Browsing by Subject "Conservation laws"
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Item Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games(2014-04-18) Tomov, VladimirIn this dissertation we consider two major subjects. The primary topic is the Entropy Viscosity method for Lagrangian hydrodynamics, the goal of which is to solve numerically the Euler equations of compressible gas dynamics. The second topic is concerned with applications of second order central differencing schemes to the Mean Field Games equations. The Entropy Viscosity method discretizes all kinematic and thermodynamic variables by high-order finite elements and solves the resulting discrete problem on a computational mesh that moves with the material velocity. The method is based on two major concepts. The first one is producing high order convergence rates for smooth solutions even with active viscosity terms. This is achieved by using high order finite element spaces and, more importantly, entropy based viscosity coefficients that clearly distinguish between smooth and singular regions. The second concept is providing control over oscillations around contact discontinuities as well as oscillations in shock regions. Achieving this requires adding extra viscosity terms in a way that the resulting system is still in agreement with generalized entropy inequalities, the minimum principle on the specific entropy and the general requirements for artificial tensor viscosities like orthogonal transformation invariance, radial symmetry, Galilean invariance, etc. We define a fully-discrete finite element algorithm and present numerical results on model Lagrangian hydro problems. We also discuss possible extensions of the method, e.g. length scale independent viscosity coefficients, incorporating mass diffusion into the mesh motion, and handling of different materials. In addition we present approaches to the different stages of arbitrary Lagrangian-Eulerian (ALE) methods, which can be used to extend the Entropy Viscosity method. That is, we discuss mesh relaxation by harmonic smoothing schemes, advection based solution remap, and multi-material zones treatment. The Mean Field Games (MFG) equations describe situations in which a large number of individual players choose their optimal strategy by considering global (but limited) incentive information that is available to everyone. The resulting system consists of a forward Hamilton-Jacobi equation and a backward convection-diffusion equation. We propose fully discrete explicit second order staggered finite difference schemes for the two equations and combine these schemes into a fixed point iteration algorithm. We discuss the second order accuracy of both schemes, their interaction in time, memory issues resulting from the forward-backward coupling, stopping criteria for the fixed point iteration, and parallel performance of the method.Item A fragmentation model for sprays and L² stability estimates for shockes solutions of scalar conservation laws using the relative entropy method(2010-05) Leger, Nicholas Matthew; Vasseur, Alexis F.; Arbogast, Todd J.; Gamba, Irene M.; Vishik, Mikhail M.; Raman, VenkatramananWe present a mathematical study of two conservative systems in fluid mechanics. First, we study a fragmentation model for sprays. The model takes into account the break-up of spray droplets due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system. Next, with an interest in understanding hydrodynamical limits in discontinuous regimes, we study a classical model for shock waves. Specifically, we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L² norm of a perturbed solution relative to the shock wave is bounded above by the L² norm of the initial perturbation. Finally, we include some preliminary relative entropy estimates which are suitable for a study of shock wave solutions to n x n systems of conservation laws having a convex entropy.