基于静动态重构的高阶DG/FV混合格式在二维非结构网格中的推广

通过比较间断Galerkin有限元方法(DGM)和有限体积方法(FVM),提出"静态重构"和"动态重构"的概念,进一步建立基于静动态"混合重构"算法的三阶DG/FV混合格式.在DG/FV混合格式中,单元平均值和一阶导数由DGM方法"动态重构",二阶导数利用FVM方法"静态重构";在此基础上,构造高阶多项式插值函数,得到...     

Extending Seventh-Order Dissipative Compact Scheme Satisfying Geometric Conservation Law to Large Eddy Simulation on Curvilinear Grids

Seventh-order hybrid cell-edge and cell-node dissipative compact scheme (HDCS-E8T7) is extended to a new implicit large eddy simulation named HILES on stretched and curvilinear meshes. Although the conception of HILES is similar to that of monotone integrated LES (MILES), i.e., truncation error of the discretization scheme itself is employed to model the effects of unresolved scales, HDCS-E8T7 is a new high-order finite difference scheme, which can eliminate the surface conservation law (SCL) errors and has inherent dissipation. The capability of HILES is tested by solving several benchmark cases. In the case of flow past a circular cylinder, the solutions of HILES fulfilling the SCL have good agreement with the corresponding experiment data, however, the flow field is gradually contaminated when the SCL error is enlarged. With the help of fulling the SCL, ability of HILES for handling complex geometry has been enhanced. The numerical solutions of flow over delta wing demonstrate the potential of HILES in simulating turbulent flow on complex configuration.     

DGM-FD: A finite difference scheme based on the discontinuous Galerkin method applied to wave propagation

In this paper we formulate a numerical method that is high order with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the discontinuous Galerkin method (DGM) applied to the hyperbolic equation, resulting in finite difference type schemes applicable to non-uniform grids. The schemes will be referred to as DGM-FD schemes. These schemes inherit naturally some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Stability of the schemes with boundary closures is investigated and validated. Proposed scheme is demonstrated by numerical examples including the linearized acoustic waves and solutions of non-linear Burger’s equation and the flat-plate boundary layer problem. For non-linear equations, proposed flux finite difference formula requires no explicit upwind and downwind split of the flux. This is in contrast to existing upwind finite difference schemes in the literature.     

Noise Prediction in Subsonic Flow Using Seventh-Order Dissipative Compact Scheme on Curvilinear Mesh

In this paper, we investigate the performance of the seventh-order hybrid cell-edge and cell-node dissipative compact scheme (HDCS-E8T7) on curvilinear mesh for noise prediction in subsonic flow. In order to eliminate the errors due to surface conservation law (SCL) is dissatisfied with curvilinear meshes, the symmetrical conservative metric method (SCMM) is adopted to calculate the grid metric derivatives for the HDCS-E8T7. For the simulation of turbulence flow which may have main responsibility for the noise radiation, the new high-order implicit large eddy simulation (HILES) based on the HDCS-E8T7 is employed. Three typical cases, i.e., scattering of acoustic waves by multiple cylinder, sound radiated from a rod-airfoil and subsonic jet noise from nozzle, are chosen to investigate the performance of the new scheme for predicting aeroacoustic problem. The results of scattering of acoustic waves by multiple cylinder indicate that the HDCS-E8T7 satisfying the SCL has high resolution for the aeroacoustic prediction. The potential of the HDCS-E8T7 for aeroacoustic problems on complex geometry is shown by the predicting sound radiated from a rod-airfoil configuration. Moreover, the subsonic jet noise from nozzle has been successfully predicted by the HDCS-E8T7.     

Nonuniform time-step Runge–Kutta discontinuous Galerkin method for Computational Aeroacoustics

With many superior features, Runge–Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge–Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge–Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge–Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.     

A PLIC–VOF method suited for adaptive moving grids

This paper presents a new variant of the volume-of-fluid (VOF) color function C advection algorithm based on the piecewise linear interface construction (PLIC) method suitable for use on general moving grids. From several existing methods for reconstructing the linear interface we adopted the least squares volume-of-fluid interface reconstruction algorithm (LVIRA) which can be easily implemented on general grids. The distinguishing step in the advection algorithm that takes into account the grid movement is the construction of the donating region containing the fluid passing through corresponding cell-faces in a single time-step. The donating regions are constructed utilizing fluid velocity in cell corners relative to grid (corner) velocities. The method is conservative as it complies with the space conservation law (SCL) and requires a proper definition of the grid velocities and fluxes due to the grid movement. The accuracy of the presented advection algorithm is assessed with standard test cases. It is comparable with other PLIC based algorithms on fixed grids, while the applicability on adaptive moving grids enables a considerable reduction in the number of grid cells.     

