Björn Sjögreen

On Godunov-type methods near low densities

Abstract When the energy of a flow is largely kinetic, many conservative differencing schemes may fail by predicting non-physical states with negative density or internal energy. We describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the Godunov method is positively conservative. It is also shown that no scheme whose interface flux derives from a linearised Riemann solution can be positively conservative. Classes of data that will bring about the failure of such schemes are described. However, the Harten-Lax-van Leer (HLLE) scheme is positively conservative under certain conditions on the numerical wavespeeds, and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed.

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier-Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field (@?.B) numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature.

Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for High Order Methods

The recently developed essentially fourth-order or higher low dissipative shock-capturing scheme of Yee, Sandham, and Djomehri [25] aimed at minimizing numerical dissipations for high speed compressible viscous flows containing shocks, shears and turbulence. To detect non-smooth behavior and control the amount of numerical dissipation to be added, Yee et al. employed an artificial compression method (ACM) of Harten [4] but utilize it in an entirely different context than Harten originally intended. The ACM sensor consists of two tuning parameters and is highly physical problem dependent. To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed. The new sensors are derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch off the extra numerical dissipation outside shock layers. The non-dissipative spatial base scheme of arbitrarily high order of accuracy can be maintained without compromising its stability at all parts of the domain where the solution is smooth. Two types of redundant non-orthogonal wavelet basis functions are considered. One is the B-spline wavelet (Mallat and Zhong [14]) used by Gerritsen and Olsson [3] in an adaptive mesh refinement method, to determine regions where refinement should be done. The other is the modification of the multiresolution method of Harten [5] by converting it to a new, redundant, non-orthogonal wavelet. The wavelet sensor is then obtained by computing the estimated Lipschitz exponent of a chosen physical quantity (or vector) to be sensed on a chosen wavelet basis function. Both wavelet sensors can be viewed as dual purpose adaptive methods leading to dynamic numerical dissipation control and improved grid adaptation indicators. Consequently, they are useful not only for shock-turbulence computations but also for computational aeroacoustics and numerical combustion. In addition, these sensors are scheme independent and can be stand-alone options for numerical algorithms other than the Yee et al. scheme.

Nonlinear filters for efficient shock computation

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

The Convergence Rate of Finite Difference Schemes in the Presence of Shocks

Finite difference approximations generically have

Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

{\cal O}(1)

Gene expression profiling - Clusters of possibilities

pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.

Multiway real-time PCR gene expression profiling in yeast Saccharomyces cerevisiae reveals altered transcriptional response of ADH-genes to glucose stimuli

We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Ystrom, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967], we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.

Conservative and Non-Conservative Interpolation between Overlapping Grids for Finite Volume Solutions of Hyperbolic Problems

Advances in qPCR technology allow studies of increasingly large systems comprising many genes and samples. The increasing data sizes allow expression profiling both in the gene and the samples dimension while also putting higher demands on sound statistical analysis and expertise to handle and interpret its results. We distinguish between exploratory and confirmatory statistical studies. In this paper we demonstrate several techniques available for exploratory studies on a system of Xenopus laevis development from egg to tadpole. Techniques include hierarchical clustering, heatmap, principal component analysis and self-organizing maps. We stress that even though exploratory studies are excellent for generating hypotheses, results have not been proven statistically significant until an independent confirmatory study has been performed. An exploratory study may certainly be valuable in its own right, and there are often not enough resources to report both an exploratory and a confirmatory study at the same time. However, exploratory and confirmatory studies are intimately connected and we would like to raise that awareness among qPCR practitioners. We suggest that scientific reports should always have a hypothesis focus. Reports are either hypothesis generating, from an exploratory study, or hypothesis validating, from a confirmatory study, or both. In either case, we suggest the generated or validated hypotheses be specifically stated.