Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent non-physical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD (CTVD) schemes were tested, one is third-order accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a two-dimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.     

用隐式方法和WENO格式计算悬停旋翼跨声速无粘流场

寻找一种能够准确计算以涡为主要特征的复杂流场和克服尾迹耗散问题的数值方法,一直是旋翼空气动力学研究的热点和难点。本文发展了一种基于高阶迎风格式计算悬停旋翼无粘流场的隐式数值方法。无粘通量采用Roe通量差分分裂格式,为提高精度,使用五阶WENO格式进行左右状态插值,并与MUSCL插值进行比较。为提高收敛到定常解的效率,时间推进采用LU-SGS隐式方法。用该方法对一跨声速悬停旋翼无粘流场进行了数值计算,数值结果表明WENO-Roe的激波分辨率高于MUSCL-Roe,体现出了格式精度的提高对计算结果的改善,LU-SGS隐式方法的计算效率比5步Runge-Kutta显式方法的高。     

Non-oscillatory shock-capturing finite element methods for the one-dimensional compressible Euler equations

A class of shock-capturing Petrov–Galerkin finite element methods that use high-order non-oscillatory interpolations is presented for the one-dimensional compressible Euler equations. Modified eigenvalues which employ total variation diminishing (TVD), total variation bounded (TVB) and essentially non-oscillatory (ENO) mechanisms are introduced into the weighting functions. A one-pass Euler explicit transient algorithm with lumped mass matrix is used to integrate the equations. Numerical experiments with Burgers' equation, the Riemann problem and the two-blast-wave interaction problem are presented. Results indicate that accurate solutions in smooth regions and sharp and non-oscillatory solutions at discontinuities are obtainable even for strong shocks.     

Improved artificial dissipation schemes for the Euler equations

The influence of artificial dissipation schemes on the accuracy and stability of the numerical solution of compressible flow is extensively examined. Using an implicit central difference factored scheme, an improved form of artificial dissipation is introduced which highly reduces the errors due to numerical viscosity. A function of the local Mach number is used to scale the amount of numerical damping added into the solution according to the character of the flow in several flow regimes. The resulting scheme is validated through several inviscid flow test cases.     

NND schemes and numerical simulation of axial symmetric free jet flows

Through a study on one-dimensional Navier-Stokes equations, it was found that the spurious oscillations occuring near shock waves with finite difference equations are related to the dispersion term in the corresponding modified differential equations. If the sign of dispersion coefficient is properly adjusted so that the sign changes across shock waves, the undesirable oscillations can be totally suppressed. Based on this finding, the non-oscillatory, containing no free parameters and dissipative shheme (NND scheme) is developed. This scheme is one of “TVD”. The axisymmetric free jet flows are simulated numerically using this scheme. The results obtained by the present scheme are compared with the experimental picture. It is shown that the agreement is very good, and that this scheme has advantages of high resolution for capturing shocks and contact discontinuities. Project supported by National Science Foundation of China     

GASEOUS FLOW ORIGINATED BY A GUNSHOT,I.

SUMMARY

The numerical simulation of the gaseous flow originated by a gunshot both during the projectile motion within the muzzle and out of it is considered. The numerical solution of the Euler equations is obtained via a second order accurate ENO scheme. A special algorithm of cells reorganization is developed for computing the projectile motion out of the muzzle. It reveals an interesting property of the ENO schemes and Roe's approximate solution of automatically capturing discontinuities by means of the choice of the corresponding grid velocity. The algorithm developed describes sufficiently well all gasdynamics phenomena which accompany the gunshot.     

Linear stability of WENO schemes coupled with explicit Runge–Kutta schemes

This paper focuses on the results of the linear stability analysis of the finite‐difference weighted essentially non‐oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order‐optimised WENO schemes of the sixth and eighth order and the bandwidth‐optimised WENO schemes of the third and fourth order are considered. Several explicit Runge–Kutta schemes including the recently published strong stability‐preserving explicit Runge–Kutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the Courant–Friedrichs–Lewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high‐order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive multi-resolution central-upwind schemes for systems of conservation laws

In this article, we develop an adaptive scheme for solving systems of hyperbolic conservation laws. In this scheme nonlinear shock and linear contact waves will be treated differently. The proposed scheme uses the Kurganov central-upwind scheme. Fourth-order non-oscillatory reconstruction is employed near shock only while the unlimited fifth-order reconstruction is used for smooth regions and linear contact waves. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multi-resolution technique. The advantages of the scheme are high resolution and computationally efficient since limiters are used only for shocks. Numerical experiments with one- and two-dimensional problems are presented which show the robustness of the proposed scheme.     

