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Simulation of Inviscid Compressible Flows Using PDE Transform

Published online by Cambridge University Press:  03 June 2015

Langhua Hu ,
Siyang Yang  and
Guo-Wei Wei
Langhua Hu*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA
Siyang Yang*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA
Guo-Wei Wei*
Affiliation:
Department of Mathematics, Michigan State University, MI 48824, USA Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA Center for Mathematical Molecular Biosciences, Michigan State University, MI 48824, USA
*
Email:hulanghu.at.msu.edu@gmail.com
Email:yangsiyang@gmail.com
Corresponding author.Email:wei@math.msu.edu
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Abstract

The solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity of the physical situations. The present work introduces the use of the partial differential equation (PDE) transform, paired with the Fourier pseudospectral method (FPM), as a new approach for hyperbolic conservation law problems. The PDE transform, based on the scheme of adaptive high order evolution PDEs, has recently been applied to decompose signals, images, surfaces and data to various target functional mode functions such as trend, edge, texture, feature, trait, noise, etc. Like wavelet transform, the PDE transform has controllable time-frequency localization and perfect reconstruction. A fast PDE transform implemented by the fast Fourier Transform (FFT) is introduced to avoid stability constraint of integrating high order PDEs. The parameters of the PDE transform are adaptively computed to optimize the weighted total variation during the time integration of conservation law equations. A variety of standard benchmark problems of hyperbolic conservation laws is employed to systematically validate the performance of the present PDE transform based FPM. The impact of two PDE transform parameters, i.e., the highest order and the propagation time, is carefully studied to deliver the best effect of suppressing Gibbs’ oscillations. The PDE orders of 2-6 are used for hyperbolic conservation laws of low oscillatory solutions, while the PDE orders of 8-12 are often required for problems involving highly oscillatory solutions, such as shock-entropy wave interactions. The present results are compared with those in the literature. It is found that the present approach not only works well for problems that favor low order shock capturing schemes, but also exhibits superb behavior for problems that require the use of high order shock capturing methods.

