Abstract
With an increase in model resolution, compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor computers. To provide high-quality numerical ocean simulation on overset grids, we tried a novel formulation of the fourth-order multi-moment constrained finite volume scheme to simulate continuous and discontinuous problems in the Cartesian coordinate. Utilizing some degrees of freedom over each cell and derivatives at the cell center, we obtained a two-dimensional (2D) cubic polynomial from which point values on the extended overlap can achieve fourth-order accuracy. However, this interpolation causes a lack of conservation because the flux between the regions are no longer equal; thus, a flux correction is implemented to ensure conservation. A couple of numerical experiments are presented to evaluate the numerical scheme, which confirms its approximately fourth-order accuracy in conservative transportation on overset grid. The test cases reveal that the scheme is effective to suppress numerical oscillation in discontinuous problems, which may be powerful for salinity advection computing with a sharp gradient.
Similar content being viewed by others
References
Baba, Y., Takahashi, K., and Sugimura, T., 2010. Dynamical core of an atmospheric general circulation model on a yinyang grid. Monthly Weather Review, 138 (10): 3988–4005, DOI: https://doi.org/10.1175/2010MWR3375.1.
Bastian, P., 2014. A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure. Computational Geosciences, 18 (5): 779–796, DOI: https://doi.org/10.1007/s10596-014-9426-y.
Chen, C. G., and Xiao, F., 2008. Shallow water model on cubedsphere by multi-moment finite volume method. Journal of Computational Physics, 227 (10): 5019–5044, DOI: https://doi.org/10.1016/j.jcp.2008.01.033.
Chen, C. G., Li, X. L., Shen, X. S., and Xiao, F., 2014. Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids. Journal of Computational Physics, 271: 191–223, DOI: https://doi.org/10.1016/j.jcp.2013.10.026.
Deng, X., Sun, Z. Y., Xie, B., Yokoi, K., Chen, C. G., and Xiao, F., 2017a. A non-oscillatory multi-moment finite volume scheme with boundary gradient switching. Journal of Scientific Computing, 72: 1–23, DOI: https://doi.org/10.1007/s10915-017-0392-0.
Deng, X., Xie, B., and Xiao, F., 2017b. A finite volume multimoment method with boundary variation diminishing principle for Euler equation on three-dimensional hybrid unstructured grids. Computers & Fluids, 153: 85–101, DOI: https://doi.org/10.1016/j.compfluid.2017.05.007.
Hsu, T. W., Doong, D. J., Hsieh, K. J., and Liang, S. J., 2015. Numerical study of monsoon effect on green island wake. Journal of Coastal Research, 31 (5): 1141–1150, DOI: https://doi.org/10.2112/JCOASTRES-D-14-00206.1.
Ii, S., and Xiao, F., 2007. CIP/multi-moment finite volume method for Euler equations, a semi-Lagrangian characteristic formulation. Journal of Computational Physics, 222 (2): 849–871, DOI: https://doi.org/10.1016/j.jcp.2006.08.015.
Ii, S., and Xiao, F., 2009. High order multi-moment constrained finite volume method. Part I: Basic formulation. Journal of Computational Physics, 228 (10): 3669–3707, DOI: https://doi.org/10.1016/j.jcp.2009.02.009.
Ii, S., and Xiao, F., 2010. A global shallow water model using high order multi-moment constrained finite volume method and icosahedral grid. Journal of Computational Physics, 229 (5): 1774–1796, DOI: https://doi.org/10.1016/j.jcp.2009.11.008.
Ii, S., Shimuta, M., and Xiao, F., 2005. A 4th-order and single-cell-based advection scheme on unstructured grids using multimoments. Computer Physics Communications, 173 (1–2): 17–33, DOI: https://doi.org/10.1016/j.cpc.2005.07.003.
Imai, Y., Aoki, T., and Takizawa, K., 2008. Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics. Journal of Computational Physics, 227 (4): 2263–2285, DOI: https://doi.org/10.1016/j.jcp.2007.11.031.