A class of hybrid DG/FV methods for conservation laws II: Two-dimensional cases

By comparing the discontinuous Galerkin (DG) methods, the k-exact finite volume (FV) methods and the lift collocation penalty (LCP) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ was introduced for higher-order numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods was presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent face neighboring cells. In this follow-up paper, the hybrid DG/FV schemes are extended onto two-dimensional unstructured and hybrid grids. The two-dimensional linear and non-linear scalar conservation law and Euler equations are considered. Some typical cases are tested to demonstrate the performance of the hybrid DG/FV method, and the numerical results show that they can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy in the same mesh.     

A class of hybrid DG/FV methods for conservation laws I: Basic formulation and one-dimensional systems

By comparing the discontinuous Galerkin (DG) and the finite volume (FV) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ is introduced for high-order numerical methods. Based on the new concept, a class of hybrid DG/FV schemes is presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of a piecewise polynomial solution are computed locally in a cell by the DG method based on Taylor basis functions (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and its adjacent neighboring cells. The hybrid DG/FV methods can greatly reduce CPU time and memory required by the traditional DG methods with the same order of accuracy on the same mesh, and they can be extended directly to unstructured and hybrid grids in two and three dimensions similar to the DG and/or FV methods. The hybrid DG/FV methods are applied to one-dimensional conservation law, including linear and non-linear scalar equation and Euler equations. In order to capture the strong shock waves without spurious oscillations, a simple shock detection approach is developed to mark ‘trouble cells’, and a moment limiter is adopted for higher-order schemes. The numerical results demonstrate the accuracy, and the super-convergence property is shown for the third-order hybrid DG/FV schemes. In addition, by analyzing the eigenvalues of the semi-discretized system in one dimension, we discuss the spectral properties of the hybrid DG/FV schemes to explain the super-convergence phenomenon.     

Piston error characteristics of the self-phase-controlled stimulated Brillouin scattering phase conjugate mirrors

It was reported by Kong et al. that the self-phase-controlled stimulated Brillouin scattering phase conjugate mirror (SBS-PCM) is useful for the beam combination laser amplifier system, which generally introduces a phase delay (piston error) between combined beams along separate optical paths. In this work, we have investigated the piston error characteristics of two schemes for self-phase-controlled SBS-PCMs, a collinear scheme and a triangular scheme. Experimental results show that the piston errors of the reflected beams are 2kδ and kδ for the former and the latter scheme, respectively, where k is the wave number and δ is the optical path difference (OPD) variation introduced into the pass.     

四阶紧致差分格式对静力模式中垂直网格计算频散性的影响

为了说明四阶紧致差分格式在大气和海洋数值模式中的潜在价值,提出一种通用方法,推导静力线性斜压适应方程组在微分和差分情况下的频散关系,水平尺度分100 km,10 km和1 km三种情况,从频率、水平群速和垂直群速方面,对采用二阶中央差和四阶紧致差分格式情况下,非跳点网格(N网格)、Lorenz网格(L网格)、Charney-Phillips网格(CP网格)、Lorenz时间跳点网格(LTS网格)和Charney-Phillips时间跳点网格(CPTS网格)的计算特性进行比较,发现采用高精度的四阶紧致差分格式总体上可以明显减少上述三种水平尺度波动在N网格、CP网格、L网格和CPTS网格上的频率、水平群速和垂直群速误差,但对LTS网格,采用四阶紧致差分格式,会使得计算水平群速和垂直群速误差变大.     

Self‐Force in 1D Electrostatic Particle‐in‐Cell Codes for NonEquidistant Grids

Effects of non‐equidistant grids on momentum conservation is studied for simple test cases of an electrostatic 1D PIC code. The aim is to reduce the errors in energy and momentum conservation. Assuming an exact Poisson solver only numerical errors for the particle mover are analysed. For the standard electric field calculation using a central‐difference scheme, artificial electric fields at the particle position are generated in the case when the particle is situated next to a cell size change. This is sufficient to destroy momentum conservation. A modified electric field calculation scheme is derived to reduce this error. Independent of the calculation scheme additional fake forces in a two‐particle system are found which result in an error in the total kinetic energy of the system. This contribution is shown to be negligible for many particle systems. To test the accuracy of the two electric field calculation schemes numerical tests are done to compare with an equidistant grid set‐up. All tests show an improved momentum conservation and total kinetic energy for the modified calculation scheme of the electric field. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A family of dynamic finite difference schemes for large-eddy simulation

A family of dynamic low-dispersive finite difference schemes for large-eddy simulation is developed. The dynamic schemes are constructed by combining Taylor series expansions on two different grid resolutions. The schemes are optimized dynamically during the simulation according to the flow physics and dispersion errors are minimized through the real-time adaption of the dynamic coefficient. In case of DNS-resolution, the dynamic schemes reduce to the standard Taylor-based finite difference schemes with formal asymptotic order of accuracy. When going to LES-resolution, the schemes seamlessly adapt to dispersion-relation preserving schemes. The schemes are tested for large-eddy simulation of Burgers’ equation and numerical errors are investigated as well as their interaction with the subgrid model. Very good results are obtained.     

