A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme for the Stokes and incompressible Navier–Stokes equations is proposed. The original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least-squares sense. The collocation conditions and the interface conditions lead to overdetermined systems. These systems are directly solved by QR decomposition of the underlying matrices. By numerical simulations it is shown that the direct method leads to better results than the approach with normal equations. Furthermore, it is shown that the condition numbers can be reduced by introducing the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Finally, our scheme is successfully applied to the regularized and lid-driven cavity flow problems.     

Numerical simulation of core convection by a multi-layer semi-implicit spherical spectral method

A semi-implicit multi-layer spherical spectral method for simulating stellar core convection is described. The fully compressible three-dimensional hydrodynamic equations with rotation and energy generation are solved. Prognostic variables are expressed as finite sums of spherical harmonics in the horizontal directions and handled by the finite difference method in the radial direction. The stratified approximation is used to simplify the nonlinearity to quadratic. A multi-layer scheme is employed to overcome the time step problem arising from shrinking grid sizes in the physical space near the center of the star. Despite of the different spectral truncations in different layers, round-off conservation of the total mass and total angular momentum of the whole domain can be maintained, and were confirmed numerically. The code is parallelized; with 12 processors the speedup factor is about 9. The solutions of model core convection with and without rotation are discussed.     

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach     

A conservative numerical scheme for solving an autonomous Hamiltonian system

A new numerical scheme is proposed for solving Hamilton’s equations that possesses the properties of symplecticity. Just as in all symplectic schemes known to date, in this scheme the conservation laws of momentum and angular momentum are satisfied exactly. A property that distinguishes this scheme from known schemes is proved: in the new scheme, the energy conservation law is satisfied for a system of linear oscillators. The new numerical scheme is implicit and has the third order of accuracy with respect to the integration step. An algorithm is presented by which the accuracy of the scheme can be increased up to the fifth and higher orders. Exact and numerical solutions to the two-body problem, calculated by known schemes and by the scheme proposed here, are compared.     

Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids

The development of a compact fourth-order finite volume method for solutions of the Navier–Stokes equations on staggered grids is presented. A special attention is given to the conservation laws on momentum control volumes. A higher-order divergence-free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes. Three forms of the nonlinear correction for staggered grids are proposed and studied. The accuracy of each approximation is assessed comparatively in Fourier space. The importance of higher-order approximations of pressure is discussed and numerically demonstrated. Fourth-order accuracy of the complete scheme is illustrated by the doubly-periodic shear layer and the instability of plane-channel flow. The efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations. The proposed scheme is highly accurate and efficient. At the same level of accuracy, the fourth-order scheme can be ten times faster than the second-order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

A second-order unsplit Godunov scheme for cell-centered MHD: The CTU-GLM scheme

We assess the validity of a single step Godunov scheme for the solution of the magnetohydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by Dedner et al. (2002) [11]. The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.     

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.     

解流体力学方程组的一种隐式完全守恒差分格式

对Lagrange非守恒流体力学方程组给出了一种隐式完全守恒差分格式,既保证了质量、动量和总能量守恒的差分近似,又能满足内能与动能的平衡特性,提高了数值解的精度。并用该格式对两个可压缩理想流体模型进行了数值计算,并与其它差分格式作了比较。     

Time-dependent aerodynamic non-LTE radiative transfer

A method is developed for solving simultaneously in one dimension the equation of transfer for non-LTE spectral line radiation and the time-dependent equations specifying conservation of mass, energy and linear momentum. In particular, we illustrate the method on a ‘simple’ time-dependent problem in which a pulsating disturbance at some point in a model homogeneous atmosphere propagates towards the surface and steepens into a shock. The resulting emergent intensities show rather dramatic changes over very small time intervals due to the effect of the velocity, density and temperature distributions on the radiative absorption properties of the gas, and thus emphasises the need to solve the above-mentioned four basic equations if one is to obtain physically realistic model atmospheres experiencing initial disturbances.     

Multi-Symplectic Wavelet Collocation Method for Maxwell's Equations

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell's equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.     

无结构网格有限体积方法及在热对流中的应用研究

首次将无结构三角网格的有限体积方法和压强连接半隐式算法相结合,用于求解非平行壁管道中的热对流问题。并由此分析了化学汽相淀积薄膜生长的均匀性问题。计算结果对于分析一类管道中热和动量输运现象均有普遍指导意义。     

General Navier–Stokes-like momentum and mass-energy equations

A new system of general Navier–Stokes-like equations is proposed to model electromagnetic flow utilizing analogues of hydrodynamic conservation equations. Such equations are intended to provide a different perspective and, potentially, a better understanding of electromagnetic mass, energy and momentum behaviour. Under such a new framework additional insights into electromagnetism could be gained. To that end, we propose a system of momentum and mass-energy conservation equations coupled through both momentum density and velocity vectors.     

