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High Order Boundary Conditions for High Order Finite Difference Schemes on Curvilinear Coordinates Solving Compressible Flows

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Abstract

In the present paper, high order boundary conditions for high order finite difference schemes on curvilinear coordinates are implemented and analyzed. The finite difference scheme employed in the present paper is a conservative scheme on a finite volume type grid arrangement for which the implementation of the boundary conditions is characterized by a proper evaluation of the numerical fluxes at the boundaries. Simple extrapolation procedures dealing with both the inviscid and viscous fluxes at the boundary are proposed. Specifically, for the inviscid fluxes, a characteristic-variable-based extrapolation procedure is adopted, which effectively reduces the numerical reflection at the far-field boundaries. The emphasis of the present paper is to design straightforward and efficient numerical boundary closure procedures which are capable of achieving high order of accuracy near the boundaries. Special attention is paid to the influence of the extrapolation accuracy on the overall accuracy of the scheme. Numerical results show that the finite difference scheme can achieve its nominal order accuracy when the extrapolation accuracy is sufficiently high.

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Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant No. 11302250).

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Authors and Affiliations

  1. Xi’an Research Institute of High-tech, Xi’an, 710025, Shaanxi, China

    Zhen-sheng Sun, Bai-lin Zha & Shi-ying Zhang

  2. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Zhen-sheng Sun & Yu-xin Ren

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  1. Zhen-sheng Sun
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Correspondence to Yu-xin Ren.

Appendix: The Extrapolation Formulas for the BCECV

Appendix: The Extrapolation Formulas for the BCECV

See Tables 3, 4 and 5.

Table 3 \({\phi _{1/2}} = \sum \nolimits _{m = 1}^p {{\alpha _m}{\phi _m}} +O{\left( {\Delta x} \right) ^p}, \, p\) is the order accuracy of the extrapolation formulas
Full size table
Table 4 \({\phi _0} = {\beta _0}\phi _{1/2}^* + \sum \nolimits _{m = 1}^{q-1} {{\beta _m}{\phi _m}} + O{\left( {\Delta x} \right) ^q}, \, n\) is the order accuracy of the extrapolation formulas
Full size table
Table 5 \({\phi _{0}} = \sum \nolimits _{m = 1}^q {{\gamma _m}{\phi _m}} + O{\left( {\Delta x} \right) ^q}, \, q\) is the order accuracy of the extrapolation formulas
Full size table

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Sun, Zs., Ren, Yx., Zha, Bl. et al. High Order Boundary Conditions for High Order Finite Difference Schemes on Curvilinear Coordinates Solving Compressible Flows. J Sci Comput 65, 790–820 (2015). https://doi.org/10.1007/s10915-015-9988-4

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