Lie group analysis and evolution of weak waves for certain hyperbolic system of partial differential equations

In the present thesis, we study the applications of Lie group theory to system of quasilinear hyperbolic partial differential equations (PDEs), which are governed by many physical phenomena and having various important physical significance in the real life. Our primary objective in this thesis is to identify the symmetries of system of PDEs in order to obtain certain classes of group invariant solutions. The investigations carried out in this thesis are confined to the applications of Lie group method to the system of quasilinear hyperbolic PDEs arising in magnetogasdynamics, two phase flows and other scientific fields. We organize the whole thesis into 7 chapters, described as follows. First chapter is introductory and deals with a short background history of Lie group of transformations and symmetries along with some of their important features which are of great importance in the work of proceeding chapters and the motivation behind our interest. In the second chapter, we obtain exact solutions to the quasilinear system of PDEs, describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities. The next chapter deals with system of PDEs, governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid in the presence of magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuity

Asymptotic limit analysis for numerical models of atmospheric frontogenesis

Accurate prediction of the future state of the atmosphere is important throughout society, ranging from the weather forecast in a few days time to modelling the effects of a changing climate over decades and generations. The equations which govern how the atmosphere evolves have long been known; these are the Navier-Stokes equations, the laws of thermodynamics and the equation of state. Unfortunately the nonlinearity of the equations prohibits analytic solutions, so simplified models of particular flow phenomena have historically been, and continue to be, used alongside numerical models of the full equations. In this thesis, the two-dimensional Eady model of shear-driven frontogenesis (the creation of atmospheric fronts) was used to investigate how errors made in a localised region can affect the global solution. Atmospheric fronts are the boundary of two different air masses, typically characterised by a sharp change in air temperature and wind direction. This occurs across a small length of O(10 km), whereas the extent of the front itself can be O(1000 km). Fronts are a prominent feature of mid-latitude weather systems and, despite their narrow width, are part of the large-scale, global solution. Any errors made locally in the treatment of fronts will therefore affect the global solution. This thesis uses the convergence of the Euler equations to the semigeostrophic equations, a simplified model which is representative of the large-scale flow, including fronts. The Euler equations were solved numerically using current operational techniques. It was shown that highly predictable solutions could be obtained, and the theoretical convergence rate maintained, even with the presence of near-discontinuous solutions given by intense fronts. Numerical solutions with successively increased resolution showed that the potential vorticity, which is a fundamental quantity in determining the large-scale, balanced flow, approached the semigeostrophic limit solution. Regions of negative potential vorticity, indicative of local areas of instability, were reduced at high resolution. In all cases, the width of the front reduced to the grid-scale. While qualitative features of the limit solution were reproduced, a stark contrast in amplitude was found. The results of this thesis were approximately half in amplitude of the limit solution. Some attempts were made at increasing the intensity of the front through spatial- and temporal-averaging. A scheme was proposed that conserves the potential vorticity within the Eady model.Open Acces

GIZMO: A New Class of Accurate, Mesh-Free Hydrodynamic Simulation Methods

We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, SPH, and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both smoothed-particle hydrodynamics (SPH) and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume 'overlap.' We implement and test a parallel, second-order version of the method with self-gravity & cosmological integration, in the code GIZMO: this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require 'artificial diffusion' terms; and allows the fluid elements to move with the flow so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages vs. SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced 'particle noise' & numerical viscosity, more accurate sub-sonic flow evolution, & sharp shock-capturing. Advantages vs. non-moving meshes include: automatic adaptivity, dramatically reduced advection errors & numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of 'grid alignment' effects. We can, for example, follow hundreds of orbits of gaseous disks, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize 'grid noise.' These differences are important for a range of astrophysical problems.Comment: 57 pages, 33 figures. MNRAS. A public version of the GIZMO code, user's guide, test problem setups, and movies are available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm

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

Hyperbolic Techniques in Modelling, Analysis and Numerics

Several research areas are flourishing on the roots of the breakthroughs in conservation laws that took place in the last two decades. The meeting played a key role in providing contacts among the different branches that are currently developing. All the invitees shared the same common background that consists of the analytical and numerical techniques for nonlinear hyperbolic balance laws. However, their fields of applications and their levels of abstraction are very diverse. The workshop was the unique opportunity to share ideas about analytical issues like the fine-structure of singular solutions or the validity of entropy solution concepts. It turned out that generalized hyperbolic techniques are able to handle the challenges posed by new applications. The design of efficient structure preserving methods turned out to be the major line of development in numerical analysis

Local-in-time structure-preserving finite-element schemes for the Euler-Poisson equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling, respectively. The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy. A detailed discussion of algorithmic details is given, as well as proofs of the claimed properties. We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme

Effective diffusion of scalar fields in a chaotic flow

Copyright © 2008 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluids 20 (2008) and may be found at http://link.aip.org/link/?PHFLE6/20/107103/1The advection of a tracer field in a fluid flow can create complex scalar structures and increase the effect of weak diffusion by orders of magnitude. One tool to quantify this is to measure the flux of scalar across contour lines of constant scalar. This gives a diffusion equation in area coordinates with an effective diffusion that depends on the structure of the scalar field and, in particular, takes large values when scalar contours become very extended. The present paper studies the properties of this effective diffusion using a mixture of analytical and numerical tools. First the presence of hyperbolic stationary points, that is, saddles, in the scalar concentration field is investigated analytically, and it is shown that these give rise to singular spikes in the effective diffusion. This is confirmed in numerical simulations in which complex scalar fields are generated using a time-periodic flow. Issues of numerical resolution are discussed and results are given on the dependence of the effective diffusion on grid resolution and discretization in area or scalar values. These simulations show complex dependence of the effective diffusion on time as saddle points appear and disappear in the scalar field. It is found that time averaging (in the presence of an additional scalar source term) removes this dependence to leave robust results for the effective diffusion

Modélisation hydrodynamique du schéma d'allumage par choc pour la fusion par confinement inertiel

The shock ignition concept in inertial confinement fusion uses an intense power spike at the end of an assembly laser pulse. the key feature of shock ignition are the generation of a high ablation pressure, the shock pressure amplification by at least a factor of a hundred in the cold fuel shell and the shock coupling to the hot-spot. in this theses, new semi-analytical hydrodynamic models are developed to describe the ignitor shock from its generation up to the moment of fuel ignition. A model is developed to describe a spherical concerging shock wave in a pre-heated hotspot. The self-similar solution developed by Guderley is perturbed over the shock Mach number Ms >>1. The first order correction accounts for the effects of the shock strength. An analytical ignition criterion is defined in terms of the shock strength ans th hot-spot areal density. The ignition threshold is higher when the initial Mach number of the shock is lower. A minimal shock pressure of 20 Gbar is needed when it enters the hot-spot. The shock dynamics in the imploding shell is the analyzed. The shock is propagating into a non inertial medium with a high radial pressure gradient and an averall pressure increase with time. The collision with a returning shock coming from the assembly phase enhances further the ignitor shock pressure. The analytica theory allows to des cribe the shock pressure and strength evolution in a typical shock ignition implosion. It is demonstrated that, in the case of the HiPER target design, a generation shock pressure near the ablation zone on the order of 300-400 Mbar is needed. An analysis of experiments on the strong shock generation performed on the OMEGA laser facility is presented. It is sown that a shock presssure close to 300 Mbar near the ablation zone has been reached with an absorbed laser intensity up to 2 x 10(15) W:cm-2 and a laser wavelength of 351 nm. This value is two times higher than the one expected from collisional laser absorption only. That significant pressure enhancement is explained by contribution of hot-electrons generated by non-linear laser/plasma interaction in the corona. The proposed analytical models allow to optimize the shock ignition scheme, including the inuence of the implosion parameters. Analytical, numerical and experimental results are mutualy consistent.Le schéma d'allumage par choc pour la fusion par confinement inertiel utilise une impulsion laser intense à la fin d'une phase d'assemblage de combustible. Les paramètres clefs de ce schéma sont la génération d'une haute pression d'ablation, l'amplification de la pression du choc généré par un facteur supérieur à cent et le couplage du choc avec le point chaud de la cible. Dans cette thèse, de nouveaux modèles semi-analytiques sont développés afin de décrire le choc d'allumage depuis sa génération jusqu'à l'allumage du combustible. Tout d'abord, un choc sphérique convergent dans le coeur pré-chauffé de la cible est décrit. Le modèle est obtenu par perturbation de la solution auto-semblable de Guderley en tenant compte du nombre de Mach du choc élevé mais fini. La correction d'ordre un tient compte de l'effet de la force du choc. Un critère d'allumage analytique est exprimé en fonction de la densité surfacique du point chaud et de la pression du choc d'allumage. Le seuil d'allumage est plus élevé pour un nombre de Mach faible. Il est montré que la pression minimale du choc, lorsqu'il entre dans le coeur de la cible, est de 20Gbar. La dynamique du choc dans la coquille en implosion est ensuite analysée. Le choc se propage dans un milieu non inertiel avec un fort gradient de pression et une augmentation temporelle générale de la pression. La pression du choc est amplifiée plus encore durant la collision avec une onde de choc divergente provenant de la phase d'assemblage. Les modèles analytiques développés permettent une description de la pression et de la force du choc dans une simulation typique de l'allumage par choc. Il est démontré que, dans le cas d'une cible HiPER, une pression initiale du choc de l'ordre de 300 Mbar dans la zone d'ablation est nécessaire. Il est proposé une analyse des expériences sur la génération de chocs forts avec l'installation laser OMEGA. Il est montré qu'une pression du choc proche de 300Mbar est atteinte près de la zone d'ablation avec une intensité laser absorbée de l'ordre de 2 X 10(15) W.cm-2 et une longueur d'onde de 351 nm. Cette valeur de la pression est deux fois plus importante que la valeur attendue en considérant une absorption collisionnelle de l'énergie laser. Cette importante différence est expliquée par la contribution d'électrons supra-thermiques générés durant l'interaction laser/plasma dans la couronne. Les modèles analytiques proposés permettent une optimisation de l'allumage par choc lorsque les paramètres de la phase d'assemblage, sont pris en compte. Les diverses approches analytiques, numériques et expérimentales sont cohérentes entre-elles