Jiang, G. S., and Shu, C. W., 1996. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126: 202–228, DOI: https://doi.org/10.1006/jcph.1996.0130.
Jin, P., Deng, X., and Xiao, F., 2018. A direct ALE multi-moment finite volume scheme for the compressible Euler equations. Communication in Computational Physics, 24 (5): 1300–1325, DOI: https://doi.org/10.4208/cicp.OA-2017-0189.
Kageyama, A., and Sato, T., 2004. The ‘yin-yang grid’: An overset grid in spherical geometry. Geochemistry Geophysics Geosystems, 5 (9): Q09005, DOI: https://doi.org/10.1029/2004GC000734.
Li, X. H., and Peng, X. D., 2018. Long-term integration of a global non-hydrostatic atmospheric model on an aqua planet. Journal of Meteorological Research, 32 (4): 517–533, DOI: CNKI:SUN:QXXW.0.2018-04-001.
Li, X. H., Peng, X. D., and Li, X. L., 2015a. An improved dynamic core for a non-hydrostatic model system on the yinyang grid. Advances in Atmospheric Sciences, 32 (5): 648–658, DOI: https://doi.org/10.1007/s00376-014-4120-5.
Li, X. L., Chen, D. H., Peng, X. D., Takahashi, K., and Xiao, F., 2008. A multimoment finite-volume shallow-water model on the yin yang overset spherical grid. Monthly Weather Review, 136 (8): 3066, DOI: https://doi.org/10.1175/2007mwr2206.1.
Li, X. L., Chen, D. H., Peng, X. D., Xiao, F., and Shen, X. S., 2006. Implementation of the semi-Lagrangian advection scheme on a quasi-uniform overset grid on a sphere. Advances in Atmospheric Sciences, 23 (5): 792–801, DOI: https://doi.org/10.1007/s00376-006-0792-9.
Li, X. L., Chen, C. G., Xiao, F., and Shen, X. S., 2015b. A high-order multi-moment constrained finite-volume global shallow-water model on the yin-yang grid. Quarterly Journal of the Royal Meteorological Society, 141 (691): 2090–2102, DOI: https://doi.org/10.1002/qj.2504.
Liu, X. D., Osher, S., and Chan, T., 1994. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 115 (1): 200–212, DOI: https://doi.org/10.1006/jcph.1994.1187.
Onodera, N., Aoki, T., and Kobayashi, H., 2011. Large-eddy simulation of turbulent channel flows with conservative IDO scheme. Journal of Computational Physics, 230: 5787–5805, DOI: https://doi.org/10.1016/j.jcp.2011.04.004.
Peng, X. D., Xiao, F., and Takahashi, K., 2006. Conservative constraint for a quasi-uniform overset grid on the sphere. Quarterly Journal of the Royal Meteorological Society, 132 (616): 979–996, DOI: https://doi.org/10.1256/qj.05.18.
Peng, X. D., Xiao, F., Takahashi, K., and Yabe, T., 2004. Conservative CIP transport in meteorological models. JSME International Journal (Series B), 47 (4): 725–734, DOI: https://doi.org/10.1299/jsmeb.47.725.
Qaddouri, A., and Lee, V., 2011. The Canadian Global Environmental Multiscale model on the Yin-Yang grid system. Quarterly Journal of the Royal Meteorological Society, 137 (660): 1913–1926, DOI: https://doi.org/10.1002/qj.873.
Spiteri, R. J., and Ruuth, S. J., 2002. A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM Journal on Numerical Analysis, 40 (2): 469–491, DOI: https://doi.org/10.1137/S0036142901389025.
Sun, Y. Z., Wang, Z. J., and Liu, Y., 2007. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Communications in Computational Physics, 2 (2): 310–333, DOI: https://doi.org/10.2514/6.2006-301.
