222 research outputs found
Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws
It is known that HLL-type schemes are more dissipative than schemes based on
characteristic decompositions. However, HLL-type methods offer greater
flexibility to large systems of hyperbolic conservation laws because the
eigenstructure of the flux Jacobian is not needed. We demonstrate in the
present work that several HLL-type Riemann solvers are provably entropy stable.
Further, we provide convex combinations of standard dissipation terms to create
hybrid HLL-type methods that have less dissipation while retaining entropy
stability. The decrease in dissipation is demonstrated for the ideal MHD
equations with a numerical example.Comment: 6 page
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
A multidimensional grid-adaptive relativistic magnetofluid code
A robust second order, shock-capturing numerical scheme for multi-dimensional
special relativistic magnetohydrodynamics on computational domains with
adaptive mesh refinement is presented. The base solver is a total variation
diminishing Lax-Friedrichs scheme in a finite volume setting and is combined
with a diffusive approach for controlling magnetic monopole errors. The
consistency between the primitive and conservative variables is ensured at all
limited reconstructions and the spatial part of the four velocity is used as a
primitive variable. Demonstrative relativistic examples are shown to validate
the implementation. We recover known exact solutions to relativistic MHD
Riemann problems, and simulate the shock-dominated long term evolution of
Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication
A direct primitive variable recovery scheme for hyperbolic conservative equations: the case of relativistic hydrodynamics
In this article we develop a Primitive Variable Recovery Scheme (PVRS) to
solve any system of coupled differential conservative equations. This method
obtains directly the primitive variables applying the chain rule to the time
term of the conservative equations. With this, a traditional finite volume
method for the flux is applied in order avoid violation of both, the entropy
and "Rankine-Hugoniot" jump conditions. The time evolution is then computed
using a forward finite difference scheme. This numerical technique evades the
recovery of the primitive vector by solving an algebraic system of equations as
it is often used and so, it generalises standard techniques to solve these kind
of coupled systems. The article is presented bearing in mind special
relativistic hydrodynamic numerical schemes with an added pedagogical view in
the appendix section in order to easily comprehend the PVRS. We present the
convergence of the method for standard shock-tube problems of special
relativistic hydrodynamics and a graphical visualisation of the errors using
the fluctuations of the numerical values with respect to exact analytic
solutions. The PVRS circumvents the sometimes arduous computation that arises
from standard numerical methods techniques, which obtain the desired primitive
vector solution through an algebraic polynomial of the charges.Comment: 19 pages, 6 figures, 2 tables. Accepted for publication in PLOS ON
Hybridisable Discontinuous Galerkin Formulation of Compressible Flows
This work presents a review of high-order hybridisable discontinuous Galerkin
(HDG) methods in the context of compressible flows. Moreover, an original
unified framework for the derivation of Riemann solvers in hybridised
formulations is proposed. This framework includes, for the first time in an HDG
context, the HLL and HLLEM Riemann solvers as well as the traditional
Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their
superiority with respect to Roe in supersonic cases due to their positivity
preserving properties. In addition, HLLEM specifically outstands in the
approximation of boundary layers because of its shear preservation, which
confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A
comprehensive set of relevant numerical benchmarks of viscous and inviscid
compressible flows is presented. The test cases are used to evaluate the
competitiveness of the resulting high-order HDG scheme with the aforementioned
Riemann solvers and equipped with a shock treatment technique based on
artificial viscosity.Comment: 60 pages, 31 figures. arXiv admin note: substantial text overlap with
arXiv:1912.0004
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