Computer Methods in Applied Mechanics and Engineering
A new way for constructing high accuracy shock-capturing generalized compact difference schemes
Introduction
Considerable progress has been made over the past two decades on developing high-order accuracy and high-resolution schemes for calculating flow fields with shocks. In the calculation of complex flow fields containing shock waves, most attention has been paid to the shock-capturing method. It is well known that first-order accurate schemes are too diffusive, but classical central higher-order schemes, while giving better resolution of the discontinuities of the solution, exhibit spurious oscillations around such points (e.g., the Lax–Wendroff scheme, the Fromm scheme, and the warming-beam scheme). In order to capture shock waves smoothly without spurious oscillations, the development of non-oscillatory dissipative schemes containing no free parameters with high resolution has much been emphasized recently.
On the other hand, many physical phenomena possess a wide range of length and time scales, turbulent fluid flows being a common example. Direct numerical simulations of these processes require all the relevant scales to be properly represented in the numerical model. Due to the above reasons, the accuracy requirement in large-eddy simulations and direct numerical simulations of turbulence, computational electro-magnetics, and computational aeroacoustics has become a pacing item for technology development.
The TVD and NND schemes are widely used for capturing shock waves clearly. Some more robust schemes such as ENO, ENN, UENO and WENO schemes have been proposed in order to improve the resolution of capturing contact discontinuities as well as shocks. However, for all these schemes, the higher the order of accuracy the higher the number of discretization points involved. It will result in difficulties of boundary treatment, and it is impossible to keep the same order of accuracy on the near boundary points as on the inner points [1].
To avoid using a too larger point-stencil, compact (or Pade) schemes have also been considered. Compact method considers as unknowns at each discretization point not only the value of the function itself uj, but also those of its first or second derivatives uj(1) or uj(2). The system is closed by considering relationships between the function and its first or second derivatives in some consecutive discretization points. These relationships are called auxiliary relationships. The idea of such a method is rather old, as explained by Collatz [2], but only recently it has found application to flow problems. A pioneering work in this direction is due to Hirsh [3] for the incompressible Navier–Stokes equations. In [4] Lele developed central compact finite difference schemes with spectral-like resolution. Compared to the traditional finite difference approximations the compact difference schemes provide a better representation of the shorter length scales. However, the extension of centered compact schemes to compressible flows is difficult because of the lack of numerical dissipation, except perhaps for direct numerical simulations of simple problems in which all the shock structures can be perfectly resolved (see [4]).
Halt [5] pointed out that further development of the compact schemes needs to investigate the non-oscillatory shock-capturing technique. For this reason, many upwind compact schemes have been proposed [6], [7], [8], [9]. In order to get higher order accurate compact difference schemes with smaller stencil dimension the super compact schemes have been presented by Ma and Fu [10]. Recently a family of finite difference schemes for the first and second derivatives of smooth functions called the coupled-derivative schemes has been presented by Mahesh [11].
A new class of finite difference schemes called the generalized compact schemes (GC schemes) has been proposed in [12].
We combined the GC schemes with the UENO schemes and WENO schemes separately to construct the three-point GC–UENO schemes [13] and three-point GC–WENO–FCT schemes [14] for calculating the flow field with shocks. In this paper, a new way is presented to construct high order accuracy and high-resolution GC schemes. The schemes constructed by using this way could satisfy some principles and demands prescribed in advance to ensure some desired properties to the schemes, such as the principle about suppression of the oscillations, the principle of stability, the order of accuracy and number of scheme points, etc. As two examples, a three-point third-order compact scheme and a three-point fifth-order GC scheme satisfying the principle about suppression of the oscillations and the principle of stability are constructed [15], [16]. Numerical results of typical problems indicate good shock-capturing capability.
Section snippets
Auxiliary equations of the generalized compact difference schemes
Consider the ordinary differential equation as an example:In the solution interval [a,b], the coordinates of nodes are denoted by {xi}, the mesh size is denoted by h, and the values u(xi) is denoted by {ui} (i=1,2,…,N). Define L(s) as a linear combination of function ui and its consecutive derivatives up to order m in neighboring discretized grid points from i+k1 to i+k2:The GC difference schemes with arbitrary order of accuracy can be
Principles about the construction of high-order accuracy schemes [17]
When constructing high-order accuracy schemes, two problems should be paid attention to: (1) The modified equation of the difference equation is different from the original differential equation. Their exact solutions are different also. There are no spurious oscillations in the original equation, but there may be spurious oscillations in the exact solution of difference equation if the difference scheme is not designed properly. In lower-order schemes, such oscillations are not very small, so
Construction of the three-point fifth-order accuracy generalized compact scheme satisfying the principle about suppression of the oscillations [16]
A three-point fifth-order accuracy GC scheme satisfying the principle about suppression of the oscillations and the principle of stability is constructed by author. The procedure is described as follows.
Consider the linear wave equation:Its semi-discretized equation is:The interior scheme is expressed by the following two linearly independent auxiliary equations:
Numerical examples
In this section some 2D IBVP which can be considered as model problems for typical phenomena in the computation of flows governed by the Euler equations are calculated by using the three-point fifth-order GC scheme satisfying the principle about suppression of the oscillations and the principle of stability presented in this paper. The third-order TVD Runge–Kutta time discretization developed by Shu and Osher [20] is implemented for time integration.
Conclusion
A new class of finite difference schemes called the GC schemes has been proposed by author. The schemes can be upwind or symmetric and may be considered an extension of the standard Pade schemes described in [4]. In this paper, a general way is presented to construct high order accuracy and high-resolution GC schemes through the modified equation. The schemes constructed by using this way could satisfy some principles and demands prescribed in advance to ensure some desired properties to the
Acknowledgements
This work is supported by the National Science Foundation and also by the Ph.D. Foundation of the National Education Commission.
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