A new family of high-order compact upwind difference schemes with good spectral resolution

This paper presents a new family of high-order compact upwind difference schemes. Unknowns included in the proposed schemes are not only the values of the function but also those of its first and higher derivatives. Derivative terms in the schemes appear only on the upwind side of the stencil. One can calculate all the first derivatives exactly as one solves explicit schemes when the boundary conditions of the problem are non-periodic. When the proposed schemes are applied to periodic problems, only periodic bi-diagonal matrix inversions or periodic block-bi-diagonal matrix inversions are required. Resolution optimization is used to enhance the spectral representation of the first derivative, and this produces a scheme with the highest spectral accuracy among all known compact schemes. For non-periodic boundary conditions, boundary schemes constructed in virtue of the assistant scheme make the schemes not only possess stability for any selective length scale on every point in the computational domain but also satisfy the principle of optimal resolution. Also, an improved shock-capturing method is developed. Finally, both the effectiveness of the new hybrid method and the accuracy of the proposed schemes are verified by executing four benchmark test cases.     

A shock-detecting sensor for filtering of high-order compact finite difference schemes

A new shock-detecting sensor for properly switching between a second-order and a higher-order filter is developed and assessed. The sensor is designed based on an order analysis. The nonlinear filter with the proposed sensor ensures damping of the high-frequency waves in smooth regions and at the same time removes the Gibbs oscillations around the discontinuities when using high-order compact finite difference schemes. In addition, a suitable scaling is proposed to have dissipation proportional to the shock strength and also to minimize the effects of the second-order filter on the very small scales. Several numerical experiments are carried out and the accuracy of the nonlinear filter with the proposed sensor is examined. In addition, some comparisons with other filters and sensors are made.     

A 5th order monotonicity-preserving upwind compact difference scheme

Based on an upwind compact difference scheme and the idea of monotonicity-preserving, a 5th order monotonicity-preserving upwind compact difference scheme (m-UCD5) is proposed. The new difference scheme not only retains the advantage of good resolution of high wave number but also avoids the Gibbs phenomenon of the original upwind compact difference scheme. Compared with the classical 5th order WENO difference scheme, the new difference scheme is simpler and small in diffusion and computation load. By emplo...     

Curvilinear finite-volume schemes using high-order compact interpolation

During the last years, the need of high fidelity simulations on complex geometries for aeroacoustics predictions has grown. Most of high fidelity numerical schemes, in terms of low dissipative and low dispersive effects, lie on finite-difference (FD) approach. But for industrial applications, FD schemes are less robust compared to finite-volume (FV) ones. Thus the present study focuses on the development of a new compact FV scheme for two- and three-dimensional applications.The proposed schemes are formulated in the physical space and not in the computational space as it is the case in most of the known works. Therefore, they are more appropriate for general grids. They are based on compact interpolation to approximate interface-averaged field values using known cell-averaged values. For each interface, the interpolation coefficients are determined by matching Taylor series expansions around the interface center. Two types of schemes can be distinguished. The first one uses only the curvilinear abscissa along a mesh line to derive a sixth-order compact interpolation formulae while the second, more general, uses coordinates in a spatial three-dimensional frame well chosen. This latter is formally sixth-order accurate in a preferred direction almost orthogonal to the interface and at most fourth-order accurate in transversal directions.For non-linear problems, different approaches can be used to keep the high-order scheme. However, in the present paper, a MUSCL-like formulation was sufficient to address the presented test cases.All schemes have been modified to treat multiblock and periodic interfaces in such a way that high-order accuracy, stability, good spectral resolution, conservativeness and low computational costs are guaranteed. This is a first step to insure good scalability of the schemes although parallel performances issues are not addressed. As high frequency waves, badly resolved, could be amplified and then destabilize the scheme, compact filtering operators have been used.Numerous test cases as the linear convection of a Gaussian wave, the convection of a Lamb–Oseen vortex and the diffraction of an acoustic wave on a plane have been realized to validate the schemes. The most efficient schemes are shown to be at least fifth-order accurate on linear and non-linear convection problems. They are also less dissipative and less dispersive on non-uniform curvilinear grids than schemes using implicit interpolation with constant coefficients of the same order on uniform cartesian grids.     

