Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for the all proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods

Pore-scale Direct Numerical Simulation of Flow and Transport in Porous Media

This dissertation presents research on the pore-scale simulation of flow and transport in porous media and describes the application of a new numerical approach based on the discontinuous Galerkin (DG) finite elements to pore-scale modelling. In this approach, the partial differential equations governing the flow at the pore-scale are solved directly where the main advantage is that it does not require a body fitted grid and works on a structured partition of the domain. Furthermore this approach is locally mass conservative, a desirable property for transport simulation. This allows the investigation of pore-scale processes and their effect on macroscopic behaviour more efficiently. The Stokes flow in two and three dimensional disordered packing was solved and the flow field was used in a random-walk particle tracking model to simulate the transport through the packing. The permeabilities were computed and asymptotic behaviour of solute dispersion for a wide range of Péclet numbers was studied. The simulated results agree well with the data reported in the literature, which indicates that the approach chosen here is well suited for pore-scale simulation

On the Advective Component of Active Flux Schemes for Nonlinear Hyperbolic Conservation Laws

A new class of numerical methods called Active Flux (AF) is investigated for nonlinear hyperbolic conservation laws. The AF method is designed specifically to address the aspect that most modern compressible flow methods fail to do; the multidimensionality aspect. It addresses the shortcoming by employing a two stage update process. In the first stage, a nonconservative form of the system is introduced to provide the flexibility to pursue distinct numerical approaches for flow processes with differing physics. Because each process is treated separately, the numerical method can be appropriately formed to reflect each type of physics and to provide the maximal stability. The method is completed with the conservation update to produce a third-order accurate scheme. The AF advection scheme is founded on the characteristic tracing method, a semi-Lagrangian method, which has long been used for developing numerical methods for hyperbolic problems. The first known AF method for advection, Scheme V by van Leer, is revisited as a part of the development of the scheme. Details of Scheme V are examined closely, and new improvements are made for the multidimensional nonlinear advection scheme. A detailed study of the nonlinear system of equations is made possible by the pressureless Euler system, which is the advective component of the Euler system. It serves as a stepping stone for the Euler system, and all necessary details of the nonlinear system are explored. Lastly, an extension to the Euler system is presented where a novel nonlinear operator splitting method is introduced to correctly blend the contributions of the nonlinear advection and acoustic processes. The AF method, as a result, produces a maximally stable, third-order accurate method for the multidimensional Euler system. Some guiding principles of limiting are presented. Because two types of flow feature are separately treated, the limiting process must also be kept separate. Advective problems obeying natural bounding principles are treated differently from acoustic problems with no explicit bounding principles. Distinct limiting approaches are explored along with discussions.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138695/1/jmaeng_1.pd

A Quasi-Hamiltonian Discretization of the Thermal Shallow Water Equations

International audienceThe rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model