Coupled axial and flexural vibration of 1-D members

The method of characteristics is used to model vibrations of one-dimensional members in the presence of coupling between axial and flexural waves. Only small deflections are considered and so the couple F_xδ_y is large only when the axial force F_x is large. Alternative numerical schemes are presented for the integration of terms describing coupling and a suitable explicit scheme is identified. The analysis is used to illustrate the influence of axial loads on frequencies and amplitudes of lateral oscillations of cantilever structures. Strong parametric behaviour is reproduced for a single-storey shear structure, including unstable resonance at double the linear natural frequency.     

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.     

Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows

In the present work, errors generated in computations of compressible multi-material flows using shock-capturing schemes are examined, specifically pressure oscillations (when the specific heats ratio is variable), but also temperature spikes and species conservation errors. These numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms. Though temperature errors are irrelevant to solutions to the Euler equations, it is shown that they have the potential to lead to problems when physical diffusion is included, i.e., for the Navier–Stokes equations. These errors are studied analytically and numerically by considering the one-dimensional advection of isolated material discontinuities. A methodology preventing such errors for weighted essentially non-oscillatory (WENO) schemes is presented, in which modified WENO weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors. Pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats. Several multi-dimensional problems with various discontinuities (shocks, material interfaces and contact discontinuities), including the single-mode Richtmyer–Meshkov instability, and turbulence are considered to test the method.     

A high-order low-dispersion symmetry-preserving finite-volume method for compressible flow on curvilinear grids

A new high-order finite-volume method is presented that preserves the skew symmetry of convection for the compressible flow equations. The method is intended for Large-Eddy Simulations (LES) of compressible turbulent flows, in particular in the context of hybrid RANS–LES computations. The method is fourth-order accurate and has low numerical dissipation and dispersion. Due to the finite-volume approach, mass, momentum, and total energy are locally conserved. Furthermore, the skew-symmetry preservation implies that kinetic energy, sound-velocity, and internal energy are all locally conserved by convection as well. The method is unique in that all these properties hold on non-uniform, curvilinear, structured grids. Due to the conservation of kinetic energy, there is no spurious production or dissipation of kinetic energy stemming from the discretization of convection. This enhances the numerical stability and reduces the possible interference of numerical errors with the subgrid-scale model. By minimizing the numerical dispersion, the numerical errors are reduced by an order of magnitude compared to a standard fourth-order finite-volume method.     

Solid-state NMR/NQR and first-principles study of two niobium halide cluster compounds

Two hexanuclear niobium halide cluster compounds with a [Nb₆X₁₂]²⁺ (X=Cl, Br) diamagnetic cluster core, have been studied by a combination of experimental solid-state NMR/NQR techniques and PAW/GIPAW calculations. For niobium sites the NMR parameters were determined by using variable B_o field static broadband NMR measurements and additional NQR measurements. It was found that they possess large positive chemical shifts, contrary to majority of niobium compounds studied so far by solid-state NMR, but in accordance with chemical shifts of ⁹⁵Mo nuclei in structurally related compounds containing [Mo₆Br₈]⁴⁺ cluster cores. Experimentally determined δ_iso(⁹³Nb) values are in the range from 2400 to 3000 ppm. A detailed analysis of geometrical relations between computed electric field gradient (EFG) and chemical shift (CS) tensors with respect to structural features of cluster units was carried out. These tensors on niobium sites are almost axially symmetric with parallel orientation of the largest EFG and the smallest CS principal axes (V_zz and δ₃₃) coinciding with the molecular four-fold axis of the [Nb₆X₁₂]²⁺ unit. Bridging halogen sites are characterized by large asymmetry of EFG and CS tensors, the largest EFG principal axis (V_zz) is perpendicular to the X-Nb bonds, while intermediate EFG principal axis (V_yy) and the largest CS principal axis (δ₁₁) are oriented in the radial direction with respect to the center of the cluster unit. For more symmetrical bromide compound the PAW predictions for EFG parameters are in better correspondence with the NMR/NQR measurements than in the less symmetrical chlorine compound. Theoretically predicted NMR parameters of bridging halogen sites were checked by ^79/81Br NQR and ³⁵Cl solid-state NMR measurements.     

Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations

In this paper, we propose explicit multi-symplectic schemes for Klein–Gordon–Schrödinger equation by concatenating suitable symplectic Runge–Kutta-type methods and symplectic Runge–Kutta–Nyström-type methods for discretizing every partial derivative in each sub-equation. It is further shown that methods constructed in this way are multi-symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions. In the aim of the commonly practical applications, a novel 2-order one-parameter family of explicit multi-symplectic schemes through such concatenation is constructed, and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods. Furthermore, some high-order explicit multi-symplectic schemes of such category are given as well, good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments.