Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy

In this paper we design a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes of G.-S. Jiang and C.-W. Shu (1996) and X.-D. Liu, S. Osher, and T. Chan (1994). Used by themselves, the schemes may not always be monotonicity preserving but coupled with the monotonicity preserving bounds of A. Suresh and H. T. Huynh (1997) they perform very well. The resulting monotonicity preserving weighted essentially non-oscillatory (MPWENO) schemes have high phase accuracy and high order of accuracy. The higher-order members of this family are almost spectrally accurate for smooth problems. Nevertheless, they, have robust shock capturing ability. The schemes are stable under normal CFL numbers. They are also efficient and do not have a computational complexity that is substantially greater than that of the lower-order members of this same family of schemes. The higher accuracy that these schemes offer coupled with their relatively low computational complexity makes them viable competitors to lower-order schemes, such as the older total variation diminishing schemes, for problems containing both discontinuities and rich smooth region structure. We describe the MPWENO schemes here as well as show their ability to reach their designed accuracies for smooth flow. We also examine the role of steepening algorithms such as the artificial compression method in the design of very high order schemes. Several test problems in one and two dimensions are presented. For multidimensional problems where the flow is not aligned with any of the grid directions it is shown that the present schemes have a substantial advantage over lower-order schemes. It is argued that the methods designed here have great utility for direct numerical simulations and large eddy simulations of compressible turbulence. The methodology developed here is applicable to other hyperbolic systems, which is demonstrated by showing that the MPWENO schemes also work very well on magnetohydrodynamical test problems.     

An Efficient Algorithm for Truncating Spatial Domain in Modeling Light Scattering by Finite-Difference Technique

The finite-difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry. In practice, this method requires that the spatial domain for the computation of near-field be truncated. An absorbing boundary condition must be imposed in conjunction with this truncation. The performance of this boundary condition is essential to the stability of numerical computations and the reliability of results. In the present study, a new boundary condition, referred to as the mixed T algorithm, has been developed, which is a generalization of the transmitting boundary condition originally developed by Liao and co-workers. The present algorithm does not require spatial interpolation for wave values at interior grid points. In addition, it produces two minima of spurious reflections at small and large incident angles, allowing efficient absorption of the scattered waves at the boundary for large incident angles. When the third-order mixed T algorithm is used, the reflection coefficient of the boundary is less than 1% for incident angles from 0° to about 70°. We find that the numerical instability associated with the transmitting boundary condition is caused by the location-dependent amplitude of outgoing waves in the vicinity of the boundary. For this reason, the mixed T algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation. When the stabilized algorithm is applied, the near-field within the truncated domain can be computed by using single-precision arithmetic without overflows for more than 10⁵steps in the time-marching iteration. Finally, the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a TM wave excited by a sinusoidal point source, simultaneous simulation of the wave propagation in small and large domains, and the scattering of a TM wave by an infinite circular cylinder.     

Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method

In many realistic fluid-dynamical simulations the specification of the boundary conditions, the error sources, and the number of time steps to reach a steady state are important practical considerations. In this paper we study these issues in the case of the lattice-BGK model. The objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice-BGK method and to introduce some new ideas useful in practical simulations. We begin with an evaluation of the widely used bounce-back boundary condition in staircase geometries by simulating flow in an inclined tube. It is shown that the bounce-back scheme is first-order accurate in space when the location of the non-slip wall is assumed to be at the boundary nodes. Moreover, for a specific inclination angle of 45 degrees, the scheme is found to be second-order accurate when the location of the non-slip velocity is fitted halfway between the last fluid nodes and the first solid nodes. The error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls. Next, a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented. A good agreement between these two boundary conditions has been found in the creeping-flow regime. For higher Reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries. Furthermore, two widely used 3D models, namelyD₃Q₁₅andD₃Q₁₉, are analysed. It is shown that theD₃Q₁₅model may induce artificial checkerboard invariants due to the connectivity of the lattice. Finally, a new iterative method, which significantly reduces the saturation time, is presented and validated on different benchmark problems.     

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics

We present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio–Wu Riemann problems, the two-dimensional Kelvin–Helmholtz instability problems, and the two-dimensional Orszag–Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.     

Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier–Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourth-order differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

Half-Moment Closure for Radiative Transfer Equations

In this paper a moment method for radiative transfer equations is considered which has been developed and investigated using different approaches. Problems appearing for this moment system for boundary value problems using Maxwell-type boundary conditions are described. A new method based on the consideration of positive and negative half fluxes is developed and shown to overcome the above problems. Moreover, a numerical scheme and numerical results for the new moment system are presented.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

Partially implicit peer methods for the compressible Euler equations

When cut cells are used for the representation of orography in numerical weather prediction models this leads to very small cells. On one hand this results in very harsh time step restrictions for explicit methods due to the CFL criterion. On the other hand cut cells only appear in a small region of the domain. Therefore we consider a partially implicit method: In cut cells the Jacobian incorporates advection, diffusion and acoustics while in the full cells of the free atmosphere only the acoustic part is used, i.e. the method is linearly implicit in the cut cell regions and semi-explicit in the free regions and computes with time step sizes restricted only by the CFL condition in the free atmosphere. Furthermore we use a simplified Jacobian in the cut cell regions in order to save storage and gain computational efficiency. While the method retains the order independently of the Jacobian we present a linear stability theory which takes the effects of the simplifications of the Jacobian on stability into account. The presented method is as stable and accurate as the underlying split-explicit method but furthermore it can compute with cut cells with nearly no additional effort.     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.     

GAMMA simulations of stray field responses: slice thickness and pulse calibration

A stray field (STRAFI) module has been added to the GAMMA magnetic resonance simulation platform in order to facilitate computational investigations of NMR experiments in large static field gradients that are on the order of 50 T/m. The package has been used to examine system response during echo trains generated by the application of shaped pulses. The associated echo amplitude maxima and effective slice thickness are presented. A new accurate method for STRAFI pulse calibration based on relative echo amplitudes is proposed.     

A Residual-Based Compact Scheme for the Compressible Navier–Stokes Equations

A simple and efficient time-dependent method is presented for solving the steady compressible Euler and Navier–Stokes equations with third-order accuracy. Owing to its residual-based structure, the numerical scheme is compact without requiring any linear algebra, and it uses a simple numerical dissipation built on the residual. The method contains no tuning parameter. Accuracy and efficiency are demonstrated for 2-D inviscid and viscous model problems. Navier–Stokes calculations are presented for a shock/boundary layer interaction, a separated laminar flow, and a transonic turbulent flow over an airfoil.     

Modeling Three-Dimensional Multiphase Flow Using a Level Contour Reconstruction Method for Front Tracking without Connectivity

Three-dimensional multiphase flow and flow with phase change are simulated using a simplified method of tracking and reconstructing the phase interface. The new level contour reconstruction technique presented here enables front tracking methods to naturally, automatically, and robustly model the merging and breakup of interfaces in three-dimensional flows. The method is designed so that the phase surface is treated as a collection of physically linked but not logically connected surface elements. Eliminating the need to bookkeep logical connections between neighboring surface elements greatly simplifies the Lagrangian tracking of interfaces, particularly for 3D flows exhibiting topology change. The motivation for this new method is the modeling of complex three-dimensional boiling flows where repeated merging and breakup are inherent features of the interface dynamics. Results of 3D film boiling simulations with multiple interacting bubbles are presented. The capabilities of the new interface reconstruction method are also tested in a variety of two-phase flows without phase change. Three-dimensional simulations of bubble merging and droplet collision, coalescence, and breakup demonstrate the new method's ability to easily handle topology change by film rupture or filamentary breakup. Validation tests are conducted for drop oscillation and bubble rise. The susceptibility of the numerical method to parasitic currents is also thoroughly assessed.     

A representation of curved boundaries for the solution of the Navier–Stokes equations on a staggered three-dimensional Cartesian grid

A method is presented for representing curved boundaries for the solution of the Navier–Stokes equations on a non-uniform, staggered, three-dimensional Cartesian grid. The approach involves truncating the Cartesian cells at the boundary surface to create new cells which conform to the shape of the surface. We discuss in some detail the problems unique to the development of a cut cell method on a staggered grid. Methods for calculating the fluxes through the boundary cell faces, for representing pressure forces and for calculating the wall shear stress are derived and it is verified that the new scheme retains second-order accuracy in space. In addition, a novel “cell-linking” method is developed which overcomes problems associated with the creation of small cells while avoiding the complexities involved with other cell-merging approaches. Techniques are presented for generating the geometric information required for the scheme based on the representation of the boundaries as quadric surfaces. The new method is tested for flow through a channel placed oblique to the grid and flow past a cylinder at Re=40 and is shown to give significant improvement over a staircase boundary formulation. Finally, it is used to calculate unsteady flow past a hemispheric protuberance on a plate at a Reynolds number of 800. Good agreement is obtained with experimental results for this flow.     

