Craig Michoski

Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

We study a family of generalized slope limiters in two dimensions for Runge-Kutta discontinuous Galerkin (RKDG) solutions of advection-diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiters advantages and disadvantages. We then introduce a series of coupled p-enrichment schemes that may be used as standalone dynamic p-enrichment strategies, or may be augmented via any in the family of variable-in-p slope limiters presented.

A discontinuous Galerkin method for viscous compressible multifluids

Abstract We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier–Stokes system of equations. The system presented has a functional viscosity ν which depends on the pressure p = p ( ρ , μ i ) of the flow, with the density ρ and the local concentration μ i . High order Runge–Kutta time-discretization techniques are employed, and different methods of dealing with arbitrary coupled boundary conditions are discussed. Analysis of the energy consistency of the scheme is performed in addition to inspection of the relative error of the solution compared to exact analytic test cases. Finally several examples, comparisons, generalizations and physical applications are presented.

Discontinuous Galerkin methods for plasma physics in the scrape-off layer of tokamaks

A new parallel discontinuous Galerkin solver, called ArcOn, is developed to describe the intermittent turbulent transport of filamentary blobs in the scrape-off layer (SOL) of fusion plasma. The model is comprised of an elliptic subsystem coupled to two convection-dominated reaction-diffusion-convection equations. Upwinding is used for a class of numerical fluxes developed to accommodate cross product driven convection, and the elliptic solver uses SIPG, NIPG, IIPG, Brezzi, and Bassi-Rebay fluxes to formulate the stiffness matrix. A novel entropy sensor is developed for this system, designed for a space-time varying artificial diffusion/viscosity regularization algorithm. Some numerical experiments are performed to show convergence order on manufactured solutions, regularization of blob/streamer dynamics in the SOL given unstable parameterizations, long-time stability of modon (or dipole drift vortex) solutions arising in simulations of drift-wave turbulence, and finally the formation of edge mode turbulence in the scrape-off layer under turbulent saturation conditions.

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection–diffusion–reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.

Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called fixed tolerance schemes, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called dioristic schemes, rely on time-evolving bounds on the local variation in the solution. Each class of p-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in ‘‘areas of highest interest.’’ The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested on three model systems. The first is an academic exact system where basic analysis is easily performed. Then we discuss a pair of application model problems arising in coastal hydrology. The first being a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. And the second, a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR A COMPRESSIBLE MULTIPHASE NAVIER–STOKES MISCIBLE FLUID-FLOW PROBLEM IN DIMENSION n = 1

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L2 norm of the space derivative of the density, via a new kind of entropy.

Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry

We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.

Artificial boundary layers in discontinuous Galerkin solutions to shallow water equations in channels

In this work, we consider the application of Discontinuous Galerkin (DG) solutions to open channel flow problems, governed by two-dimensional shallow water equations (SWE), with solid curved wall boundaries on which the no-normal flow boundary conditions are prescribed. A commonly used approach consists of straightforwardly imposing the no-normal flow condition on the linear approximation of curved walls. Numerical solutions indicate clearly that this approach could lead to unfavorable results and that a proper treatment of the no-normal flow condition on curved walls is crucial for an accurate DG solution to the SWE. In the test case used, errors introduced through the commonly used approach result in artificial boundary layers of one-grid-size thickness in the velocity field and a corresponding over-prediction of the surface elevation in the upstream direction. These significant inaccuracies, which render the coarse mesh solution unreliable, appear in all DG schemes employed including those using linear, quadratic, and cubic DG polynomials. The issue can be alleviated by either using an approach accounting for errors introduced by the geometric approximation or an approach that accurately represents the geometry.

Stability of Nonlinear Convection–Diffusion–Reaction Systems in Discontinuous Galerkin Methods

In this work we provide an extension of the classical von Neumann stability analysis for high-order accurate discontinuous Galerkin methods applied to generalized nonlinear convection–reaction–diffusion systems. We provide a partial linearization under which a sufficient condition emerges that guarantees stability in this context. The stability behavior of these systems is then closely analyzed relative to Runge–Kutta Chebyshev (RKC) and strong stability preserving (RKSSP) temporal discretizations over a nonlinear system of reactive compressible gases arising in the study of atmospheric chemistry.

Effect of chaos on plasma filament dynamics and turbulence in the scrape-off layer

Naturally occurring error fields as well as resonant magnetic perturbations applied for stability control are known to cause magnetic field-line chaos in the scrape-off layer (SOL) region of tokamaks. Here, 2D simulations with the BOUT++ simulation framework are used to investigate the effect of the field-line chaos on the SOL and in particular on its width and peak particle flux. The chaos enters the SOL dynamics only through the connection length, which is evaluated using a Poincare map. The variation of experimentally relevant quantities, such as the SOL gradient length scale and the intermittency of the particle flux in the SOL, is described as a function of the strength of the magnetic perturbation. It is found that the effect of the chaos is to broaden the profile of the sheath-loss coefficient, which is proportional to the inverse connection length. That is, the SOL transport in a chaotic field is equivalent to that in a model where the sheath-loss coefficient is replaced by its average over the unperturbed flux surfaces. The model does not include the effects of chaotic features other than the parallel connection length.