A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system. This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.     

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

In [14], Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However, it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work [8] to the first order control volume scheme of Maire in [14] to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.     

A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the "initial" cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.     

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.     

Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

We develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.     

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme's order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.     

A Priori Subcell Limiting Approach for the FR/CPR Method on Unstructured Meshes

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) methods on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on modal energy coefficients is proposed to detect troubled cells, where discontinuities exist. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted (NNW) interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR method. Numerical investigations show that the proposed subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.     

An Efficient Positivity-Preserving Finite Volume Scheme for the Nonequilibrium Three-Temperature Radiation Diffusion Equations on Polygonal Meshes

This paper develops an efficient positivity-preserving finite volume scheme for the two-dimensional nonequilibrium three-temperature radiation diffusion equations on general polygonal meshes. The scheme is formed as a predictor-corrector algorithm. The corrector phase obtains the cell-centered solutions on the primary mesh, while the predictor phase determines the cell-vertex solutions on the dual mesh independently. Moreover, the flux on the primary edge is approximated with a fixed stencil and the nonnegative cell-vertex solutions are not reconstructed. Theoretically, our scheme does not require any nonlinear iteration for the linear problems, and can call the fast nonlinear solver (e.g. Newton method) for the nonlinear problems. The positivity, existence and uniqueness of the cell-centered solutions obtained on the corrector phase are analyzed, and the scheme on quasi-uniform meshes is proved to be $L^2$- and $H^1$-stable under some assumptions. Numerical experiments demonstrate the accuracy, efficiency and positivity of the scheme on various distorted meshes.     

Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

The purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks, T. J. R. Hughes, "An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves", Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.     

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method. The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the resulting microdynamic evolution equation. The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form. The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion. The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented. It is found the Knudsen minimum can be captured in all three statistics studied. The results also indicate distinct characteristics of the effects of quantum statistics.     

Application of the LS-STAG Immersed Boundary/ Cut-Cell Method to Viscoelastic Flow Computations

This paper presents the extension of a well-established Immersed Boundary (IB)/cut-cell method, the LS-STAG method (Y. Cheny & O. Botella, J. Comput. Phys. Vol. 229, 1043-1076, 2010), to viscoelastic flow computations in complex geometries. We recall that for Newtonian flows, the LS-STAG method is based on the finite-volume method on staggered grids, where the IB boundary is represented by its level-set function. The discretization in the cut-cells is achieved by requiring that global conservation properties equations be satisfied at the discrete level, resulting in a stable and accurate method and, thanks to the level-set representation of the IB boundary, at low computational costs.In the present work, we consider a general viscoelastic tensorial equation whose particular cases recover well-known constitutive laws such as the Oldroyd-B, White-Metzner and Giesekus models. Based on the LS-STAG discretization of the Newtonian stresses in the cut-cells, we have achieved a compatible velocity-pressure-stress discretization that prevents spurious oscillations of the stress tensor. Applications to popular benchmarks for viscoelastic fluids are presented: the four-to-one abrupt planar contraction flows with sharp and rounded re-entrant corners, for which experimental and numerical results are available. The results show that the LS-STAG method demonstrates an accuracy and robustness comparable to body-fitted methods.     

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However, such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.     

A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations

A monotone cell-centered finite volume scheme for diffusion equations on tetrahedral meshes is established in this paper, which deals with tensor diffusion coefficients and strong discontinuous diffusion coefficients. The first novelty here is to propose a new method of interpolating vertex unknowns (auxiliary unknowns) with cell-centered unknowns (primary unknowns), in which a sufficient condition is given to guarantee the non-negativity of vertex unknowns. The second novelty of this paper is to devise a modified Anderson acceleration, which is based on an iterative combination of vertex unknowns and will be denoted as AA-Vertex algorithm, in order to solve the nonlinear scheme efficiently. Numerical testes indicate that our new method can obtain almost second order accuracy and is more accurate than some existing methods. Furthermore, with the same accuracy, the modified Anderson acceleration is much more efficient than the usual one.     

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases

With discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-differential-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolution in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.     

Evaluation of Three Lattice Boltzmann Models for Particulate Flows

A comparative study is conducted to evaluate three types of lattice Boltzmann equation (LBE) models for fluid flows with finite-sized particles, including the lattice Bhatnagar-Gross-Krook (BGK) model, the model proposed by Ladd [Ladd AJC, J. Fluid Mech., 271, 285-310 (1994); Ladd AJC, J. Fluid Mech., 271, 311-339 (1994)], and the multiple-relaxation-time (MRT) model. The sedimentation of a circular particle in a two-dimensional infinite channel under gravity is used as the first test problem. The numerical results of the three LBE schemes are compared with the theoretical results and existing data. It is found that all of the three LBE schemes yield reasonable results in general, although the BGK scheme and Ladd's scheme give some deviations in some cases. Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes. Regarding the computational efficiency, it is found that the BGK scheme is the most superior one, while the other two schemes are nearly identical. We also observe that the MRT scheme can unequivocally reduce the viscosity dependence of the wall correction factor in the simulations, which reveals the superior robustness of the MRT scheme. The superiority of the MRT scheme over the other two schemes is also confirmed by the simulation of the sedimentation of an elliptical particle.     

An Efficient,Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.     

Modeling Magma Dynamics with a Mixed Fourier Collocation - Discontinuous Galerkin Method

A high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.     

Sound Propagation Properties of the Discrete-Velocity Boltzmann Equation

As the numerical resolution is increased and the discretization error decreases, the lattice Boltzmann method tends towards the discrete-velocity Boltzmann equation (DVBE). An expression for the propagation properties of plane sound waves is found for this equation. This expression is compared to similar ones from the Navier-Stokes and Burnett models, and is found to be closest to the latter. The anisotropy of sound propagation with the DVBE is examined using a two-dimensional velocity set. It is found that both the anisotropy and the deviation between the models is negligible if the Knudsen number is less than 1 by at least an order of magnitude.     

Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

High order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost as first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains the ν-th order (ν≤6) approximations of the α-th derivative (α>0) or integral (α<0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with α∈(1,2) for time dependent problem. By weighting and shifting Lubich's 2nd order discretization scheme, in [Chen & Deng, SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD operators there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.