A discontinuous Galerkin method for the Vlasov-Poisson system

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86 figure

A Conservative Discontinuous Galerkin Scheme With O(N-2) Operations In Computing Boltzmann Collision Weight Matrix

In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of >shifting symmetry> in the weight matrix to reduce the computing complexity from theoretical O(N-3) down to O(N-2), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpetiMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.Mathematic

ON SPURIOUS BEHAVIOR OF CFD SIMULATIONS

Spurious behavior in underresolved grids and:or semi-implicit temporal discretizations for four computational fluid dynamics (CFD) simulations are studied. The numerical simulations consist of (a) a 1-D chemically relaxed non-equilibrium flow model, (b) the direct numerical simulation (DNS) of 2D incompressible flow over a backward facing step, (c) a loosely coupled approach for a 2D fluid–structure interaction, and (d) a 3D unsteady compressible flow simulation of vortex breakdown on delta wings. These examples were chosen based on their non-apparent spurious behaviors that were difficult to detect without extensive grid and:or temporal refinement studies and without some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations, underresolved grids, semi-implicit procedures, loosely coupled implicit procedures, and insufficiently long-time integration in DNS are most often unavoidable. Consequently, care must be taken in both computation and in interpretation of the numerical data. The results presented confirm the important role that dynamical systems theory can play in the understanding of the non-linear behavior of numerical algorithms and in aiding the identification of the sources of numerical uncertainties in CFD

Computation of libration point orbits and manifolds using collocation methods

This thesis contains a methodology whose aim is to compute trajectories describing natural motion of the phase space in a neighborhood of Libtation points and stable/unstable manifolds which correspond to these orbits in the Restricted Three Body Problem. There are two models the Circular Restricted Three Body Problem and Elliptic Restricted Three Body Problem which are special cases of RTBP . In this paper we pay attention to CRTBP which is autonomous (depending on time). The CRTBP is the most easily understood and well-analysed in a coordinate system rotating with two large bodies. The method is based on the collocation method implemented in AUTO - 07p software and must provide an isolated periodic solution. The paper includes explanation of the collocation method, its application in case of CRTBP, numerical and graphical results of its implementation

Stability Analysis of Magnetised Neutron Stars - a Semi-Analytic Approach

The stability problem of magnetised neutron stars is addressed by applying a new semi-analytic method for stability analysis which is based on the energy variational principle. It is shown that the semi-analytic method represents a valuable tool for stability investigations, aiming at explaining the durability and long-lasting stability of magnetised neutron stars. The method provides the opportunity to constrain the interior neutron star magnetic field structure which is still widely unknown but highly interesting for all kinds of neutron star studies. This work describes the analytical and numerical setup of the method and shows applications on neutron star models with polytropic as well as non-barotropic equations of state, purely toroidally and purely poloidally magnetised stars as well as mixed magnetic fields. The validity of the Cowling approximation is tested and new physical insights are presented.Diese Arbeit befasst sich mit dem Stabilitätsproblem magnetisierter Neutronensterne, unter Verwendung einer neuen semi-analytischen Untersuchungsmethode zur Stabilitätsanalyse, welche auf dem Energievariationsprinzip basiert. Es wird gezeigt, dass die semi-analytische Methode ein nützliches Hilfsmittel darstellt, um die Ursachen von hoher Lebensdauer und anhaltender Stabilität magnetisierter Neutronensterne zu untersuchen. Die Methode ermöglicht es die bislang weitgehend unbekannte, aber für alle Arten von Neutronensternuntersuchungen hochinteressante, innere Magnetfeldstruktur von Neutronensternen einzugrenzen. In dieser Arbeit werden der analytische und numerische Aufbau der Methode sowie Anwendungen auf Neutronensternmodelle mit polytroper und nicht-barotroper Zustandsgleichung, rein toroidalem, rein poloidalem und gemischtem Magnetfeld gezeigt. Die Gültigkeit der Cowling-Näherung wird untersucht und neue physikalische Erkenntnisse werden präsentiert

Dynamics of Numerics & Spurious Behaviors in CFD Computations

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD