5,985 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 direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization
A deterministic method is proposed for solving the Boltzmann equation. The
method employs a Galerkin discretization of the velocity space and adopts, as
trial and test functions, the collocation basis functions based on weights and
roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or
full-range Hermite polynomials depending whether or not the distribution
function presents a discontinuity in the velocity space. The resulting
semi-discrete Boltzmann equation is in the form of a system of hyperbolic
partial differential equations whose solution can be obtained by standard
numerical approaches. The spectral rate of convergence of the results in the
velocity space is shown by solving the spatially uniform homogeneous relaxation
to equilibrium of Maxwell molecules. As an application, the two-dimensional
cavity flow of a gas composed by hard-sphere molecules is studied for different
Knudsen and Mach numbers. Although computationally demanding, the proposed
method turns out to be an effective tool for studying low-speed slightly
rarefied gas flows
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
Augmented resolution of linear hyperbolic systems under nonconservative form
Hyperbolic systems under nonconservative form arise in numerous applications
modeling physical processes, for example from the relaxation of more general
equations (e.g. with dissipative terms). This paper reviews an existing class
of augmented Roe schemes and discusses their application to linear
nonconservative hyperbolic systems with source terms. We extend existing
augmented methods by redefining them within a common framework which uses a
geometric reinterpretation of source terms. This results in intrinsically
well-balanced numerical discretizations. We discuss two equivalent
formulations: (1) a nonconservative approach and (2) a conservative
reformulation of the problem. The equilibrium properties of the schemes are
examined and the conditions for the preservation of the well-balanced property
are provided. Transient and steady state test cases for linear acoustics and
hyperbolic heat equations are presented. A complete set of benchmark problems
with analytical solution, including transient and steady situations with
discontinuities in the medium properties, are presented and used to assess the
equilibrium properties of the schemes. It is shown that the proposed schemes
satisfy the expected equilibrium and convergence properties
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