A characteristic-based shock-capturing scheme for hyperbolic problems

https://doi.org/10.1016/j.jcp.2007.03.007Get rights and content

Abstract

In order to suppress numerical oscillations of linear compact schemes around discontinuities, a characteristic-based flux splitting limited method is introduced instead of ENO/WENO or other shock-capturing algorithms. This method begins with upwind schemes and flux vector splittings. The upwind schemes are projected along characteristic directions in a different way, and their amplitudes are carefully controlled by a special limiter in order to meet entropy condition and to prevent non-physical oscillations. A fifth-order linear compact upwind scheme is modified by this method for solving problems involving discontinuities. The properties of the numerical algorithm are checked on some benchmark problems in one, two and three space dimensions. Numerical results show that it is high-order accurate with high resolution and oscillation-free.

Introduction

Direct numerical simulations (DNS) of turbulence and large-eddy simulations (LES) require the use of high accurate numerical schemes, which must be capable of resolving a very broad range of length scales that are often orders of magnitude apart [1]. It is generally believed that the accurate simulation of fluid flow with multiple and wide range of spatial scales and structures is a difficult task expect through spectral approximations. However, the use of spectral approximations is limited to simple geometries with generally periodic boundary conditions. Compact algorithm makes it possible to devise, on a given stencil, difference schemes that have much better resolution properties than conventional explicit difference schemes of comparable order of accuracy. Compact schemes with spectral-like resolution properties are more convenient to use than spectral and pseudo-spectral schemes, and are easier to handle, especially when non-trivial geometries are involved. The price paid is that one is required in general to invert a tri-diagonal system of linear algebra equation systems to obtain derivatives. Central compact schemes have been developed from the 70s of the last century (see for instance [2], [3]). However, centered algorithms are intrinsically non-dissipative, and cannot prevent odd–even decoupling, which gives rise to high frequency oscillations even in smooth regions. Reducing or removing such oscillations requires the introduction of dissipation terms. Asymmetric schemes with their dissipative properties are more stable. Fu and Ma [4], Adams and Shariff [5], Tolstykh and Lipavskii [6], Zhuang and Cheng [7] have developed some compact upwind (dissipative) schemes. These schemes enjoy high-order accuracy and high resolution for low wave numbers (large scale), while allowing much dissipation to high wave numbers (small scale), and the dissipation can be adjusted by a careful design. In general, compact upwind schemes can avoid odd–even decoupling and can prevent non-physical oscillations in smooth regions.

Recently, Chu and Fan [8] and Mahesh [9] have developed combined compact difference (CCD) schemes. Their ideas are basically to produce a high-order scheme by combining and solving the first and second derivatives together. The CCD schemes become more compact and more accurate than normal compact schemes. Finite-volume compact schemes have also been attempted by Gaitonde and Shang [10] and Kobayashi [11]. Nevertheless, in the transonic and supersonic flow regions when dealing with flows involving shock waves, one must use a numerical scheme which can both represent small scale structures with the minimum of numerical dissipation and capture discontinuities with the robustness that is common to Godunov-type methods. To achieve these dual objectives, high-order accurate shock-capturing schemes must be employed [12]. Unfortunately, the overall dependency characteristic of compact schemes hinders them from this purpose application, and the toughest difficulty is to capture the discontinuities smoothly in strong non-linear problems. Few attempts have been made to achieve the shock-capturing capability for compact schemes. Cockburn and Shu [13] have developed non-linearly stable compact schemes for shock calculations in 1994. They followed TVD’s idea to define a non-linear limiter based on the local mean to avoid spurious oscillations while maintaining the formal accuracy of the schemes. However, spurious oscillations were still evident in their numerical test problems for their fourth-order scheme. An extended and improved version of Cockburn and Shu’s scheme can be found in Yee’s paper [14], but no numerical tests are given. Ravichandran [15] has employed a TVD limiter combined with kinetic flux vector splitting (KFVS) method to improve the stability of compact upwind schemes, and third-order schemes were given, which is supposed to degenerate to first-order accuracy at extrema. Deng and Zhang [16], Deng and Maekawa [17] have proposed some compact non-linear schemes by employing dissipative terms and weighted interpolations, but their schemes have lost the compactness (e.g. at least seven points are needed for their fifth-order schemes). Lerat and Corre [18] have employed residual-based dissipations to suppress non-physical oscillations, and a third-order compact scheme was given for compressible flows, but the scheme is only suitable for steady flows, and its applications to unsteady problems are in development. Ma and Fu [19] have developed high-order compact schemes with the method of group velocity control. Among all those efforts, blending compact schemes with other shock-capturing schemes such as ENO/WENO schemes is most common. ENO/WENO schemes [20], [21], [22], [23], [24] show great promise for accurately treating flow discontinuities. These schemes can be used to achieve a uniformly high-order accuracy while maintaining essentially non-oscillatory behavior for piecewise smooth functions by preventing the interpolation of the field values across the discontinuities as much as possible. This is done through a reconstruction or a flux evaluation procedure to allow the interpolating stencils to shift adaptively with the local smoothness of the function. However, the numerical solutions obtained with ENO/WENO schemes in smooth regions with moderately high field gradients are not very satisfactory (worse than padé schemes [30]). One way to eliminate this disadvantage of ENO/WENO schemes is to construct a hybrid scheme in which the scheme is switched to a conventional compact scheme in smooth regions and to an ENO/WENO scheme near/across discontinuities. However, a free threshold parameter, which controls the switch between the compact scheme and the ENO/WENO scheme, needs to be tuned, and some of the hybrid compact-ENO/WENO schemes [1], [5] experience non-smooth transitions near the interfaces where the scheme switches types. Some spurious waves might be generated at these interfaces between different schemes, and these spurious waves would eventually propagate into the smooth regions as reported by Adams and Shariff [5]. Ren et al. [25] have developed characteristic-wise hybrid compact-WENO schemes, which can be regarded as an improvement of the scheme presented in [1].

