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Constrained Interpolation Profile Conservative Semi-Lagrangian Scheme Based on Third-Order Polynomial Functions and Essentially Non-Oscillatory (CIP-CSL3ENO) Scheme

Part of: Two-phase and multiphase flows Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  06 July 2017

Qijie Li ,
Syazana Omar ,
Xi Deng  and
Kensuke Yokoi
Qijie Li*
Affiliation:
School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK
Syazana Omar*
Affiliation:
School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK
Xi Deng*
Affiliation:
School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK Department of Energy Sciences, Tokyo Institute of Technology, Yokohama, 226-8502, Japan
Kensuke Yokoi*
Affiliation:
School of Engineering, Cardiff University, Cardiff, CF24 3AA, UK
*
*Corresponding author. Email addresses:YokoiK@cardiff.ac.uk (K. Yokoi), liq33@cardiff.ac.uk (Q. Li), omars2@cardiff.ac.uk (S. Omar), deng.x.aa@m.titech.ac.jp (X. Deng)
*Corresponding author. Email addresses:YokoiK@cardiff.ac.uk (K. Yokoi), liq33@cardiff.ac.uk (Q. Li), omars2@cardiff.ac.uk (S. Omar), deng.x.aa@m.titech.ac.jp (X. Deng)
*Corresponding author. Email addresses:YokoiK@cardiff.ac.uk (K. Yokoi), liq33@cardiff.ac.uk (Q. Li), omars2@cardiff.ac.uk (S. Omar), deng.x.aa@m.titech.ac.jp (X. Deng)
*Corresponding author. Email addresses:YokoiK@cardiff.ac.uk (K. Yokoi), liq33@cardiff.ac.uk (Q. Li), omars2@cardiff.ac.uk (S. Omar), deng.x.aa@m.titech.ac.jp (X. Deng)
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Abstract

We propose a fully conservative and less oscillatory multi-moment scheme for the approximation of hyperbolic conservation laws. The proposed scheme (CIP-CSL3ENO) is based on two CIP-CSL3 schemes and the essentially non-oscillatory (ENO) scheme. In this paper, we also propose an ENO indicator for the multimoment framework, which intentionally selects non-smooth stencil but can efficiently minimize numerical oscillations. The proposed scheme is validated through various benchmark problems and a comparison with an experiment of two droplets collision/separation. The CIP-CSL3ENO scheme shows approximately fourth-order accuracy for smooth solution, and captures discontinuities and smooth solutions simultaneously without numerical oscillations for solutions which include discontinuities. The numerical results of two droplets collision/separation (3D free surface flow simulation) show that the CIP-CSL3ENO scheme can be applied to various types of fluid problems not only compressible flow problems but also incompressible and 3D free surface flow problems.

Keywords

CIP-CSL schemeENO schemeconservation lawsfree surface flows

MSC classification

Secondary: 35L65: Conservation laws 65M06: Finite difference methods 65M08: Finite volume methods 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 765 - 788
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Ashgriz, N., and Poo, J. Y. Coalescence and separation in binary collisions of liquid drops. Journal of Fluid Mechanics 221 (12 1990), 183–204.CrossRefGoogle Scholar
[2] Colella, P., and Woodward, P. R. The piecewise parabolic method (PPM) for gas-dynamical simulations. Journal of Computational Physics 54, 1 (1984), 174–201.Google Scholar
[3] Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. R. Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics 71, 2 (1987), 231–303.CrossRefGoogle Scholar
[4] Hu, G., Li, R., and Tang, T. A robust WENO type finite volume solver for steady euler equations on unstructured grids. Communications in Computational Physics 9, 3 (2011), 627–648.Google Scholar
[5] Huang, C.-S., Xiao, F., and Arbogast, T. Fifth order multi-moment weno schemes for hyperbolic conservation laws. Journal of Scientific Computing 64, 2 (2015), 477–507.Google Scholar
[6] Hyman, J. M. Accurate monotonicity preserving cubic interpolation. SIAM Journal on Scientific and Statistical Computing 4, 4 (1983), 645–654.CrossRefGoogle Scholar
[7] Ii, S., and Xiao, F. CIP/multi-moment finite volume method for euler equations: A semi-lagrangian characteristic formulation. Journal of Computational Physics 222, 2 (2007), 849–871.Google Scholar
[8] Ii, S., and Xiao, F. High order multi-moment constrained finite volume method. part i: Basic formulation. Journal of Computational Physics 228, 10 (2009), 3669–3707.CrossRefGoogle Scholar
[9] Imai, Y., and Aoki, T. Accuracy study of the IDO scheme by fourier analysis. Journal of Computational Physics 217, 2 (2006), 453–472.CrossRefGoogle Scholar
[10] Jiang, G.-S., and Shu, C.-W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics 126, 1 (1996), 202–228.Google Scholar
[11] Liu, X.-D., Osher, S., and Chan, T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics 115, 1 (1994), 200–212.CrossRefGoogle Scholar
[12] Onodera, N., Aoki, T., and Yokoi, K. A fully conservative high-order upwind multi-moment method using moments in both upwind and downwind cells. International Journal for Numerical Methods in Fluids (2016). fld.4228.Google Scholar
[13] Qiu, J.-M., and Shu, C.-W. Conservative high order semi-lagrangian finite difference WENO methods for advection in incompressible flow. Journal of Computational Physics 230, 4 (2011), 863–889.Google Scholar
[14] Serna, S., and Marquina, A. Power ENO methods: a fifth-order accurate weighted power ENO method. Journal of Computational Physics 194, 2 (2004), 632–658.Google Scholar
[15] Shu, C.-W. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, 6 (1988), 1073–1084.Google Scholar
[16] Shu, C.-W. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 23–28, 1997. Springer Berlin Heidelberg, Berlin, Heidelberg, 1998, ch. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, pp. 325–432.Google Scholar
[17] Shu, C.-W., and Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77, 2 (1988), 439–471.Google Scholar
[18] Shu, C.-W., and Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. Journal of Computational Physics 83, 1 (1989), 32–78.Google Scholar
[19] Sod, G. A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27, 1 (1978), 1–31.Google Scholar
[20] Sun, Z., Teng, H., and Xiao, F. A slope constrained 4th order multi-moment finite volume method with weno limiter. Communications in Computational Physics 18 (2015), 901–930.Google Scholar
[21] Tanaka, R., Nakamura, T., and Yabe, T. Constructing exactly conservative scheme in a non-conservative form. Computer Physics Communications 126, 3 (2000), 232–243.Google Scholar
[22] Xiao, F. Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow. Journal of Computational Physics 195, 2 (2004), 629–654.Google Scholar
[23] Xiao, F., Akoh, R., and Ii, S. Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows. Journal of Computational Physics 213, 1 (2006), 31–56.Google Scholar
[24] Xiao, F., Ikebata, A., and Hasegawa, T. Numerical simulations of free-interface fluids by a multi-integrated moment method. Computers and Structures 83, 67 (2005), 409–423. Frontier of Multi-Phase Flow Analysis and Fluid-Structure Frontier of Multi-Phase Flow Analysis and Fluid-Structure.CrossRefGoogle Scholar
[25] Xiao, F., and Yabe, T. Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation. Journal of Computational Physics 170, 2 (2001), 498–522.Google Scholar
[26] Xiao, F., Yabe, T., Peng, X., and Kobayashi, H. Conservative and oscillation-less atmospheric transport schemes based on rational functions. Journal of Geophysical Research: Atmospheres 107, D22 (2002), ACL 2–1–ACL 2–11. 4609.Google Scholar
[27] Yabe, T., Tanaka, R., Nakamura, T., and Xiao, F. An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension. Monthly Weather Review 129, 2 (2001), 332–344.Google Scholar
[28] Yokoi, K. A numerical method for free-surface flows and its application to droplet impact on a thin liquid layer. Journal of Scientific Computing 35, 2 (2008), 372–396.Google Scholar
[29] Yokoi, K. A practical numerical framework for free surface flows based on CLSVOF method, multi-moment methods and density-scaled CSF model: Numerical simulations of droplet splashing. Journal of Computational Physics 232, 1 (2013), 252–271.CrossRefGoogle Scholar
[30] Yokoi, K. A density-scaled continuum surface force model within a balanced force formulation. Journal of Computational Physics 278 (2014), 221–228.CrossRefGoogle Scholar
[31] Yokoi, K., Onishi, R., Deng, X.-L., and Sussman, M. Density-scaled balanced continuum surface force model with a level set based curvature interpolation technique. International Journal of ComputationalMethods 13, 04 (2016), 1641004.Google Scholar