411 research outputs found
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
Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall
We prove the incompressible limit of compressible ideal magnetohydrodynamic
(MHD) flows in a reference domain where the magnetic field is tangential to the
boundary. Unlike the case of transversal magnetic fields, the linearized
problem of our case is not well-posed in standard Sobolev space , while the incompressible problem is still well-posed in . The key
observation to overcome the difficulty is a hidden structure contributed by
Lorentz force in the vorticity analysis, which reveals that one should trade
one normal derivative for two tangential derivatives together with a gain of
Mach number weight . Thus, the energy functional should be
defined by using the anisotropic Sobolev space . The weights of Mach
number should be carefully chosen according to the number of tangential
derivatives, such that the energy estimates are uniform in Mach number.
Besides, part of the proof is parallel to the study of compressible water
waves, so our result opens the possibility to study the incompressible limit of
free-boundary problems in ideal MHD such as current-vortex sheets.Comment: 28 page
Recommended from our members
Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)
This small collaborative workshop brought together
experts from the Sino-German project working in the field of advanced numerical methods for
hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the
convergence of numerical methods and proper solution concepts were addressed as well
Recommended from our members
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion and related topics.
These topics included both inviscid and viscous fluids in two and three dimensions. Some talks addressed
aspects of fluid dynamics such as the construction of wild weak solutions, compressible shock formation,
inviscid limit and behavior of boundary layers, as well as both polymer/fluid and structure/fluid interaction
Discontinuous Galerkin Spectral Element Methods for Astrophysical Flows in Multi-physics Applications
In engineering applications, discontinuous Galerkin methods (DG) have been proven to be a powerful and flexible class of high order methods for problems in computational fluid dynamics. However, the potential benefits of DG for applications in astrophysical contexts is still relatively unexplored in its entirety. To this day, a decent number of studies surveying DG for astrophysical flows have been conducted. But the adoption of DG by the astrophysics community is just beginning to gain traction and integration of DG into established, multi-physics simulation frameworks for comprehensive astrophysical modeling is still lacking. It is our firm believe, that the full potential of novel approaches for numerically solving the fluid equations only shows under the pressure of real-world simulations with all aspects of multi-physics, challenging flow configurations, resolution and runtime constraints, and efficiency metrics on high-performance systems involved. Thus, we see the pressing need to propel DG from the well-trodden path of cataloguing test results under "optimal" laboratory conditions towards the harsh and unforgiving environment of large-scale astrophysics simulations. Consequently, the core of this work is the development and deployment of a robust DG scheme solving the ideal magneto-hydrodynamics equations with multiple species on three-dimensional Cartesian grids with adaptive mesh refinement. We chose to implement DG within the venerable simulation framework FLASH, with a specific focus on multi-physics problems in astrophysics. This entails modifications of the vanilla DG scheme to make it fit seamlessly within FLASH in such a way that all other physics modules can be naturally coupled without additional implementation overhead. A key ingredient is that our DG scheme uses mean value data organized into blocks - the central data structure in FLASH. Having the opportunity to work on mean values, allows us to rely on a rock-solid, monotone Finite Volume (FV) scheme as "backup" whenever the high order DG method fails in cases when the flow gets too harsh. Finding ways to combine the two schemes in a fail-safe manner without loosing primary conservation while still maintaining high order accuracy for smooth, well-resolved flows involves a series of careful considerations, which we document in this thesis. The result of our work is a novel shock capturing scheme - a hybrid between FV and DG - with smooth transitions between low and high order fluxes according to solution smoothness estimators. We present extensive validations and test cases, specifically its interaction with multi-physics modules in FLASH such as (self-)gravity and radiative transfer. We also investigate the benefits and pitfalls of integrating end-to-end entropy stability into our numerical scheme, with special focus on highly compressible turbulent flows and shocks. Our implementation of DG in FLASH allows us to conduct preliminary yet comprehensive astrophysics simulations proving that our new solver is ready for assessments and investigations by the astrophysics community
Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence
We consider the closure problem for turbulence in the dry convective atmospheric boundary
layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large
plumes in the well mixed middle part up to the inversion that separates the CBL from the
stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF
approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02)
that additionally includes a term for background turbulence. Thus an exact solution is derived
and all higher order moments (HOMs) are explained by second order moments, correlation
coefficients and the skewness. The solution provides a proof of the extended universality
hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi-
normality of FOM). This refined hypothesis states that CBL turbulence can be considered as
result of a linear interpolation between the Gaussian and the very skewed turbulence regimes.
Although the extended universality hypothesis was confirmed by results of field
measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained
unexplained. These are now answered by the new model including the reasons of the
universality of the functional form of the HOMs, the significant scatter of the values of the
coefficients and the source of the magic of the linear interpolation. Finally, the closures
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predicted by the model are tested against measurements and LES data. Some of the other
issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area
coverage parameters of plumes (so called filling factors) with HOM will be discussed also
Rigorous results on conserved and dissipated quantities ideal MHD turbulence
We review recent mathematical results on the theory of ideal MHD turbulence. On the one hand, we explain a mathematical version of Taylor's conjecture on magnetic helicity conservation, both for simply and multiply connected domains. On the other hand, we describe how to prove the existence of weak solutions conserving magnetic helicity but dissipating cross helicity and energy in 3D Ideal MHD. Such solutions are bounded. In fact, we show that as soon as we are below the critical L-3 integrability for magnetic helicity conservation, there are weak solutions which do not preserve even magnetic helicity. These mathematical theorems rely on understanding the mathematical relaxation of MHD which is used as a model of the macroscopic behaviour of solutions of various nonlinear partial differential equations. Thus, on the one hand, we present results on the existence of weak solutions consistent with what is expected from experiments and numerical simulations, on the other hand, we show that below certain thresholds, there exist pathological solutions which should be excluded from physical grounds. It is still an outstanding open problem to find suitable admissibility conditions that are flexible enough to allow the existence of weak solutions but rigid enough to rule out physically unrealistic behaviour.Peer reviewe
Topics in Magnetohydrodynamics
To understand plasma physics intuitively one need to master the MHD behaviors. As sciences advance, gap between published textbooks and cutting-edge researches gradually develops. Connection from textbook knowledge to up-to-dated research results can often be tough. Review articles can help. This book contains eight topical review papers on MHD. For magnetically confined fusion one can find toroidal MHD theory for tokamaks, magnetic relaxation process in spheromaks, and the formation and stability of field-reversed configuration. In space plasma physics one can get solar spicules and X-ray jets physics, as well as general sub-fluid theory. For numerical methods one can find the implicit numerical methods for resistive MHD and the boundary control formalism. For low temperature plasma physics one can read theory for Newtonian and non-Newtonian fluids etc
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