基于气动声学问题的高阶非线性紧致格式

文章基于线性中心紧致差分格式, 通过非线性加权插值的方法来求解网格中心处的函数值.这类格式保持了原有中心紧致差分格式的高阶精度和低耗散特性, 同时其分辨率也非常高, 由于其非线性插值的机制, 使得这类格式能够捕捉强激波, 所以这类新的高阶非线性紧致格式是一种较好的模拟湍流和气动声学等多尺度问题的方法.      

Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence

The Osher–Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag–Tang 2D vortex problem. Total-variation-bounded (TVB) behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley–Leverett test simulations.     

Hybrid weighted essentially non-oscillatory schemes with different indicators

A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property.     

Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi–Dirac and Gauss–Fermi statistics

We compare three thermodynamically consistent Scharfetter–Gummel schemes for different distribution functions for the carrier densities, including the Fermi–Dirac integral of order 1/2 and the Gauss–Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter–Gummel scheme requires the solution of an integral equation. Since one cannot solve this integral equation analytically, several modified Scharfetter–Gummel schemes have been proposed, yielding explicit flux approximations to the implicit generalized flux. The two state-of-the-art modified fluxes used in device simulation software are the diffusion-enhanced flux and the inverse activity coefficient averaging flux. We would like to study which of these two modified schemes approximates the implicit flux better. To achieve this, we propose a new method to solve the integral equation numerically based on Gauss quadrature and Newton’s method. This numerical procedure provides a highly accurate reference flux, enabling us to compare the quality of the two modified Scharfetter–Gummel schemes. We extend previous results (Farrell in J Comput Phys 346:497–513, 2017a) showing that the diffusion-enhanced ansatz leads to considerably lower flux errors for the Blakemore approximation to the physically more relevant Fermi–Dirac and Gauss–Fermi statistics.     

求解多维欧拉方程的二阶非结构网格混合旋转Riemann求解器

将基于旋转近似Riemann求解器的二阶精度迎风型有限体积方法推广到非结构网格,采用基于网格中心的有限体积法,梯度的计算采用基于节点的方法引入更多的控制体模板,限制器的构造采用与非结构化网格相适应的形式.在求解Riemann问题时,沿具有一定物理意义的两个迎风方向,即控制体界面两侧速度差矢量方向及与之正交的方向.能够完全消除基于Riemann求解器的通量差分裂格式存在的激波不稳定或"红斑"现象.为减小计算量,采用HLL和Roe FDS混合旋转格式.     

An iterative multiscale finite volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).     

Heat conduction in relativistic neutral gases revisited

The kinetic theory of dilute gases to first order in the gradients yields linear relations between forces and fluxes. The heat flux for the relativistic gas has been shown to be related not only to the temperature gradient but also to the density gradient in the representation where number density, temperature and hydrodynamic velocity are the independent state variables. In this work we show the calculation of the corresponding transport coefficients from the full Boltzmann equation and compare the magnitude of the relativistic correction.     

Turbulent scalar fluxes in H2-air premixed flames at low and high Karlovitz numbers

The statistical behaviour and closure of sub-grid scalar fluxes in the context of turbulent premixed combustion have been assessed based on an a priori analysis of a detailed chemistry Direct Numerical Simulation (DNS) database consisting of three hydrogen-air flames spanning the corrugated flamelets (CF), thin reaction zones (TRZ) and broken reaction zones (BRZ) regimes of premixed turbulent combustion. The sub-grid scalar fluxes have been extracted by explicit filtering of DNS data. It has been found that the conventional gradient hypothesis model is not appropriate for the closure of sub-grid scalar flux for any scalar in the context of a multispecies system. However, the predictions of the conventional gradient hypothesis exhibit a greater level of qualitative agreement with DNS data for the flame representing the BRZ regime. The aforementioned behaviour has been analysed in terms of the properties of the invariants of the anisotropy tensor in the Lumley triangle. The flames in the CF and TRZ regimes are characterised by a pronounced two-dimensional anisotropy due to strong heat release whereas a three-dimensional and more isotropic behaviour is observed for the flame located in the BRZ regime. Two sub-grid scalar flux models which are capable of predicting counter-gradient transport have been considered for a priori DNS assessment of multispecies systems and have been analysed in terms of both qualitative and quantitative agreements. By combining the latter two sub-grid scalar flux closures, a new modelling strategy is suggested where one model is responsible for properly predicting the conditional mean accurately and the other model is responsible for the correlations between model and unclosed term. Detailed physical explanations for the observed behaviour and an assessment of existing modelling assumptions have been provided. Finally, the classical Bray–Moss–Libby theory for the scalar flux closure has been extended to address multispecies transport in the context of large eddy simulations.     

