Multi-dimensional conservative semi-Lagrangian method of characteristics CIP for the shallow water equations. (English) Zbl 1166.76045

Summary: A new characteristic approach that guarantees conservative property is proposed and applied to shallow water equations. CIP-CSL (Constrained Interpolation Profile/Conservative Semi-Lagrangian) interpolation is applied to the CIP method of characteristics in order to enhance the mass conservation. Although the characteristic formulation is originally derived from non-conservative form, the present scheme achieves complete mass conservation by solving mass conservation simultaneously and reflecting conserving mass in interpolation profile. Compared to the CIP method of characteristics, the present method has height error less by several orders of magnitude. By the enhanced conservation property, the present scheme is applicable to nonlinear problem such as shock problems. Furthermore, application to two dimensions including the Coriolis term is straightforward with directional splitting technique.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Arakawa, A.; Lamb, V. R., A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. Weather Rev., 109, 18-36 (1981)
[2] Lauritzen, P. H.; Kaas, E.; Machenhauer, B., A mass-conservative semi-implicit semi-Lagrangian limited-area shallow-water model on the sphere, Mon. Weather Rev., 134, 1205-1221 (2006)
[3] Staniforth, A.; Côté, J., Semi-Lagrangian integration schemes for atmospheric models – a review, Mon. Weather Rev., 119, 2206-2223 (1991)
[4] Erbes, G., A semi-Lagrangian method of characteristics for the shallow-water equations, Mon. Weather Rev., 121, 3443-3452 (1993)
[5] Yabe, T.; Aoki, T., A universal solver for hyperbolic equations by cubic-polynomial interpolation. I. One-dimensional solver, Comput. Phys. Commun., 66, 219-232 (1991) · Zbl 0991.65521
[6] Yabe, T.; Ishikawa, T.; Wang, P. Y., A universal solver for hyperbolic equations by cubic-polynomial interpolation. II. Two- and three-dimensional solver, Comput. Phys. Commun., 66, 233-242 (1991) · Zbl 0991.65522
[7] Ogata, Y.; Yabe, T., Multi-dimensional semi-Lagarangian characteristic approach to the shallow water equations by the CIP method, Int. J. Comput. Eng. Sci., 5, 3, 699-730 (2004)
[8] Leonard, B. P.; Lock, A. P.; Macvean, M. K., Conservative explicit unstricted-time-step multidimensional constancy-preserving advection schemes, Mon. Weather Rev., 124, 2588-2606 (1996)
[9] Zerroukat, M.; Wood, N.; Staniforth, A., SLICE: a semi-Lagrangian inherently conserving and efficient scheme for transport problems, Quart. J. R. Meteorol. Soc., 128, 2801-2820 (2002)
[10] Thuburn, J., A fully implicit, mass-conserving, semi-Lagrangian scheme for the \(f\)-plane shallow-water equations, Int. J. Numer. Method Fluid, 56, 1047-1059 (2008) · Zbl 1225.76217
[11] Nair, R. D.; Machenhauer, B., The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere, Mon. Weather Rev., 130, 649-667 (2002)
[12] Kaas, E., A simple and efficient locally mass conserving semi-Lagrangian transport scheme, Tellus, 60A, 305-320 (2008)
[13] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J. R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 474-499 (2006) · Zbl 1088.76037
[14] Lukáčová-Medvid’ová, M.; Noelle, S.; Kraft, M., Well-balanced finite volume evolution Galerkin methods for the shallow water equations, J. Comput. Phys., 221, 122-147 (2007) · Zbl 1123.76041
[15] Akoh, R.; Ii, S.; Xiao, F., A CIP/multi-moment finite volume method for shallow water equations with source terms, Int. J. Numer. Method Fluid, 56, 2245-2270 (2007) · Zbl 1135.76035
[16] Yabe, T.; Xiao, F.; Utsumi, T., Constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169, 556-593 (2001) · Zbl 1047.76104
[17] Yabe, T.; Tanaka, R.; Nakamura, T.; Xiao, F., An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension, Mon. Weather Rev., 129, 332-344 (2001)
[18] Tanaka, R.; Nakamura, T.; Yabe, T., Constructing exactly conservative scheme in non-conservative form, Comput. Phys. Commun., 126, 232-243 (2000) · Zbl 0959.65097
[19] Nakamura, T.; Tanaka, R.; Yabe, T., Multi-dimensional conservative scheme in non-conservative form, CFD J., 9, 437-453 (2001)
[20] Nakamura, T.; Tanaka, R.; Yabe, T.; Takizawa, K., Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, J. Comput. Phys., 174, 171-207 (2001) · Zbl 0995.65094
[21] Stoker, J., Water Waves: The Mathematical Theory with Applications (1957), Interscience · Zbl 0078.40805
[22] Utsumi, T.; Kunugi, T.; Aoki, T., Stability and accuracy of the cubic interpolated propagation scheme, Comput. Phys. Commun., 101, 9-20 (1997)
[23] Lin, S-J.; Rood, R. B., Multidimensional flux-form semi-Lagrangian transport schemes, Mon. Weather Rev., 124, 2046-2070 (1996)
[24] Shoucri, Magdi M., Note: numerical solution of the shallow water equations, J. Comput. Phys., 63, 240-245 (1986) · Zbl 0596.76020
[25] Ogata, Y.; Yabe, T., Shock capturing with improved numerical viscosity in primitive Euler representation, Comput. Phys. Commun., 119, 179-193 (1999) · Zbl 1175.76112
[26] Yabe, T.; Mizoe, H.; Takizawa, K.; Moriki, H.; Im, H-N.; Ogata, Y., Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme, J. Comput. Phys., 194, 57-77 (2004) · Zbl 1049.76051
[27] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Springer · Zbl 0888.76001
[28] Warming, R. F.; Beam, R. M., Upwind second-order difference schemes and applications in aerodynamics flows, AIAA J., 14, 9, 1241-1249 (1976) · Zbl 0364.76047
[29] Takewaki, H.; Yabe, T., Cubic-interpolated pseudo particle (CIP) method: application to nonlinear or multi-dimensional problems, J. Comput. Phys., 70, 355-372 (1987) · Zbl 0624.65079
[30] Dritschel, D. G.; Polvani, L. M.; Mohebalhojeh, A. R., The contour-advective semi-Lagrangian algorithm for the shallow water equations, Mon. Weather Rev., 127, 1551-1565 (1999)
[31] Sadourny, R., Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Mon. Weather Rev., 100, 136-144 (1972)
[32] Chen, C.; Xiao, F., Shallow water model on cubed-sphere by multi-moment finite volume method, J. Comput. Phys., 227, 5019-5044 (2008) · Zbl 1388.86003
[33] LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys., 146, 346-365 (1998) · Zbl 0931.76059
[34] Zhou, J. G.; Causon, D. M.; Mingham, C. G.; Ingram, D. M., The surface gradient method for the treatment of source terms in the shallow-water equations, J. Comput. Phys., 168, 1-25 (2001) · Zbl 1074.86500
[35] Vukovic, S.; Sopta, L., ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, J. Comput. Phys., 179, 593-621 (2002) · Zbl 1130.76389
[36] Xing, Y.; Shu, C. W., High order finite difference WENO schemes for a class of hyperbolic systems with source terms, J. Comput. Phys., 208, 206-227 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.