Application of high-resolution magic-angle spinning NMR spectroscopy to define the cell uptake of MRI contrast agents

A new method, based on proton high-resolution magic-angle spinning ((1)H HR-MAS) NMR spectroscopy, has been employed to study the cell uptake of magnetic resonance imaging contrast agents (MRI-CAs). The method was tested on human red blood cells (HRBC) and white blood cells (HWBC) by using three gadolinium complexes, widely used in diagnostics, Gd-BOPTA, Gd-DTPA, and Gd-DOTA, and the analogous complexes obtained by replacing Gd(III) with Dy(III), Nd(III), and Tb(III) (i.e., complexes isostructural to the ones of gadolinium but acting as shift agents). The method is based on the evaluation of the magnetic effects, line broadening, or induced lanthanide shift (LIS) caused by these complexes on NMR signals of intra- and extracellular water. Since magnetic effects are directly linked to permeability, this method is direct. In all the tests, these magnetic effects were detected for the extracellular water signal only, providing a direct proof that these complexes are not able to cross the cell membrane. Line broadening effects (i.e., the use of gadolinium complexes) only allow qualitative evaluations. On the contrary, LIS effects can be measured with high precision and they can be related to the concentration of the paramagnetic species in the cellular compartments. This is possible because the HR-MAS technique provides the complete elimination of bulk magnetic susceptibility (BMS) shift and the differentiation of extra- and intracellular water signals. Thus with this method, the rapid quantification of the MRI-CA amount inside and outside the cells is actually feasible.     

Optimization of residual water signal removal by HLSVD on simulated short echo time proton MR spectra of the human brain

Suppression of the residual water signal from proton magnetic resonance (MR) spectra recorded in human brain is a prerequisite to an accurate quantification of cerebral metabolites. Several postacquisition methods of residual water signal suppression have been reported but none of them provide a complete elimination of the residual water signal, thereby preventing reliable quantification of brain metabolites. In the present study, the elimination of the residual water signal by the Hankel Lanczos singular value decomposition method has been evaluated and optimized to provide fast automated processing of spectra. Model free induction decays, reproducing the proton signal acquired in human brain localized MR spectroscopy at short echo times (e.g., 20 ms), have been generated. The optimal parameters in terms of number of components and dimension of the Hankel data matrix allowing complete elimination of the residual water signal are reported.     

Higher-order surface solitons in one-dimensional Bessel optical lattices

We demonstrate the existence of higher-order solitons occurring at an interface separating two one-dimensional (1D) Bessel optical lattices with different orders or modulation depths in a defocusing medium. We show that, in contrast to homogeneous waveguides where higher-order solitons are always unstable, the Bessel lattices with an interface support branches of higher-order structures bifurcating from the corresponding linear modes. The profiles of solitons depend remarkably on the lattice parameters and the stability can be enhanced by increasing the lattice depth and selecting higher-order lattices. We also reveal that the interface model with defocusing saturable Kerr nonlinearity can support stable multi-peaked solitons. The uncovered phenomena may open a new way for soliton control and manipulation.     

A subspace time-domain algorithm for automated NMR spectral normalization

Recently, two methods have been proposed for quantitatively comparing NMR spectra of control and treated samples, in order to examine the possible occurring variations in cell metabolism and/or structure in response to numerous physical, chemical, and biological agents. These methods are the maximum superposition normalization algorithm (MaSNAl) and the minimum rank normalization algorithm (MiRaNAl). In this paper a new subspace-based time-domain normalization algorithm, denoted by SuTdNAl (subspace time-domain normalization algorithm), is presented. By the determination of the intersection of the column spaces of two Hankel matrices, the common signal poles and further on the components having proportionally varying amplitudes are detected. The method has the advantage that it is computationally less intensive than the MaSNAl and the MiRaNAl. Furthermore, no approximate estimate of the normalization factor is required. The algorithm was tested by Monte Carlo simulations on a set of simulation signals. It was shown that the SuTdNAl has a statistical performance similar to that of the MiRaNAl, which itself is an improvement over the MaSNAl. Furthermore, two samples of known contents are compared with the MiRaNAl, the SuTdNAl, and an older method using a standard. Finally, the SuTdNAl is tested on a realistic simulation example derived from an in vitro measurement on cells.     

Boundary data immersion method for Cartesian-grid simulations of fluid-body interaction problems

A new robust and accurate Cartesian-grid treatment for the immersion of solid bodies within a fluid with general boundary conditions is described. The new approach, the Boundary Data Immersion Method (BDIM), is derived based on a general integration kernel formulation which allows the field equations of each domain and the interfacial conditions to be combined analytically. The resulting governing equation for the complete domain preserves the behavior of the original system in an efficient Cartesian-grid method, including stable and accurate pressure values on the solid boundary. The kernel formulation allows a detailed analysis of the method, and it is demonstrated that BDIM is consistent, obtains second-order convergence relative to the kernel width, and is robust with respect to the grid and boundary alignment. Formulation for no-slip and free slip boundary conditions are derived and numerical results are obtained for the flow past a cylinder and the impact of blunt bodies through a free surface. The BDIM predictions are compared to analytic, experimental and previous numerical results confirming the properties, efficiency and efficacy of this new boundary treatment for Cartesian grid methods.