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ₃ have no effects on the SCL, no extra errors can be introduced as δ₃ = δ₂. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ₁ which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ₁ to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.     

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.     

An implicit high-order spectral difference approach for large eddy simulation

The filtered fluid dynamic equations are discretized in space by a high-order spectral difference (SD) method coupled with large eddy simulation (LES) approach. The subgrid-scale stress tensor is modelled by the wall-adapting local eddy-viscosity model (WALE). We solve the unsteady equations by advancing in time using a second-order backward difference formulae (BDF2) scheme. The nonlinear algebraic system arising from the time discretization is solved with the nonlinear lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. In order to study the sensitivity of the method, first, the implicit solver is used to compute the two-dimensional (2D) laminar flow around a NACA0012 airfoil at Re = 5 × 10⁵ with zero angle of attack. Afterwards, the accuracy and the reliability of the solver are tested by solving the 2D “turbulent” flow around a square cylinder at Re = 10⁴ and Re =  2.2 × 10⁴. The results show a good agreement with the experimental data and the reference solutions.     

A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2× 2 matrix modified Korteweg--de Vries (mKdV) spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting.     

A family of charged compact objects with anisotropic pressure

Utilizing an ansatz developed by Maurya et al. we present a class of exact solutions of the Einstein–Maxwell field equations describing a spherically symmetric compact object. A detailed physical analysis of these solutions in terms of stability, compactness and regularity indicates that these solutions may be used to model strange star candidates. In particular, we model the strange star candidate Her X-1 and show that our solution conforms to observational data to an excellent degree of accuracy. An interesting and novel phenomenon which arises in this model is the fact that the relative difference between the electromagnetic force and the force due to the pressure anisotropy changing sign within the stellar interior. This may be an additional mechanism required for stability against cracking of the stellar object.     

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.     

A systematic methodology for constructing high-order energy stable WENO schemes

A third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme developed by the authors of this paper [N.K. Yamaleev, M.H. Carpenter, Third-order energy stable WENO scheme, J. Comput. Phys. 228 (2009) 3025–3047] was proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, a systematic approach is presented that enables “energy stable” modifications for existing WENO schemes of any order. The technique is demonstrated by developing a one-parameter family of fifth-order upwind-biased ESWENO schemes including one sixth-order central scheme; ESWENO schemes up to eighth order are presented in the Appendix. We also develop new weight functions and derive constraints on their parameters, which provide consistency, much faster convergence of the high-order ESWENO schemes to their underlying linear schemes for smooth solutions with arbitrary number of vanishing derivatives, and better resolution near strong discontinuities than the conventional counterparts.     

高精度有限差分法模拟Kelvin-Helmholtz不稳定性

 WENO有限差分格式有较高的分辨精度,适合复杂流场的计算,在国际上被广泛采用。本文利用WENO有限差分格式求解2维守恒型欧拉方程,实现了对无粘流体中Kelvin-Helmholtz不稳定性的数值模拟。速度剪切方向采用周期边界条件;扰动增长方向采用嵌边出流边界条件,一个不稳定波长分布64个网格。数值模拟给出的扰动幅值线性增长率与线性稳定性分析给出的结果很好符合,显示了该格式的有效性和精度。数值模拟给出了清晰的密度等值线,表明该方法还具有较好的界面变形捕捉能力。     

高精度有限差分法模拟Kelvin-Helmholtz不稳定性

WENO有限差分格式有较高的分辨精度,适合复杂流场的计算,在国际上被广泛采用。本文利用WENO有限差分格式求解2维守恒型欧拉方程,实现了对无粘流体中Kelvin-Helmholtz不稳定性的数值模拟。速度剪切方向采用周期边界条件;扰动增长方向采用嵌边出流边界条件,一个不稳定波长分布64个网格。数值模拟给出的扰动幅值线性增长率与线性稳定性分析给出的结果很好符合,显示了该格式的有效性和精度。数值模拟给出了清晰的密度等值线,表明该方法还具有较好的界面变形捕捉能力。     

