A comparison of spectral element and finite difference methods using statically refined nonconforming grids for the MHD island coalescence instability problem

A recently developed spectral-element adaptive refinement incompressible magnetohydrodynamic (MHD) code [Rosenberg, Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)] is applied to simulate the problem of MHD island coalescence instability (MICI) in two dimensions. MICI is a fundamental MHD process that can produce sharp current layers and subsequent reconnection and heating in a high-Lundquist number plasma such as the solar corona [Ng and Bhattacharjee, Phys. Plasmas, 5, 4028 (1998)]. Due to the formation of thin current layers, it is highly desirable to use adaptively or statically refined grids to resolve them, and to maintain accuracy at the same time. The output of the spectral-element static adaptive refinement simulations are compared with simulations using a finite difference method on the same refinement grids, and both methods are compared to pseudo-spectral simulations with uniform grids as baselines. It is shown that with the statically refined grids roughly scaling linearly with effective resolution, spectral element runs can maintain accuracy significantly higher than that of the finite difference runs, in some cases achieving close to full spectral accuracy.Comment: 19 pages, 17 figures, submitted to Astrophys. J. Supp

An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics: a staggered dissipation-control differencing algorithm.

A new unsplit staggered mesh algorithm (USM) that solves multidimensional magnetohydrodynamics (MHD) on a staggered mesh is introduced and studied. Proper treatments of multidimensional flow problems are required for MHD simulations to avoid unphysical results that can even introduce numerical instability. The research work in this dissertation, which is based on an approach that combines the high-order Godunov method and the constrained transport (CT) scheme, uses such multidimensional consideration in a spatial reconstruction-evolution step. The core problem of MHD simulation is the nonlinear evolution of solutions using well-designed algorithms that maintain the divergence-free constraint of the magnetic field components. The USM algorithm proposed in this dissertation ensures the solenoidal constraint by using Stokes' Theorem as applied to a set of induction equations. In CT-type of MHD schemes, one solves the discrete induction equations to proceed temporal evolutions of the staggered magnetic fields using electric fields. The accuracy of the computed electric fields therefore directly influence the solution quality of the magnetic fields. To meet this end, an accurate and improved electric field construction (IEC) scheme has been introduced as one of the essential parts of the current dissertation work. Another important feature in this work is a development of a new algorithm that solves the induction equations with an added capability that controls numerical (anti)dissipations of the magnetic fields. This staggered dissipation-control differencing algorithm (SDDA) makes use of extra dissipation terms, for which their derivations are established from modified equations of the induction equations. A series of comparison studies in a suite of numerical results of the USM-IEC-SDDA scheme will show a great deal of qualitative improvements in many stringent multidimensional MHD test problems

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

Multiphysics CFD modelling of incompressible flows at Low and Moderate Reynolds Numbers

In this Ph.D. thesis, a novel high-resolution Godunov-type numerical procedure has been developed for solving the unsteady, incompressible Navier-Stokes equations for constant and variable density flows. The proposed FSAC-PP approach encompasses both artificial compressibility (AC) and fractional step (FS) pressure-projection (PP) methods of Chorin [3, 4] in a unified solution concept. To take advantage of different computational strategies, the FS and AC methods have been coupled (FSAC formulation), and further a PP step has been employed at each pseudo-time step. To provide time-accurate solutions, the dual-time stepping procedure is utilized. Taking the advantage of the hyperbolic nature of the inviscid part of the AC formulation, high-resolution characteristics-based (CB) Godunov-type scheme is employed to discretize the non-linear advective fluxes. Highorder of accuracy is achieved by using from first- up to ninth-order interpolation schemes. Time integration is obtained from a fourth-order Runge-Kutta scheme. A non-linear fullmultigrid, full-approximation storage (FMG-FAS) acceleration technique has been further extended to the FSAC-PP solution method to increase the efficiency and decrease the computational cost of the developed method and simulations. Cont/d

Multiscale structure of turbulent channel flow and polymer, dynamics in viscoelastic turbulence

This thesis focuses on two important issues in turbulence theory of wall-bounded flows. One is the recent debate on the form of the mean velocity profile (is it a log-law or a power-law with very weak power exponent?) and on its scalings with Reynolds number. In particular, this study relates the mean flow profile of the turbulent channel flow with the underlying topological structure of the fluctuating velocity field through the concept of critical points, a dynamical systems concept that is a natural way to quantify the multiscale structure of turbulence. This connection gives a new phenomenological picture of wall-bounded turbulence in terms of the topology of the flow. This theory validated against existing data, indicates that the issue on the form of the mean velocity profile at the asymptotic limit of infinite Reynolds number could be resolved by understanding the scaling of turbulent kinetic energy with Reynolds number. The other major issue addressed here is on the fundamental mechanism(s) of viscoelastic turbulence that lead to the polymer-induced turbulent drag reduction phenomenon and its dynamical aspects. A great challenge in this problem is the computation of viscoelastic turbulent flows, since the understanding of polymer physics is restricted to mechanical models. An effective numerical method to solve the governing equation for polymers modelled as nonlinear springs, without using any artificial assumptions as usual, was implemented here for the first time on a three-dimensional channel flow geometry. The superiority of this algorithm is depicted on the results, which are much closer to experimental observations. This allowed a more detailed study of the polymer-turbulence dynamical interactions, which yields a clearer picture on a mechanism that is governed by the polymer-turbulence energy transfers