利用多小波自适应格式求解流体力学方程

高阶计算格式的高精度、高分辨率对提高复杂流场的计算水平有重要的意义,为了提高AUSMPW格式对流场计算中激波等间断的分辨率,减小数值振荡,在原有AUSMPW格式的基础之上,利用多小波对函数进行多尺度分解,并采取阈值的方法生成自适应网格,提出了一种新的基于多小波自适应算法的AUSMPW格式,理论上可以达到任意阶精度. 将所得的压强、密度与原格式、TVD格式及WENO格式的计算结果进行了比较分析. 结果表明改进后的AUSMPW格式较原格式具有更高的分辨率、更强的捕捉间断的能力及更低的数值耗散.      

Methods for extending high-resolution schemes to non-linear systems of hyperbolic conservation laws

In extending high-resolution methods from the scalar case to systems of equations there are a number of options available. These options include working with either conservative or primitive variables, characteristic decomposition, two-step methods, or component-wise extension. In this paper, several of these options are presented and compared in terms of economy and solution accuracy. The characteristic extension with either conservative or primitive variables produces excellent results with all the problems solved. Using primitive variables, the two-step formulation produces high-quality results in a more economical manner. This method can also be extended to multiple dimensions without resorting to dimensional splitting. Proper selection of limiters is also important in non-characteristic extension to systems.     

Shock wave deformation in shock-vortex interactions

An initially planar shock wave can undergo significant distortion to its shape along with changes in its strength during the period of its interaction with a compressible vortex. This phenomenon is studied by numerically simulating the shock wave-vortex interaction with a high resolution shock-capturing scheme. Incident shock waves of various Mach numbers are made to interact with a compressible vortex and the dependence of the shock wave distortion on the strength of the incident shock wave is studied in detail. It is known that the type of complex shock structure formed in the later stages of a compressible vortex-shock wave interaction is dependent on the Mach number of the incident shock wave. A simple physical model based on the principle of shock wave reflection is proposed to explain this complex shock structure formation and its dependence on the relative strengths of the interacting vortex and shock wave. Received July 28, 1997 / Accepted November 17, 1997     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.     

TVD schemes for open channel flow

The Saint Venant equations for modelling flow in open channels are solved in this paper, using a variety of total variation diminishing (TVD) schemes. The performance of second- and third-order-accurate TVD schemes is investigated for the computation of free-surface flows, in predicting dam-breaks and extreme flow conditions created by the river bed topography. Convergence of the schemes is quantified by comparing error norms between subsequent iterations. Automatically calculated time steps and entropy corrections allow high CFL numbers and smooth transition between different conditions. In order to compare different approaches with TVD schemes, the most accurate of each type was chosen. All four schemes chosen proved acceptably accurate. However, there are important differences between the schemes in the occurrence of clipping, overshooting and oscillating behaviour and in the highest CFL numbers allowed by a scheme. These variations in behaviour stem from the different orders and inherent properties of the four schemes. © 1998 John Wiley & Sons, Ltd.     

The designing approach of difference schemes by controlling the remainder‐effect

In this paper, based on the idea of the ‘modified partial differential equation’, a new designing approach to explicit finite difference schemes for the Burgers equation and PDE is proposed. The approach differs from other constructured methods in such a way that it considers the requests of the numerical dissipation and dispersion coefficients first. This method is much more constructional and directional. The results of numerical tests indicate that the method is quite successful. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparisons of TVD schemes for turbulent transonic projectile aerodynamics computations with a two-equation model of turbulence

The development of a computer program to solve the axisymmetric full Navier--Stokes equations with k-ε two-equation model of turbulence using various total variation diminishing (TVD) schemes is the primary interest of this study. The computations are performed for the turbulent, transonic, viscous flow over a projectile with/without supporting sting at zero angle of attack. The predicted results, as well as the convergence characteristics, by various TVD schemes are compared with each other. The results show that the TVD schemes of higher-order accuracy do have influence on the regions of high gradients such as shock, base corner and base flow. However, the schemes of third-order accuracy do not necessarily improve the agreement with measured data (which is not available on the base) than that of second-order accuracy, but surely generate apparent different result of base flow. The supporting sting on the projectile base will complicate the base flow and the existence of the sting will slightly shift the shock location and slightly change the flow field after the shock. More iteration steps are needed to get the converged results in the computation for the projectile with sting.     

Short Communication: Application of WENO (Weighted Essentially Non-oscillatory) Approach to Navier–Stokes Computations

The direct implementation of the essentially non-oscillatory schemes for flow simulation over complex geometries sometimes results in insufficiently robust numerical algorithms. In order to overcome this difficulty, it is suggested to use the weighted essentially non-oscillatory approach for multidimensional Navier–Stokes computations. The results indicate a significant improvement in accuracy and robustness, especially for low Mach and high supersonic flows.     

Derivative artificial compression method for conservation laws with multi-dimensional extension

A new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.