A new simplified ordered upwind method for calculating quasi-potential

We present a new method for calculation of quasi-potential,which is a key concept in the large deviation theory.This method adopts the"ordered"idea in the ordered upwind algorithm and different from the finite difference upwind scheme,the first-order line integral is used as its update rule.With sufficient accuracy,the new simplified method can greatly speed up the computational time.Once the quasi-potential has been computed,the minimum action path(MAP)can also be obtained.Since the MAP is of concern in most stochastic situations,the effectiveness of this new method is checked by analyzing the accuracy of the MAP.Two cases of isotropic diffusion and anisotropic diffusion are considered.It is found that this new method can both effectively compute the MAPs for systems with isotropic diffusion and reduce the computational time.Meanwhile anisotropy will affect the accuracy of the computed MAP.     

A mass-conserved multiphase lattice Boltzmann method based on high-order difference

The Z–S–C multiphase lattice Boltzmann model [Zheng, Shu, and Chew(ZSC), J. Comput. Phys. 218, 353(2006)]is favored due to its good stability, high efficiency, and large density ratio. However, in terms of mass conservation, this model is not satisfactory during the simulation computations. In this paper, a mass correction is introduced into the ZSC model to make up the mass leakage, while a high-order difference is used to calculate the gradient of the order parameter to improve the accuracy. To verify the improved model, several three-dimensional multiphase flow simulations are carried out,including a bubble in a stationary flow, the merging of two bubbles, and the bubble rising under buoyancy. The numerical simulations show that the results from the present model are in good agreement with those from previous experiments and simulations. The present model not only retains the good properties of the original ZSC model, but also achieves the mass conservation and higher accuracy.     

A new compact difference scheme for second derivative in non-uniform grid expressed in self-adjoint form

A single-parameter family of self-adjoint compact difference (SACD) schemes is developed for discretizing the Laplacian operator in self-adjoint form. Developed implicit scheme is formally second-order accurate (with respect to truncation error) with a free parameter, α which helps control the numerical properties in the spectral plane. The SACD scheme is analyzed in the spectral plane for its resolution properties for both periodic and non-periodic problems using the matrix spectral analysis [T.K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind schemes, J. Comput. Phys. 192 (2) (2003) 677–694]. The major objective here is to identify the advantages of the new scheme over the traditional explicit second order CD2 scheme, in discretizing the Laplacian operator in self-adjoint form. This appears in Navier–Stokes equation and in other transport equations and boundary value problems (bvp) expressed in orthogonal co-ordinate systems, either in physical or in transformed plane. We also compare the developed method with the higher order compact schemes for non-uniform grids. To demonstrate the accuracy of SACD scheme we have tested it for: (i) bi-directional wave propagation problem, given by the second order wave equation and (ii) an elliptic bvp, as in the Stommel ocean model for the stream function. These examples help infer the properties of SACD scheme when solving different types of partial differential equations. Most importantly the effects of grid-stretching and choice of value of the free parameter (α) are investigated here. We also compare the present implicit compact method with explicit compact method known as the higher order compact (HOC) method.Also, the practical applications of the SACD scheme are explored by solving the Navier–Stokes equation for the cases of: (a) a flow inside a lid-driven cavity and (b) the receptivity and instability of an external adverse pressure gradient flow over a flat plate. In the former, unsteadiness of the flow is captured and in the latter, the receptivity of the flow is studied in causing flow instability by triggering Tollmien–Schlichting waves. The new scheme shows a marked improvement over the explicit scheme for low Reynolds number steady flow in lid driven cavity. Whereas for the flow in the same geometry at higher Reynolds numbers, efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures. Presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer.     

A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a Multi-Implicit Spectral Deferred Correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework and allows us to conserve mass and energy while simultaneously satisfying the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.     

求解对流扩散方程的四种差分格式的比较

利用对流扩散方程，在边界和参数存在随机扰动的情况下，考察四种差分格式的优劣，为求 解对流扩散方程提供一种可靠的差分格式，并得到通过空间加密网格的方法可以控制边界、 参数随机影响的结论. 关键词： 对流扩散方程 差分格式 随机扰动