A General-Purpose Finite-Volume Advection Scheme for Continuous and Discontinuous Fields on Unstructured Grids

We present a new general-purpose advection scheme for unstructured meshes based on the use of a variation of the interface-tracking flux formulation recently put forward by O. Ubbink and R. I. Issa (J. Comput. Phys.153, 26 (1999)), in combination with an extended version of the flux-limited advection scheme of J. Thuburn (J. Comput. Phys.123, 74 (1996)), for continuous fields. Thus, along with a high-order mode for continuous fields, the new scheme presented here includes optional integrated interface-tracking modes for discontinuous fields. In all modes, the method is conservative, monotonic, and compatible. It is also highly shape preserving. The scheme works on unstructured meshes composed of any kind of connectivity element, including triangular and quadrilateral elements in two dimensions and tetrahedral and hexahedral elements in three dimensions. The scheme is finite-volume based and is applicable to control-volume finite-element and edge-based node-centered computations. An explicit–implicit extension to the continuous-field scheme is provided only to allow for computations in which the local Courant number exceeds unity. The transition from the explicit mode to the implicit mode is performed locally and in a continuous fashion, providing a smooth hybrid explicit–implicit calculation. Results for a variety of test problems utilizing the continuous and discontinuous advection schemes are presented.     

Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices

In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the Maxwell–Boltzmann distribution. The axioma holds for both Bravais and irregular lattices, implying a single framework for LB schemes for all lattice types. By solving benchmark problems we have shown that the scheme is indeed consistent with convection diffusion. Furthermore, we have compared the performance of the LB schemes with that of finite difference and finite element schemes. The comparison shows that the LB scheme has a similar performance as the one-step second-order Lax–Wendroff scheme: it has little numerical diffusion, but has a slight dispersion error. By changing the relaxation parameter ω the dispersion error can be balanced by a small increase of the numerical diffusion.     

A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids

In this paper we present a second order finite volume method for the resolution of the bidimensional ideal MHD equations on adaptively refined triangular meshes. Our numerical flux function is based on a multidimensional extension of the Roe scheme proposed by Cargo and Gallice for the 1D MHD system. If the mesh is only composed of triangles, our scheme is proved to be weakly consistent with the condition …B=0. This property fails on a cartesian grid. The efficiency of our refinement procedure is shown on 2D MHD shock capturing simulations. Numerical results are compared in case of the interaction of a supersonic plasma with a cylinder on the adapted grid and several non-refined grids. We also present a mass loading simulation which corresponds to a 2D version of the interaction between the solar wind and a comet.     

Mass Flux Schemes and Connection to Shock Instability

We analyze numerical mass fluxes with an emphasis on their capability for accurately capturing shock and contact discontinuities. The study of mass flux is useful because it is the term common to all conservation equations and the numerical diffusivity introduced in it bears a direct consequence to the prediction of contact (stationary and moving) discontinuities, which are considered to be the limiting case of the boundary layer. We examine several prominent numerical flux schemes and analyze the structure of numerical diffusivity. This leads to a detailed investigation into the cause of certain catastrophic breakdowns by some numerical flux schemes. In particular, we identify the dissipative terms that are responsible for shock instabilities, such as the odd–even decoupling and the so-called “carbuncle phenomenon”. As a result, we propose a conjecture stating the connection of the pressure difference term to these multidimensional shock instabilities and hence a cure to those difficulties. The validity of this conjecture has been confirmed by examining a wide class of upwind schemes. The conjecture is useful to the flux function development, for it indicates whether the flux scheme under consideration will be afflicted with these kinds of failings. Thus, a class of shock-stable schemes can be identified. Interestingly, a shock-stable scheme's self-correcting capability is demonstrated with respect to carbuncle-contaminated profiles for flows at both low supersonic and high Mach numbers.     

Divergence- and Curl-Preserving Prolongation and Restriction Formulas

We present new second-order prolongation and restriction formulas which preserve the divergence and, in some cases, the curl of a discretized vector field. The formulas are suitable for adaptive and hierarchical mesh algorithms with a factor-of-2 linear resolution change. We examine both staggered and collocated discretizations for the vector field on two- and three-dimensional Cartesian grids. The new formulas can be used in combination with numerical schemes that require a divergence-free solution in some discrete sense, such as the constrained transport schemes of computational magnetohydrodynamics. We also obtain divergence-preserving interpolation functions which may be used for streamline or field line tracing.     

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.     

Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics

The proposed scheme, which is a conservative form of the interpolated differential operator scheme (IDO-CF), can provide high accurate solutions for both compressible and incompressible fluid equations. Spatial discretizations with fourth-order accuracy are derived from interpolation functions locally constructed by both cell-integrated values and point values. These values are coupled and time-integrated by solving fluid equations in the flux forms for the cell-integrated values and in the derivative forms for the point values. The IDO-CF scheme exactly conserves mass, momentum, and energy, retaining the high resolution more than the non-conservative form of the IDO scheme. A direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods. Benchmark tests of Riemann problems and lid-driven cavity flows show that the IDO-CF scheme is immensely promising in compressible and incompressible fluid dynamics studies.     

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.     

A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates

In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence.     

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.     

Unstructured Adaptive Grid Flow Simulations of Inert and Reactive Gas Mixtures

Unstructured adaptive grid flow simulation is applied to the calculation of high-speed compressible flows of inert and reactive gas mixtures. In the present case, the flowfield is simulated using the 2-D Euler equations, which are discretized in a cell-centered finite volume procedure on unstructured triangular meshes. Interface fluxes are calculated by a Liou flux vector splitting scheme which has been adapted to an unstructured grid context by the authors. Physicochemical properties are functions of the local mixture composition, temperature, and pressure, which are computed using the CHEMKIN-II subroutines. Computational results are presented for the case of premixed hydrogen–air supersonic flow over a 2-D wedge. In such a configuration, combustion may be triggered behind the oblique shock wave and transition to an oblique detonation wave is eventually obtained. It is shown that the solution adaptive procedure implemented is able to correctly define the important wave fronts. A parametric analysis of the influence of the adaptation parameters on the computed solution is performed.     

Physics-Based Preconditioning and the Newton–Krylov Method for Non-equilibrium Radiation Diffusion

An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton–Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian–free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.     

Design and implementation of a protection system for NMR spectrometers

We have implemented a scheme, SPECMON, for monitoring various parameters of a spectrometer, such as nitrogen pressure and sample temperature, and taking corrective action. The scheme is based on considerations of protection management which are of general application. Evaluation of the spectrometer state is incorporated in macros of the application software (VNMR) and is therefore very flexible. In contrast, corrective action is limited to the single one which is deemed fully safe: complete shutdown of the spectrometer and logging. Shutdown is implemented by a minor hardware modification of the spectrometer: the introduction of a second input to a relay already present for protection of the spectrometer power supply. Monitoring is handled by the host computer, and the shutdown command is transmitted via control lines of its series port, independent of the standard connection between the host computer and the NMR system console. The monitoring system (software and hardware) is unobtrusive in normal conditions, and it can be tested without affecting the operation of the spectrometer.     

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 Composite Scheme for Gas Dynamics in Lagrangian Coordinates

One cycle of a composite finite difference scheme is defined as several time steps of an oscillatory scheme such as Lax–Wendroff followed by one step of a diffusive scheme such as Lax–Friedrichs. We apply this idea to gas dynamics in Lagrangian coordinates. We show numerical results in two dimensions for Noh's infinite strength shock problem and the Sedov blast wave problem, and for several one-dimensional problems including a Riemann problem with a contact discontinuity. For Noh's problem the composite scheme produces a better result than that obtained with a more conventional Lagrangian code.     

An Extension of Alias Sampling Method for Parametrized Probability Distributions

An extension of the alias sampling technique for distribution functions depending on a number of parameters was developed. It takes advantage of modern computer architectures with large amounts of cheap memory, by using discrete representations of probability distribution functions. The sampling is done by fast interpolation techniques involving only elementary logical and arithmetical operations, allowing one to keep a higher degree of accuracy as the grids spacing is controlled by the user. By this method it is possible to obtain the value of interest by direct interpolation between the sampled values obtained with the same set of random numbers for the grid values of the parameters adjacent to the values of interest. Sampling tests carried for the case of Molière electron multi-scatter angle distribution show that this method can be successfully used in Monte Carlo codes for sampling complex probability distributions.     

Two-Level Hierarchical FEM Method for Modeling Passive Microwave Devices

In recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh-version of the finite element method. In this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep-version of the finite element analysis applied to indefinite problems. A two-levelV-cycle algorithm has been implemented and studied with a Gauss–Seidel iterative scheme used as a smoother. The convergence of the method has been investigated, and numerical results for a number of numerical examples are presented.     

Nonlinear Landau Damping in Spherically Symmetric Vlasov Poisson Systems

The Vlasov Poisson system is a partial differential equation widely used to describe collisionless plasma. It is formulated in a six-dimensional phase space, this prohibits a numerical solution on a complete phase space grid. In some applications, however, spherical symmetry is given, which introduces singularities into the Vlasov Poisson equation. We focus on such problems and propose a stable algorithm using accommodating boundaries. At first, the method is tested in the linear regime, where analytical solutions are available. Thereafter it is applied to large disturbances from equilibrium.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.