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

  • Qijie Li (a1), Syazana Omar (a1), Xi Deng (a1) (a2) and Kensuke Yokoi (a1)

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.

Corresponding author
*Corresponding author. Email addresses: (K. Yokoi), (Q. Li), (S. Omar), (X. Deng)
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[1] AshgrizN., and PooJ. Y. Coalescence and separation in binary collisions of liquid drops. Journal of Fluid Mechanics 221 (12 1990), 183204.
[2] ColellaP., and WoodwardP. R. The piecewise parabolic method (PPM) for gas-dynamical simulations. Journal of Computational Physics 54, 1 (1984), 174201.
[3] HartenA., EngquistB., OsherS., and ChakravarthyS. R. Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics 71, 2 (1987), 231303.
[4] HuG., LiR., and TangT. A robust WENO type finite volume solver for steady euler equations on unstructured grids. Communications in Computational Physics 9, 3 (2011), 627648.
[5] HuangC.-S., XiaoF., and ArbogastT. Fifth order multi-moment weno schemes for hyperbolic conservation laws. Journal of Scientific Computing 64, 2 (2015), 477507.
[6] HymanJ. M. Accurate monotonicity preserving cubic interpolation. SIAM Journal on Scientific and Statistical Computing 4, 4 (1983), 645654.
[7] IiS., and XiaoF. CIP/multi-moment finite volume method for euler equations: A semi-lagrangian characteristic formulation. Journal of Computational Physics 222, 2 (2007), 849871.
[8] IiS., and XiaoF. High order multi-moment constrained finite volume method. part i: Basic formulation. Journal of Computational Physics 228, 10 (2009), 36693707.
[9] ImaiY., and AokiT. Accuracy study of the IDO scheme by fourier analysis. Journal of Computational Physics 217, 2 (2006), 453472.
[10] JiangG.-S., and ShuC.-W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics 126, 1 (1996), 202228.
[11] LiuX.-D., OsherS., and ChanT. Weighted essentially non-oscillatory schemes. Journal of Computational Physics 115, 1 (1994), 200212.
[12] OnoderaN., AokiT., and YokoiK. 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.
[13] QiuJ.-M., and ShuC.-W. Conservative high order semi-lagrangian finite difference WENO methods for advection in incompressible flow. Journal of Computational Physics 230, 4 (2011), 863889.
[14] SernaS., and MarquinaA. Power ENO methods: a fifth-order accurate weighted power ENO method. Journal of Computational Physics 194, 2 (2004), 632658.
[15] ShuC.-W. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9, 6 (1988), 10731084.
[16] ShuC.-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. 325432.
[17] ShuC.-W., and OsherS. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77, 2 (1988), 439471.
[18] ShuC.-W., and OsherS. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. Journal of Computational Physics 83, 1 (1989), 3278.
[19] SodG. A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27, 1 (1978), 131.
[20] SunZ., TengH., and XiaoF. A slope constrained 4th order multi-moment finite volume method with weno limiter. Communications in Computational Physics 18 (2015), 901930.
[21] TanakaR., NakamuraT., and YabeT. Constructing exactly conservative scheme in a non-conservative form. Computer Physics Communications 126, 3 (2000), 232243.
[22] XiaoF. 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), 629654.
[23] XiaoF., AkohR., and IiS. 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), 3156.
[24] XiaoF., IkebataA., and HasegawaT. Numerical simulations of free-interface fluids by a multi-integrated moment method. Computers and Structures 83, 67 (2005), 409423. Frontier of Multi-Phase Flow Analysis and Fluid-Structure Frontier of Multi-Phase Flow Analysis and Fluid-Structure.
[25] XiaoF., and YabeT. Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation. Journal of Computational Physics 170, 2 (2001), 498522.
[26] XiaoF., YabeT., PengX., and KobayashiH. 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.
[27] YabeT., TanakaR., NakamuraT., and XiaoF. An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension. Monthly Weather Review 129, 2 (2001), 332344.
[28] YokoiK. 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), 372396.
[29] YokoiK. 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), 252271.
[30] YokoiK. A density-scaled continuum surface force model within a balanced force formulation. Journal of Computational Physics 278 (2014), 221228.
[31] YokoiK., OnishiR., DengX.-L., and SussmanM. Density-scaled balanced continuum surface force model with a level set based curvature interpolation technique. International Journal of ComputationalMethods 13, 04 (2016), 1641004.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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