Some Mandatory Benchmark Tests for Stability and Accuracy of High-Order Finite Difference Schemes


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In Computational Fluid Dynamics (CFD) we have to deal with various types of phenomena with strong discontinuities at different length scales like turbulence or hypersonic flows. It is essential that the numerical techniques must provide accurate approximation to calculate the time and space-derivatives. This paper focuses on the behavior of the solution of some known numerical schemes in comparison with analytical solution provided here for the first time, applied on one-dimensional conservation law with different fluxes and non-usual different initial conditions. The results are very interesting in the sense that methods which report very good results on classical tests schemes were not able to accurately predict solutions that admit discontinuities and sharp gradients inside.



Edited by:

Adrian Olaru




S. Dănăilă et al., "Some Mandatory Benchmark Tests for Stability and Accuracy of High-Order Finite Difference Schemes", Applied Mechanics and Materials, Vol. 859, pp. 52-56, 2017

Online since:

December 2016




* - Corresponding Author

[1] S. Dănăilă, C. Berbente, Metode numerice în mecanica fluidelor, Ed. Academiei, (2003).

[2] A. Bogoi, An introduction to differential equations, Ed. Monitorul Oficial, (2014).

[3] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys, 103, 16, (1992).

[4] C.K.W. Tam and J.C. Webb. Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics. J. Comput. Phys, Vol. 107, 1993, pp.262-281.

DOI: 10.1006/jcph.1993.1142

[5] Kim J W and Lee D J 1996 Optimized compact finite difference schemes with maximum resolution. AIAA J. 34 887-893.

DOI: 10.2514/3.13164

[6] W. J. Zhu, Aero-Acoustic Computations of Wind Turbines, PHd thesis, Technical University of Denmark, Septembrie (2007).

[7] Zhong, X. High-order finite-difference schemes for numerical simulations of hypersonic boundary-layer transition. J. Comp. Phys. 144, 662–709, (1998).

DOI: 10.1006/jcph.1998.6010

[8] X. Liu, S. Zhang, H. Zhang, C-W. Shu, A new class of central compact schemes with spectral-like resolution I: Linear schemes, J. Comput. Phys, 248 (2013) 235-256.

DOI: 10.1016/

[9] Pirozzoli S. Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J Comp Phys, 178(1): 81–117, (2002).

DOI: 10.1006/jcph.2002.7021

[10] C. -W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77, 439 , (1988).

DOI: 10.1016/0021-9991(88)90177-5

[11] C. -W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, ICASE Report 97-65.

DOI: 10.1007/bfb0096355

[12] A.K. Henrick, T.D. Aslam, J.M. Powers, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys, 207 (2) , 542–567, (2005).

DOI: 10.1016/

[13] Ghosh, D., and Baeder, J. D., Compact Reconstruction Schemes with Weighted ENO Limiting for Hyperbolic Conservation Laws, SIAM Journal on Scientific Computing, Vol. 34, No. 3, p.1678–1706, (2012).

DOI: 10.1137/110857659

[14] C. Bogey, C. Bailly A Family of Low Dispersive and Low Dissipative Explicit Schemes for Computing the Aerodynamic Noise", AIAA-paper 2002-2509, (2002).

DOI: 10.2514/6.2002-2509

[15] Hu, F. Q., Hussaini, M. Y., and Manthey, J. L., Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, J. Comput. Phys, Vol. 124, No. 1, 1996, pp.177-191.

DOI: 10.1006/jcph.1996.0052

[16] Hardin, J., Ristorcelli, J. R., and Tam, C., editors, ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA), Hampton, Virginia, October, (1995).

[17] A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303, (1987).

DOI: 10.1007/978-3-642-60543-7_12

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