Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games
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In 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.
Entropy Viscosity Method
Mean Field Games
Tomov, Vladimir (2014). Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games. Doctoral dissertation, Texas A & M University. Available electronically from