Peculiarities in the Formation of X-Ray Fluxes by Waveguide–Resonators of Different Construction

The phenomenon of total external reflection (TER) of quasi-monochromatic X-ray radiation fluxes on a material interface and the effect of waveguide–resonator propagation of these fluxes in nanosize extended slit clearance, as well as a device operating on the basis of this effect—a planar X-ray waveguide–resonator—are briefly described. Experimental data on the formation of an X-ray flux by a composite X-ray waveguide–resonator are presented, and a model describing the decrease in the angular divergence of the formed flux without a decrease in the integral intensity is proposed. The model is based on the conception of partial angular tunneling of the radiation flux in the gap between two consequently mounted and mutually adjusted waveguide–resonators; the tunneling is implemented due to the interaction between interference fields of standing X-ray waves excited by the radiation transported by the slit clearance of these waveguide–resonators.     

A new class of gas-kinetic relaxation schemes for the compressible Euler equations

Starting from the gas-kinetic model, a new class of relaxation schemes for the Euler equations is presented. In contrast to the Riemann solver, these schemes provide a multidimensional dynamical gas evolution model, which combines both Lax-Wendroff and kinetic flux vector splitting schemes, and their coupling is based on the fact that a nonequilibrium state will evolve into an equilibrium state along with the increase of entropy. The numerical fluxes are constructed without getting into the details of the particle collisions. The results for many well-defined test cases are presented to indicate the robustness and accuracy of the current scheme.     

High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics

A class of high-order kinetic flux vector splitting schemes are presented for solving ideal quantum gas dynamics based on quantum statistical mechanics. The collisionless quantum Boltzmann equation approach is adopted and both Bose–Einstein and Fermi–Dirac gases are considered. The formulas for the split flux vectors are derived based on the general three-dimensional distribution function in velocity space and formulas for lower dimensions can be directly deduced. General curvilinear coordinates are introduced to treat practical problems with general geometry. High-order accurate schemes using weighted essentially non-oscillatory methods are implemented. The resulting high resolution kinetic flux splitting schemes are tested for 1D shock tube flows and shock wave diffraction by a 2D wedge and by a circular cylinder in ideal quantum gases. Excellent results have been obtained for all examples computed.     

Very simple,carbuncle-free,boundary-layer-resolving,rotated-hybrid Riemann solvers

In this paper, we propose new Euler flux functions for use in a finite-volume Euler/Navier–Stokes code, which are very simple, carbuncle-free, yet have an excellent boundary-layer-resolving capability, by combining two different Riemann solvers into one based on a rotated Riemann solver approach. We show that very economical Euler flux functions can be devised by combining the Roe solver (a full-wave solver) and the Rusanov/HLL solver (a fewer-wave solver), based on a rotated Riemann solver approach: a fewer-wave solver automatically applied in the direction normal to shocks to suppress carbuncles and a full-wave solver applied, again automatically, across shear layers to avoid an excessive amount of dissipation. The resulting flux functions can be implemented in a very simple and economical manner, in the form of the Roe solver with modified wave speeds, so that converting an existing Roe flux code into the new fluxes is an extremely simple task. They require only 7–14% extra CPU time and no problem-dependent tuning parameters. These new rotated fluxes are not only robust for shock-capturing, but also accurate for resolving shear layers. This is demonstrated by an extensive series of numerical experiments with standard finite-volume Euler and Navier–Stokes codes, including various shock instability problems and also an unstructured grid case.     

