the moving element interfaces through approximate
Riemann solvers, and a one-step finite volume scheme for the time update which
is directly based on the integral form of the conservation equations in
space-time. The inclusion of the LTS algorithm requires a number of crucial
extensions, such as a proper scheduling criterion for the time update of each
element and for each node; a virtual projection of the elements contained in
the reconstruction stencils of the element that has to perform the WENO
reconstruction; and the proper computation of the fluxes through the space-time
boundary surfaces that will inevitably contain hanging nodes in time due to the
LTS algorithm. We have validated our new unstructured Lagrangian LTS approach
over a wide sample of test cases solving the Euler equations of compressible
gasdynamics in two space dimensions, including shock tube problems, cylindrical
explosion problems, as well as specific tests typically adopted in Lagrangian
calculations, such as the Kidder and the Saltzman problem. When compared to the
traditional global time stepping (GTS) method, the newly proposed LTS algorithm
allows to reduce the number of element updates in a given simulation by a
factor that may depend on the complexity of the dynamics, but which can be as
large as 4.7.Comment: 31 pages, 13 figure
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