Keywords

35K4165N9976L0576N15Partial differential equation transformhyperbolic conservation lawsFourier pseudospectral methodadaptive lowpass filtersGibbs’ oscillations
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1201 - 1238
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Anderson, D. A., Tannehill, J. C., and Pletcher, R. H.Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York, 1984.Google Scholar
[2]Bates, P. W., Wei, G. W., and Zhao, S.Minimal molecular surfaces and their applications. J. Comput. Chem., 29(3):380–91,2008.Google Scholar
[3]Bianco, F., Puppo, G., and Russo, G.High-order central differencing for hyperbolic conservation laws. SIAM J. Sci. Comput., pages 294–322,1999.Google Scholar
[4]Cai, W. and Shu, C. W.Uniform high-order spectral methods for one- and two-dimensional euler equations. J. Comput. Phys., 104:427–443,1993.Google Scholar
[5]Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A.Spectral methods in fluid dynamics. Springer-Verlag, Berlin, 1988.Google Scholar
[6]Chen, D., Wei, G. W., Cong, X., and Wang, G.Computational methods for optical molecular imaging. Commun. Numer. Meth. Engng., 25:1137–1161,2009.Google Scholar
[7]Chen, Z., Baker, N. A., and Wei, G. W.Differential geometry based solvation models I: Eulerian formulation. J. Comput. Phys., 229:8231–8258,2010.Google Scholar
[8]Chen, Z., Baker, N. A., and Wei, G. W.Differential geometry based solvation models II: Lagrangian formulation. J. Math. Biol., 63:1139–1200,2011.Google Scholar
[9]Didas, S., Weickert, J., and Burgeth, B.Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vis., 35(3):208–226,2009.Google Scholar
[10]Don, W. S.Numerical study of pseudospectral methods in shock wave applications. J. Comput. Phys, 110:103–111,1994.Google Scholar
[11]Don, W. S. and Gottlieb, D.Spectral simulation of supersonic reactive flows. SIAM J. Numer Anal., 35:2370–2384,1998.CrossRefGoogle Scholar
[12]Fornberg, B.A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, 1996.Google Scholar
[13]Gottlieb, D. and Hesthaven, J. S.Spectral methods for hyperbolic problems. J. Comput. Appl. Math., 128:83–131,2001.Google Scholar
[14]Gottlieb, D., Lustman, L., and Orszag, S. A.Spectral calculations of one-dimensional inviscid compressible flow. SIAM J. Sci. Stat. Comput., 2:296–310,1981.CrossRefGoogle Scholar
[15]Gottlieb, D., Shu, C. W., Solomonoff, A., and Vandeven, H.On the Gibbs phenomenon I. J. Comput. Appl. Math., 43:81–98,1992.Google Scholar
[16]Gottlieb, D. and Tadmor, E.Recovering pointwise values of discontinuous data within spectral accuracy. Progress and Supercomputing in Computational Fluid Dynamics, E. M. Murman and S. S. Abarbanel eds., Birkhauser, pages 357–375,1985.Google Scholar
[17]Grahs, T. and Sonar, T.Entropy-controlled artificial anisotropic diffusion for the numerical solution of conservation laws based on algorithms from image processing. J. Vis. Commun. Image R., 13:176–194,2002.Google Scholar
[18]Gu, Y. and Wei, G. W.Conjugated filter approach for shock capturing. Commun. Numer. Meth. Engng., 19:99–110,2003.Google Scholar
[19]Gu, Y., Zhou, Y. C., and Wei, G. W.Conjugate filters with spectral-like resolution for 2D incompressible flows. Comput. Fluids, 33:777–794,2003.Google Scholar
[20]Harten, A., Engquist, B., Osher, S., and Chakravarthy, S.Uniform high-order accurate essentially non-oscillatory schemes, III. J. Comput. Phys., 131:3–47,1997.Google Scholar
[21]Hu, G., Li, R., and Tang, T.A robust weno type finite volume solver for steady euler equations on unstructured grids. Commun. Comput. Phys., 9:627–648,2011.Google Scholar
[22]Hussaini, M. Y., Kopriva, D. A., Salas, M. D., and Zang, T. A.Spectral methods for the euler equation: Part I - Fourier methods and shock-capturing. AIAA J., 23(No. 1):64–70,1985.Google Scholar
[23]Jiang, G. S. and Shu, C. W.Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126(1):202–228,1996.Google Scholar
[24]Jin, S. and Levermore, C. D.Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys., 126:449–467,1996.Google Scholar
[25]Johansen, S. T., Wu, J. Y., and Shyy, W.Filter-based unsteady rans computations. Int. J. Heat Fluid Flow, 25:10–21,2004.Google Scholar
[26]Krasny, R.A study of singularity formation in a vortex sheet by the point vortex approximation. J. Fluid Mech., 167:65–93,1986.CrossRefGoogle Scholar
[27]Kurganov, A. and Levy, D.A third-order semidiscrete central scheme for conservation laws and convention diffusion equations. SIAM J. Sci. Comput., 22:1461–1488,2000.Google Scholar
[28]Lee, S., Lele, S. K., and Moin, P.Interaction of isotropic turbulence with shock waves: Effect of shock strength. J. Fluid Mech., 340:225–247,1997.Google Scholar
[29]LeVeque, R. J. and Pelanti, M.A class of approximate riemann solvers and their relation to relaxation schemes. J. Comput. Phys., 172:574–591,2001.Google Scholar
[30]Li, J., Li, Q., and Xu, K.Comparison of the generalized riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations. J. Comput. Phys., 230:5080–5099,2011.Google Scholar
