A high-resolution,hybrid compact-WENO scheme with minimized dispersion and controllable dissipation

In this paper,a high-resolution,hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique.Firstly,a suffcient condition for a family of tri-diagonal compact schemes to have independent dispersion and dissipation is derived.Then,a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed.Finally,the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme.Moreover,the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme.Several test cases are carried out to verify the highresolution as well as the robust shock-capturing capabilities of the proposed scheme.     

Third order residual distribution schemes for the Navier–Stokes equations

We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P² Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.     

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.     

一种高精度的修正Hermite-ENO格式

设计一种基于三单元具有六阶精度的修正Hermite-ENO格式（CHENO）,求解一维双曲守恒律问题.CHENO格式利用有限体积法进行空间离散,在空间层上,使用ENO格式中的Newton差商法自适应选择模板.在重构半节点处的函数值及其一阶导数值时,利用Taylor展开给出修正Hermite插值使其提高到六阶精度,并设计了间断识别法与相应的处理方法以抑制间断处的虚假振荡;在时间层上采用三阶TVD Runge-Kutta法进行函数值及一阶导数值的推进.其主要优点是在达到高阶精度的同时具有紧致性.数值实验表明对一维双曲守恒律问题的求解达到了理论分析结果,是有效可行的.     

Explicit Runge–Kutta residual distribution schemes for time dependent problems: Second order case

In this paper, we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems, starting from a given residual distribution (RD) discrete approximation of the steady operator. We review the existing knowledge on consistent RD mass matrices and highlight the relations between different definitions. We then introduce our explicit approach which is based on three main ingredients: first recast the RD discretization as a stabilized Galerkin scheme, then use a shifted time discretization in the stabilization operator, and lastly apply high order mass lumping on the Galerkin component of the discretization. The discussion is particularly relevant for schemes of the residual distribution type 18 and 3 which we will use for all our numerical experiments. However, similar ideas can be used in the context of residual-based finite volume discretizations such as the ones proposed in 14 and 12. The schemes are tested on a wide variety of classical problems confirming the theoretical expectations.     

A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics

We present a highly robust second order accurate scheme for the Euler equations and the ideal MHD equations. The scheme is of predictor–corrector type, with a MUSCL scheme following as a special case. The crucial ingredients are an entropy stable approximate Riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure. For multidimensional MHD, a new discrete form of the Powell source terms is vital to ensure the stability properties. The numerical examples show that the scheme has superior stability compared to standard schemes, while maintaining accuracy. In particular, the method can handle very low values of pressure (i.e. low plasma β$\beta$ or high Mach numbers) and low mass densities.     

Simulating thermohydrodynamics by finite difference solutions of the Boltzmann equation

The formulation of a consistent thermohydrodynamics with a discrete model of the Boltzmann equation requires the representation of the velocity moments up to the fourth order. Space-filling discrete sets of velocities with increasing accuracy were obtained using a systematic approach in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev. E, 73 (5), n. 056702, 2006). These sets of velocities are suitable for collision-propagation schemes, where the discrete velocity and physical spaces are coupled and the Courant number is unitary. The space-filling requirement leads to sets of discrete velocities which can be large in thermal models. In this work, although the discrete sets of velocities are also obtained with a quadrature method based on prescribed abscissas, the lattices are not required to be space-filling. This leads to a reduced number of discrete velocities for the same approximation order but requires the use of an alternative numerical scheme. The use of finite difference schemes for the advection term in the continuous Boltzmann equation has shown to have some advantages with respect to the collision-propagation LBM method by freeing the Courant number from its unitary value and reducing the discretization error. In this work, a second order Runge-Kutta method was used for the simulation of the Sod's shock tube problem, the Couette flow and the Lid-driven cavity flow. Boundary conditions without velocity slip and temperature jumps were written for these discrete Boltzmann equation by splitting the velocity distribution function into an equilibrium and a non-equilibrium part. The equilibrium part was set using the local velocity and temperature at the wall and the non-equilibrium part by extrapolating the non-equilibrium moments to the wall sites.     

An element-free Galerkin(EFG) method for numerical solution of the coupled Schrdinger-KdV equations

The present paper deals with the numerical solution of the coupled Schrdinger-KdV equations using the elementfree Galerkin(EFG) method which is based on the moving least-square approximation.Instead of traditional mesh oriented methods such as the finite difference method(FDM) and the finite element method(FEM),this method needs only scattered nodes in the domain.For this scheme,a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method.In numerical experiments,the results are presented and compared with the findings of the finite element method,the radial basis functions method,and an analytical solution to confirm the good accuracy of the presented scheme.     

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.     