Calculation of Transfer Functions of Multilayer Biotissues in the Problems of Correction of Their Fluorescence Spectra

A method for rapid calculation of a flux of stimulated fluorescence of a multilayer optically dense medium with inhomogeneous distribution of the fluorophore has been developed. The light field in the medium at the excitation wavelength of fluorescence is represented by a superposition of incident collimated, incident diffuse, and reflected diffuse fluxes. A two-stream approximation is used to describe the light field in the medium at the wavelength of emission of the fluorescence. Fluxes in adjacent elementary layers of the medium and on its surface are connected by simple matrix operators that are obtained using a combination of engineering approaches of radiation-transfer theory and single-scattering approximation. The calculations of fluorescence fluxes of a four-layer biotissue that are excited and recorded at 400–800 nm are compared with their Monte Carlo simulation with a discrepancy of 1%. The effect of the propagation medium on the fluorescence spectra of 5-ALA-induced protoporphyrin IX that are recorded from human skin was studied, and a technique for their correction that is based on measurements and quantitative analysis of the diffuse reflectance spectrum of skin was proposed.     

A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.     

Increasing the accuracy in locally divergence-preserving finite volume schemes for MHD

It is of utmost interest to control the divergence of the magnetic flux in simulations of the ideal magnetohydrodynamic equations since, in general, divergence errors tend to accumulate and render the schemes unstable. This paper presents a higher-order extension of the locally divergence-preserving procedure developed in Torrilhon [M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput. 26 (2005) 1166–1191]; a fourth-order accurate local redistribution of the numerical magnetic field fluxes of a finite volume base scheme is introduced. The redistribution ensures that a fourth-order accurate discrete divergence operator is preserved to round off errors when applied to the cell averages of the magnetic flux density. The developed procedure is applicable to generic semi-discrete finite volume schemes and its purpose is to stabilize the schemes using a local procedure that respects the accuracy of the base scheme to a greater extent than the previous second-order achievements. Numerical experiments that demonstrate the properties of the new procedure are also presented.     

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.     

A multistep flux-corrected transport scheme

A multistep flux-corrected transport (MFCT) scheme is developed to achieve conservative and monotonic tracer transports for multistep dynamical cores. MFCT extends Zalesak two-time level scheme to any multistep time-differencing schemes by including multiple high-order fluxes in the antidiffusive flux, while computing the two-time level low-order monotone solution. The multistep time-differencing scheme used in this study is the third-order Adams–Bashforth (AB3) scheme implemented in a finite-volume icosahedral shallow-water model. The accuracy of AB3 MFCT is quantified by the shape-preserving advection experiments in non-divergent flow, as well as a cosine bell whose shape changes during advection in shear flow. AB3 MFCT has been shown to be insensitive to time step size. This make AB3 MFCT an attractive transport scheme for explicit high resolution model applications with small time step. MFCT is tested in shallow-water model simulations to demonstrate that the use of MFCT maintains positive-definite tracer transport, while at the same time conserving both fluid mass and tracer mass within round-off errors in the AB3 dynamic core.     

Combining the independent pixel and point-spread function approaches to simulate the actinic radiation field in moderately inhomogeneous 3D cloudy media

A fast method is presented for gaining 3D actinic flux density fields, F_act, in clouds employing the Independent Pixel Approximation (IPA) with a parameterized horizontal photon transport to imitate radiative smoothing effects. For 3D clouds the IPA is an efficient method to simulate radiative transfer, but it suffers from the neglect of horizontal photon fluxes leading to significant errors (up to locally 30% in the present study). Consequently, the resulting actinic flux density fields exhibit an unrealistically rough and rugged structure. In this study, the radiative smoothing is approximated by applying a physically based smoothing algorithm to the calculated IPA actinic flux field.     

High-order finite-volume methods for the shallow-water equations on the sphere

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM⁺-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge–Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM⁺-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.