[31]Liu, X. D. and Tadmor, E.Third order non-oscillatory central scheme for hyperbolic conservation laws. Numer. Math., 79:397–425,1998.Google Scholar
[32]Lysaker, M., Lundervold, A., and Tai, X. C.Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE T. Image Process., 12(12):1579–1590,2003.Google Scholar
[33]McKenzie, J. F. and Westphal, K. O.Interaction of linear waves with oblique shock waves. Phys. Fluids, 11:2350–2362,1968.Google Scholar
[34]Nessyahu, H. and Tadmor, E.Non-oscillation central differencing for hyperbolic conservation laws. J. Comput. Phys., 87:408–463,1990.Google Scholar
[35]Ni, G., Jiang, S., and Xu, K.A dgbgk scheme based on weno limiters for viscous and inviscid flows. J. Comput. Phys., 227:5799–5815,2008.Google Scholar
[36]Nithiarasu, P., Zienkiewicz, O. C., Sai, B. V. K. S., Morgan, K., Codina, R., and Vazquez, M.Shock capturing viscosities for the general fluid mechanics algorithm. Int. J. Numer. Meth. Fluids, 28:1325–1353,1998.Google Scholar
[37]Qiu, J. M. and Shu, C. W.Conservative high order semi-lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys., 230:863–889,2011.Google Scholar
[38]Qiu, J. X. and Shu, C. W.An the construction, comparison, and local characteristic decomposition for high order central WENO schemes. J. Comput. Phys., 183:187–209,2002.Google Scholar
[39]Rudin, L. I., Osher, S., and Fatemi, E.Nonlinear total variation based noise removal algorithms. Physica D, 60(1-4):259–268,1992.Google Scholar
[40]Sun, Y. H., Wu, P. R., Wei, G. W., and Wang, G.Evolution-operator-based single-step method for image processing. Int. J. Biomed. Imaging, 83847:1–27,2006.Google Scholar
[41]Sun, Y. H., Zhou, Y. C., Li, S. G., and Wei, G. W.A windowed fourier pseudospectral method for hyperbolic conservation laws. J. Comput. Phys., 214:466–490,2006.Google Scholar
[42]Tadmor, E.Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal., 26:30–44,1989.Google Scholar
[43]Toro, E. F., Millington, R. C., and Titarev, V. A.Ader: Arbitrary-order non-oscillatory advection scheme. Proc. 8th International Conference on Non-linear Hyperbolic Problems, Magdeburg, Germany, March, 2000.Google Scholar
[44]Trefethen, L. N.Spectral Methods in Matlab. Oxford University, Oxford, England, 2000.Google Scholar
[45]Vandeven, H.Family of spectral filters for discontinuous problems. J. Sci. Comput., 6:159–192,1991.Google Scholar
[46]Wan, D. C., Patnaik, B. S. V., and Wei, G. W.Discrete singular convolution-finite subdomain method for the solution of incompressible viscous flows. J. Comput. Phys., 180:229–255, 2002.Google Scholar
[47]Wang, Y., Wei, G. W., and Yang, S.-Y.Partial differential equation transform - Variational formulation and Fourier analysis. Int. J. Numer. Meth. Biomed. Engng., 27:1996–2020,2011.Google Scholar
[48]Wang, Y., Wei, G. W., and Yang, S.-Y.Iterative filtering decomposition based on local spectral evolution kernel. J. Sci. Comput., 50:629–664,2012.Google Scholar
[49]Wang, Y., Wei, G. W., and Yang, S.-Y.Mode decomposition evolution equations. J. Sci. Comput., 50:495–518,2012.Google Scholar
[50]Wei, G. W.Discrete singular convolution for the solution of the Fokker-Planck equations. J. Chem. Phys., 110:8930–8942,1999.Google Scholar
[51]Wei, G. W.Generalized Perona-Malik equation for image restoration. IEEE Signal Proc. Letters, 6(7):165–167,1999.Google Scholar
[52]Wei, G. W.Oscillation reduction by anisotropic diffusions. Comput. Phys. Commun., 144:317–342,2002.Google Scholar
[53]Wei, G. W.Differential geometry based multiscale models. B. Math. Biol., 72:1562 – 1622, 2010.Google Scholar
[54]Wei, G. W. and Gu, Y.Conjugated filter approach for solving Burgers’ equation. J. Comput. Appl. Math., 149:439–456,2002.Google Scholar
[55]Wei, G. W. and Jia, Y. Q.Synchronization-based image edge detection. Europhysics Letters, 59(6):814–819,2002.Google Scholar
[56]Wei, G. W., Zhao, Y. B., and Xiang, Y.Discrete singular convolution and its application to the analysis of plates with internal supports, I theory and algorithm. Int. J. Numer. Meth. Engng., 55:913–946,2002.CrossRefGoogle Scholar
[57]Wei, G. W., Zheng, Q., Chen, Z., and Xia, K.Variational multiscale models for charge transport. SIAM Review, 54:699–754,2012.Google Scholar
[58]Xu, K.A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys., 171:289–335,2001.Google Scholar
[59]Yee, H. C., Sandham, N. D., and Djomehri, M. J.Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys., 150:199–238,1999.Google Scholar
[60]Zheng, Q., Yang, S. Y., and Wei, G. W.Molecular surface generation using PDE transform. Int. J. Numer. Meth. Biomed. Engng., 28:291–316,2012.Google Scholar
[61]Zhou, Y. C., Gu, Y., and Wei, G. W.Shock-capturing with natural high frequency oscillations. Int. J. Numer. Meth. Fluid, 41:1319–1338,2003.CrossRefGoogle Scholar
[62]Zhou, Y. C. and Wei, G. W.High resolution conjugate filters for the simulation of flows. J. Comput. Phys., 189:159–179,2003.Google Scholar