旋流排气管的一维非定常流动计算

动力机械装置中广泛存在着非定常旋流流动现象．本文根据质量、动量、能量和旋流动量矩守恒方程，建立了管内非定常旋流流动的一维计算模型，并应用特征线方法推导出了其数值计算格式，是管内非定常一维流动计算的扩展．应用于一台四缸涡轮增压柴油机旋流排气管的计算，通过与实测压力波的比较，表明计算模型有较好的计算精度．     

Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement

Least-squares spectral element solution of steady, two-dimensional, incompressible flows are obtained by approximating velocity, pressure and vorticity variable set on Gauss–Lobatto–Legendre nodes. Constrained Approximation Method is used for h- and p-type nonconforming interfaces of quadrilateral elements. Adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient. Effective use of p-refinement to overcome poor mass conservation drawback of least-squares formulation and successful use of h- and p-refinement together to solve problems with geometric singularities are demonstrated. Capabilities and limitations of the developed code are presented using Kovasznay flow, flow past a circular cylinder in a channel and backward facing step flow.     

The temperature distributions of the electrons,atoms and ions in aerodynamic radiative transfer

The time-dependent individual specie equations specifying conservation of mass, energy and linear momentum for electrons, atoms and ions are solved, together with the transfer equation for continuum radiation, for several specific situations. We assume the temperature of the atoms and ions to be equal, but different to the electron temperature. An ionising disturbance is propagated through a physically idealistic radiating medium and it is shown that the time-development of the differences in temperature distributions are such as to sufficiently affect the absorption and emission properties of the gas, and thus, the emergent flux intensities.     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

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.     

Hybrid Flux-Splitting Schemes for a Two-Phase Flow Model

In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes.     

Modeling of multicomponent-fuel drop-laden mixing layers having a multitude of species

A formulation representing multicomponent-fuel (MC-fuel) composition as a probability distribution function (PDF) depending on the molar mass is used to construct a model of a large number of MC-fuel drops evaporating in a gas flow, so as to assess the extent of fuel specificity on the vapor composition. The PDF is a combination of two Gamma PDFs, which was previously shown to duplicate the behavior of a fuel composed of many species during single drop evaporation. The conservation equations are Eulerian for the flow and Lagrangian for the physical drops, all of which are individually followed. The gas conservation equations for mass, momentum, species, and energy are complemented by differential conservation equations for the first four moments of the gas-composition PDF; all coupled to the perfect gas equation of state. Source terms in all conservation equations couple the gas phase to the drops. The drop conservation equations for mass, position, momentum, and energy are complemented by differential equations for four moments of the liquid-composition PDF. The simulations are performed for a three-dimensional mixing layer whose lower stream is initially laden with drops. Initial perturbations excite the layer to promote the double pairing of its four initial spanwise vortices to an ultimate vortex. The drop temperature is initially lower than that of the surrounding gas, initiating drop heating and evaporation. The results focus on both evolution and the state of the drops and gas when layers reach a momentum-thickness maximum past the double vortex pairing; particular emphasis is on the gas composition. Comparisons between simulations with n-decane, diesel, and three kerosenes show that at same initial Reynolds number and Stokes number distribution, a single-component fuel cannot represent MC fuels. Substantial differences among the MC-fuel vapor composition indicate that fuel specificity must be captured for the prediction of combustion.     

积分梯度法的两种格式

在二维柱坐标系下Lagrange流体力学的计算中,积分梯度法是动量方程的一种有效离散方法.积分梯度法中,IGT(Integral Gradient Total)格式不能保持柱几何下一维球对称性;IGA(Integral Gradient Average)格式可以保持一维球对称性,但当相邻网格质量相差比较大时,会得到远远脱离真实物理现象的加速度.深入研究IGA和IGT格式发现,当相邻网格边界压力取为质量加权时,即使相邻网格质量相差较大,对于一维平面和一维柱问题,IGT与IGA等价;在二维情形下,可以缩小IGT和IGA之间的差异.理论证明,IGA格式不能保持系统的动量守恒,IGT格式能保持系统的动量守恒性.数值模拟结果进一步显示了这两个格式的优缺点.