Massive Meson Fluctuation in NJL Model

Based on the self-consistent scheme beyond the mean-field approximation in the large N_c expansion, including current quark mass explicitly, a general scheme of SU(2) NJL model is developed. To ensure the quark self-energy expanded in the proper order of N_c,an approximate internal meson propagator is deduced, which is in order of O(l/N_c). In our scheme, adopting the method of external momentum expansion, all the Feynman diagrams are calculated in a unified way by only expanding the quark propagator. Our numerical results show that, different &om the mean-field approximation in which the explicitly chiral symmetry breaking is invisible, the effect of finite pion mass can be seen clearly when beyond the meanfield approximation.     

A comparison of spatial discretization schemes for differential solution methods of the radiative transfer equation

A comparison of discretization schemes required to evaluate the radiation intensity at the cell faces of a control volume in differential solution methods of the radiative transfer equation is presented. Several schemes developed using the normalized variable diagram and the total variation diminishing formalisms are compared along with essentially non-oscillatory schemes and genuinely multidimensional schemes. The calculations were carried out using the discrete ordinates method, but the analysis is equally valid for the finite-volume method. It is shown that the S schemes of the genuinely multidimensional family perform quite well, particularly in problems with discontinuous radiation intensity fields. However, they are time consuming, and so they do not always become more attractive regarding the trade-off between accuracy and computational requirements, in comparison with other high-order schemes that, although being less accurate, are also more economical.     

A high order multivariate approximation scheme for scattered data sets

We present a high order multivariate approximation scheme for scattered data sets. Each data point is represented as a Taylor series, and the high order derivatives in the Taylor series are treated as random variables. The approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares. The approximation is an interpolation when the data points are given as exact, or a nonlinear regression function when nonzero measurement errors are associated with the data points. Using this formulation, the gradient information on each data point can be used to significantly reduce the approximation error. All parameters of the approximation scheme can be computed automatically from the data points. An uncertainty bound of the approximation function is also produced by the scheme. Numerical experiments demonstrate that although this method is more computationally intensive than traditional methods, it produces more accurate approximation functions.     

采用Power限制器的PNND和PWNND格式

为高精度捕捉激波等流场结构,引入一种Power限制器,对NND格式和WNND格式进行改进,分别得到二阶PNND(Power NND)格式和三阶PWNND(Power WNND)格式.该Power类型格式通过Power限制器对相邻待选模板上的一阶导数进行限制,改善了NND格式和WNND格式在间断附近的耗散效应.对各种格式的分析表明,在间断附近采用Power限制器的格式比原格式的表现要好,耗散小且捕捉间断精度高,其中PNND格式虽然只有二阶精度,但在所有算例中与三阶WNND格式的计算结果比较接近,在个别算例中甚至优于WNND格式.最后将PWNND格式应用到二维NACA0012翼型的强迫俯仰振动的数值模拟,计算结果与实验值、参考计算值吻合较好.     

A new complex variable meshless method for transient heat conduction problems

<正>In this paper,based on the improved complex variable moving least-square(ICVMLS) approximation,a new complex variable meshless method(CVMM) for two-dimensional(2D) transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations,and the essential boundary conditions are imposed by the penalty method.As the transient heat conduction problems are related to time,the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization.Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained.In order to demonstrate the applicability of the proposed method,numerical examples are given to show the high convergence rate,good accuracy,and high efficiency of the CVMM presented in this paper.     

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

In this paper the problem $-{\rm div}(a(x,y)\nabla u)=f$ with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for $u$ and $\nabla u$ using a new scheme called "hermitian box-scheme". The design of the scheme is based on a "hermitian box", combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2$h$. The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numerical results obtained show the efficiency of the numerical scheme. This work is the extension to strongly elliptic problems of the hermitian box-scheme presented by Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).     

辐射输运菱形差分SN方程的扩散综合加速方法

讨论辐射输运菱形差分S_N方程的源迭代加速方法,在一维情形下给出与输运算子离散相容的线性多频灰体加速计算格式.数值算例表明该加速算法是健壮有效的.     

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.     

Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers

Acoustic wave propagation in a fluid with a random assortment of identical cylindrical scatterers is considered. While the leading order correction to the effective wavenumber of the coherent wave is well established at dilute areal density (n0) of scatterers, in this paper the higher order dependence of the coherent wavenumber on n0 is developed in several directions. Starting from the quasi-crystalline approximation (QCA) a consistent method is described for continuing the Linton and Martin formula, which is second order in n0, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O (n0(4)). Then, using the QCA theory as a basis, generalized self-consistent schemes are developed and compared with self-consistent schemes using other dynamic effective medium theories. It is shown that the Linton and Martin formula provides a closed self-consistent scheme, unlike other approaches.     

The effect of the 2-D Laplacian operator approximation on the performance of finite-difference time-domain schemes for Maxwell’s equations

The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl–curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.