4,543 research outputs found
Convergence of a cell-centered finite volume discretization for linear elasticity
We show convergence of a cell-centered finite volume discretization for
linear elasticity. The discretization, termed the MPSA method, was recently
proposed in the context of geological applications, where cell-centered
variables are often preferred. Our analysis utilizes a hybrid variational
formulation, which has previously been used to analyze finite volume
discretizations for the scalar diffusion equation. The current analysis
deviates significantly from previous in three respects. First, additional
stabilization leads to a more complex saddle-point problem. Secondly, a
discrete Korn's inequality has to be established for the global discretization.
Finally, robustness with respect to the Poisson ratio is analyzed. The
stability and convergence results presented herein provide the first rigorous
justification of the applicability of cell-centered finite volume methods to
problems in linear elasticity
Finite volume schemes for diffusion equations: introduction to and review of modern methods
We present Finite Volume methods for diffusion equations on generic meshes,
that received important coverage in the last decade or so. After introducing
the main ideas and construction principles of the methods, we review some
literature results, focusing on two important properties of schemes (discrete
versions of well-known properties of the continuous equation): coercivity and
minimum-maximum principles. Coercivity ensures the stability of the method as
well as its convergence under assumptions compatible with real-world
applications, whereas minimum-maximum principles are crucial in case of strong
anisotropy to obtain physically meaningful approximate solutions
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
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
Compatible finite element spaces for geophysical fluid dynamics
Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
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