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.     

An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows

A three-dimensional, incompressible, multiphase particle-in-cell method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions. A subgrid particle, normal stress model for discrete particles which is robust and eliminates the need for an implicit calculation of the particle normal stress on the grid is presented. Interpolation operators and their properties are defined which provide compact support, are conservative, and provide fast solution for a large particle population. The solution scheme allows for distributions of types, sizes, and density of particles, with no numerical diffusion from the Lagrangian particle calculations. Particles are implicitly coupled to the fluid phase, and the fluid momentum and pressure equations are implicitly solved, which gives a robust solution.     

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 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.     

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.     

A Higher-Order Boundary Treatment for Cartesian-Grid Methods

The Euler equations describe the flow phenomena of compressible inviscid gas dynamics. We simulate such flows using a higher-order Cartesian-grid method, together with a special treatment for the cells cut by the boundary of an object. A new method for the treatment of the boundary is described where these cut boundary cells are maintained as whole cells rather than as cut cells, thus avoiding stability problems. The method is second-order accurate in one dimension and higher-order accurate in two dimensions but not strictly conservative; however, we show that this error in the conservation does not lead to spurious phenomena on some representative test calculations. The advantages of the new boundary treatment are that it is higher-order accurate, that it is independent of the applied method, and that it is simple.     

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.     

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.     

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.     

On Performance of Methods with Third- and Fifth-Order Compact Upwind Differencing

The difference schemes for fluid dynamics type of equations based on third- and fifth-order Compact Upwind Differencing (CUD) are considered. To validate their properties following from a linear analysis, calculations were carried out using the inviscid and viscous Burgers' equation as well as the compressible Navier–Stokes equation written in the conservative form for curvilinear coordinates. In the latter case, transonic cascade flow was chosen as a representative example. The performance of the CUD methods was estimated by investigating mesh convergence of the solutions and comparing with the results of second-order schemes. It is demonstrated that the oscillation-free steep gradients solutions obtained without using smoothing techniques can provide considerable increase of accuracy even when exploiting coarse meshes.     

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.     

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.     

A Particle-Mesh Method for the Shallow Water Equations Near Geostrophic Balance

In this paper we outline a new particle-mesh method for rapidly rotating shallow water flows based on a set of regularized equations of motion. The time-stepping method uses an operator splitting of the equations into an Eulerian gravity wave part and a Lagrangian advection part. An essential ingredient is the advection of absolute vorticity by means of translated radial basis functions. We show that this implies exact conservation of enstrophy. The method is tested on two model problems based on the qualitative features of the solutions obtained (i.e., dispersion or smoothness of potential vorticity contours) as well as on the increase in mean divergence level.     

Wave Simulation in Frozen Porous Media

We propose a numerical algorithm for simulation of wave propagation in frozen porous media, where the pore space is filled with ice and water. The model, based on a Biot-type three-phase theory, predicts three compressional waves and two shear waves and models the attenuation level observed in rocks. Attenuation is modeled with exponential relaxation functions which allow a differential formulation based on memory variables. The wavefield is obtained using a grid method based on the Fourier differential operator and a Runge–Kutta time-integration algorithm. Since the presence of slow quasistatic modes makes the differential equations stiff, a time-splitting integration algorithm is used to solve the stiff part analytically. The modeling is second-order accurate in the time discretization and has spectral accuracy in the calculation of the spatial derivatives.     

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.     

Solid–fluid diffuse interface model in cases of extreme deformations

Diffuse interface methods have been recently proposed and successfully used for accurate compressible multi-fluid computations Abgrall [1]; Kapila et al. [20]; Saurel et al. [30]. These methods deal with extended systems of hyperbolic equations involving a non-conservative volume fraction equation and relaxation terms. Following the same theoretical frame, we derive here an Eulerian diffuse interface model for elastic solid-compressible fluid interactions in situations involving extreme deformations. Elastic effects are included following the Eulerian conservative formulation proposed in Godunov [16], Miller and Colella [23], Godunov and Romenskii [17], Plohr and Plohr [27] and Gavrilyuk et al. [14]. We apply first the Hamilton principle of stationary action to derive the conservative part of the model. The relaxation terms are then added which are compatible with the entropy inequality. In the limit of vanishing volume fractions the Euler equations of compressible fluids and a conservative hyperelastic model are recovered. It is solved by a unique hyperbolic solver valid at each mesh point (pure fluid, pure solid and mixture cell). Capabilities of the model and methods are illustrated on various tests of impacts of solids moving in an ambient compressible fluid.     

A Volume of Fluid Based Method for Fluid Flows with Phase Change

This paper presents a numerical method directed towards the simulation of flows with mass transfer due to changes of phase. We use a volume of fluid (VOF) based interface tracking method in conjunction with a mass transfer model and a model for surface tension. The bulk fluids are viscous, conducting, and incompressible. A one-dimensional test problem is developed with the feature that a thin thermal layer propagates with the moving phase interface. This test problem isolates the ability of a method to accurately calculate the thermal layers responsible for driving the mass transfer in boiling flows. The numerical method is tested on this problem and then is used in simulations of horizontal film boiling.     

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.     

Simple and efficient relaxation methods for interfaces separating compressible fluids,cavitating flows and shocks in multiphase mixtures

Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002–3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664–698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAM Journal of Multiscale and Modeling and Simulation (5) (2006) 84–115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425–467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248] is adapted to the present formulation.     

Moving Mesh Methods in Multiple Dimensions Based on Harmonic Maps

In practice, there are three types of adaptive methods using the finite element approach, namely the h-method, p-method, and r-method. In the h-method, the overall method contains two parts, a solution algorithm and a mesh selection algorithm. These two parts are independent of each other in the sense that the change of the PDEs will affect the first part only. However, in some of the existing versions of the r-method (also known as the moving mesh method), these two parts are strongly associated with each other and as a result any change of the PDEs will result in the rewriting of the whole code. In this work, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. Our efforts are to keep the advantages of the r-method (e.g., keep the number of nodes unchanged) and of the h-method (e.g., the two parts in the code are independent). A framework for adaptive meshes based on the Hamilton–Schoen–Yau theory was proposed by Dvinsky. In this work, we will extend Dvinsky's method to provide an efficient solver for the mesh-redistribution algorithm. The key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure. Each iteration step is to move the mesh closer to the harmonic map. This procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration. The numerical schemes are applied to a number of test problems in two dimensions. It is observed that the mesh-redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi-dimensional computations.