Tanaka, R., Nakamura, T., and Yabe, T., 2000. Constructing exactly conservative scheme in a non-conservative form. Computer Physics Communications, 126 (3): 232–243, DOI: https://doi.org/10.1016/s0010-4655(99)00473-7.
Taneja, A., and Higdon, J., 2018. A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs. Journal of Computational Physics, 352: 341–372, DOI: https://doi.org/10.1016/j.jcp.2017.09.059.
Wang, Z. J., 1995. A fully conservative interface algorithm for overlapped grids. Journal of Computational Physics, 122 (1): 96–106, DOI: https://doi.org/10.1006/jcph.1995.1199.
Wang, Z. J., 2004. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation. Journal of Computational Physics, 194 (2): 716–741, DOI: https://doi.org/10.1006/jcph.2002.7041.
Xiao, F., and Yabe, T., 2001. Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation. Journal of Computational Physics, 170 (2): 498–522, DOI: https://doi.org/10.1006/jcph.2001.6746.
Xiao, F., Yabe, T., Peng, X. D., and Kobayashi H., 2002. Conservative and oscillation-less atmospheric transport schemes based on rational functions. Journal of Geophysical Research, 107 (D22): 4609, DOI: https://doi.org/10.1029/2001jd001532.
Xie, B., and Xiao, F., 2014. Two and three dimensional multimoment finite volume solver for incompressible Navier-Stokes equations on unstructured grids with arbitrary quadrilateral and hexahedral elements. Computers & Fluids, 104: 40–54, DOI: https://doi.org/10.1016/j.compfluid.2014.08.002.
Xie, B., Ii, S., Ikebata, A., and Xiao, F., 2014. A multi-moment finite volume method for incompressible Navier-Stokes equations on unstructured grids: Volume-average/point-value formulation. Journal of Computational Physics, 277: 138–162, DOI: https://doi.org/10.1016/j.jcp.2014.08.011.
Yabe, T., Tanaka, R., Nakamura, T., and Xiao, F., 2001. An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension. Monthly Weather Review, 129 (2): 332–344, DOI: https://doi.org/10.1175/1520-0493(2001)1292.0.CO;2.
Zerroukat, M., and Allen, T., 2012. On the solution of elliptic problems on overset/yin-yang grids. Monthly Weather Review, 140 (8): 2756–2767, DOI: https://doi.org/10.1175/MWR-D-11-00272.1.
Zhang, S. Q., Liu, Y., Ma, X. H., Wang, H. N., Zhang, X. F., Yu, X. L., and Lu, L., 2018. The ‘two oceans and one sea’ extended range numerical prediction system with an ultra-high resolution atmosphere-ocean-land regional coupled model. Atmospheric and Oceanic Science Letters, 11: 1674–2834.
Zhang, Y. F., and Juang, H. M. H., 2013. A mass-conserving noniteration-dimensional-split semi-Lagrangian advection scheme for limited-area modelling. Quarterly Journal of the Royal Meteorological Society, 138 (669): 2118–2125, DOI: https://doi.org/10.1002/qj.1938.
Zhang, Z., Tian, J. W., Qiu, B., Zhao, W., Chang, P., Wu, D. X., and Wan, X. Q., 2016. Observed 3D structure, generation, and dissipation of oceanic mesoscale eddies in the South China Sea. Scientific Reports, 6 (1): 24349, DOI: https://doi.org/10.1038/srep24349.
Acknowledgements
The authors thank Dr. X. Deng at the Tokyo Institute of Technology for the useful discussions on the MCV scheme. We also appreciate the helpful input by Dr. X. L. Li at the China Meteorological Administration. This study was supported by grants from the National Natural Science Foundation of China (Nos. 41575103 and 91637210).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gu, J., Peng, X., Dai, Y. et al. Fourth-Order Conservative Transport on Overset Grids Using Multi-Moment Constrained Finite Volume Scheme for Oceanic Modeling. J. Ocean Univ. China 19, 747–760 (2020). https://doi.org/10.1007/s11802-020-4309-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11802-020-4309-4