As pointed by Titarev and Toro [26], the design of high-order accurate numerical schemes for hyperbolic conservation laws is a formidable task since three major difficulties have to be overcome: ensuring the conservation property, preserving the high order of accuracy in both time and space, and controlling the generation of spurious oscillations in vicinity of discontinuities. In the present study, we try to overcome the difficulties except the accuracy in time. Unlike the discontinuity-capturing methods mentioned above, a new characteristic-based method is proposed to surmount the shortcomings of high-order linear compact schemes. The organization of the paper is as follows: In Section 2, the characteristic-based method is given to suppress spurious oscillations of linear upwind schemes which are applied to get an approximation of the first spatial derivatives, and a shock-capturing scheme is formulated based on a fifth-order linear compact upwind scheme. Some numerical test cases in one-dimensional and multi-dimensional Euler systems are presented and discussed in Section 3, including some comparisons with other high-order schemes, and the results show that the characteristic-based shock-capturing compact scheme possesses the merits of the linear compact scheme, e.g. spectral-like resolution, higher-order accuracy, etc. Finally, concluding remarks are provided in Section 4.

Section snippets

Characteristic-based treatment

Consider a hyperbolic system of conservation lawsQt+F(Q)x=0or its non-conservative formQt+AQx=0where A=F/Q, and all the eigenvalues λ(k) of A are real numbers. Let L and R be the left and right eigenvector matrices of A, then A=RΛL,R=L-1 and Λ is the diagonal matrix of λ(k).

Let us discrete the space into uniform intervals of size Δx, and various quantities at xi will be identified by the subscript i. No matter what kind of algorithms are employed, the semi-discretize scheme of Eq. (1)

Numerical tests

For the motion of inviscid compressible fluids, the elements in Eq. (1) areQ=[ρ,ρu,ρe],F=[div(ρu),div(ρuu)+p,div(ρeu+pu)]where ρ is the fluid density, u is the velocity and e is the total energy, defined as the sum of the internal energy plus the kinetic energy. The system is closed by defining the pressure p through the equation of state for a perfect gas, p=ρe(γ-1), where the constant γ is the ratio of specific heats. In all our tests considered, γ=1.4.

In this paper, the temporal

Conclusions

A characteristic-based shock-capturing scheme has been formulated. Compactness, high-order accuracy and high resolution are achieved attribute to the characteristic-based method and the high-order compact algorithm. Without implementing of other shock-capturing schemes, the scheme possesses the advantages of linear compact schemes of spectral-like resolution, higher-order accuracy, and easy for boundary closure. The proposed algorithm has been shown to yield oscillation-free and high-order

Acknowledgments

This work was supported by the National Science Foundation of China under the Grant number of 10321002 and 90205010. The authors are grateful to Professor Qing Shen for his help on our English.

References (32)

Cited by (0)

View full text