Exact calculation of Fourier series in nonconforming spectral-element methods

In this note is presented a method, given nodal values on multidimensional nonconforming spectral elements, for calculating global Fourier-series coefficients. This method is ``exact'' in that given the approximation inherent in the spectral-element method (SEM), no further approximation is introduced that exceeds computer round-off error. The method is very useful when the SEM has yielded an adaptive-mesh representation of a spatial function whose global Fourier spectrum must be examined, e.g., in dynamically adaptive fluid-dynamics simulations.Comment: 7 pages, 4 figures, submitted to J. Comp. Phys. 2005 June

Adaptive mesh refinement with spectral accuracy for magnetohydrodynamics in two space dimensions

We examine the effect of accuracy of high-order spectral element methods, with or without adaptive mesh refinement (AMR), in the context of a classical configuration of magnetic reconnection in two space dimensions, the so-called Orszag-Tang vortex made up of a magnetic X-point centered on a stagnation point of the velocity. A recently developed spectral-element adaptive refinement incompressible magnetohydrodynamic (MHD) code is applied to simulate this problem. The MHD solver is explicit, and uses the Elsasser formulation on high-order elements. It automatically takes advantage of the adaptive grid mechanics that have been described elsewhere in the fluid context [Rosenberg, Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)]; the code allows both statically refined and dynamically refined grids. Tests of the algorithm using analytic solutions are described, and comparisons of the Orszag-Tang solutions with pseudo-spectral computations are performed. We demonstrate for moderate Reynolds numbers that the algorithms using both static and refined grids reproduce the pseudo--spectral solutions quite well. We show that low-order truncation--even with a comparable number of global degrees of freedom--fails to correctly model some strong (sup--norm) quantities in this problem, even though it satisfies adequately the weak (integrated) balance diagnostics.Comment: 19 pages, 10 figures, 1 table. Submitted to New Journal of Physic

Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

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

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2