Using cubic Hermite polynomials in constructing monotone semi-Lagrangian methods for advection equation

Document Type : Research Article


1 Ph.D. Student, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran

2 Professor, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran

3 Associate Professor, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran


Semi-Lagrangian methods have been widely applied in general circulation models of the atmosphere as they do not suffer from a Courant–Fredericks–Levy (CFL) constraint for computational stability. Ease of application, high accuracy and speed of execution in general circulation models are other reasons for the popularity of semi-Lagrangian methods. Two fundamenta lissues in semi-Lagrangian methods are related to the trajectory computation and interpolation from the regular grid to departure points. If sufficiently accurate schemes are used to solve for trajectories with interpolations, one can expect good performance from the semi-Lagrangian scheme in solving the equations of motion of the atmosphere. Two general methods of solving the trajectory equation are the forward and backward methods. Most semi-Lagrangian methods use backward-trajectory schemes for estimating positions of the air parcels that arrive at the grid points in the future time step. Solving the trajectory equation is carried out by iteration. In the research reported, two iterations are used for trajectory computation. The fundamental difference between the forward and backward trajectory scheme rests in the calculation of advective quantity at the departure and destination points. While in the backward solution procedure, it is necessary to make interpolation from the regular grid to departure points; in the forward scheme, it is necessary to make interpolation from the irregular grid of destination points to the regular grid.
The usually used interpolation methods in the semi-Lagrangian method include piecewise cubic Lagrange and Hermite, cascades, and monotone Hermite. Increasing the degree of polynomial interpolation leads to a higher degree of formal accuracy, but it leads to the generation of unwanted oscillation in regions with severe gradients of the transported quantities. Eliminating the unwanted oscillations is done through a variety of methods which generally increase the computational cost and reduce the accuracy of the scheme. To address the issue, in this research, a new selective monotone semi-Lagrangian method is developed and tested along with two standard methods based on the Lagrange and Hermite interpolations.
The Lagrange polynomials have been considered by researchers for the high speed of computation in operational models. The fictitious oscillations produced at the edges of sharp gradients of the advected quantities are the main shortcoming of this method. The fictitious oscillations cannot be eliminated by increasing the degree of interpolation polynomials, which can only lead to a reduction in the wavelength of the oscillations. The results presented on increasing the degree of interpolation polynomials clearly show that the removal of the fictitious oscillations requires the use of monotone polynomials for interpolation. It is important to note that the Hermite interpolation polynomials are not inherently monotone. To make them monotone, one needs to manipulate the derivatives at the grid points appropriately. This process, however, may lead to a substantial deteriration of accuracy. For this reason, in this paper, a selective interpolation method is desined to obtain the best accuracy in solution of the advection equation, while preserving monotnonicity and removing the issue with the fictitious oscillations.
In the selective method, first the interpolation is done by the non-monotonic cubic Hermite and then a properly designed slope function is calculated at each grid interval. If the slope function takes values outside the range, it indicates that a fictitious oscillation has occurred in the interpolantion. To remove the oscillation, the non-monotone interpolation is abandoned and the monotone interpolation is performed by limiting the derivative to the monotone region. This technique can minimize the error caused by the changes in the derivatives. Results are shown to demonstrate the working and superiority of the seclective montone scheme.


Main Subjects

آزادی، م.، 1373، مدل­سازی معادلات هواشناختی به‌روش­های نیمه­لاگرانژی، کاربست به معادله تاوایی فشارورد. پایان­نامه کارشناسی ارشد هواشناسی، موسسه ژئوفیزیک دانشگاه تهران.
اصفهانیان، و. و اشرفی، خ.، 1382، اعمال روش نیمه لاگرانژی–نیمه ضمنی برای حل معادلات آب کم‌عمق. نشریه دانشکده فنی، 37، (3).
محب‌الحجه، ع. ر. و مشایخی، ر.، 1383، نمایش شارش‌های تاواری و امواج گرانی در الگوریتم­های حل عددی معادلات بسیط فشارورد منطقه­ای، مجله فیزیک زمین و فضا، 30 (1)، 37-47.
محمدی، ع.، محب‌الحجه، ع. ر. و مزرعه فراهانی، م.، 1397، چندجمله‌ای درون‌یاب هرمیت درجه سوم یکنوا و کاربرد آن در تبدیل مختصات برای مدل­های پیش­بینی عددی وضع هوا، مجله انجمن ژئوفیزیک ایران، 12 (3)، 21-38.
Bermejo, R. and Staniforth, A., 1992, The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes. Monthly Weather Review, 120(11), 2622–2632.
Blossey, P. N. and Durran, D. R., 2008, Selective monotonicity preservation in scalar advection. Journal of Computational Physics, 227(10), 5160–5183.
Boris, J. P. and Book, D. L., 1973, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69.
Courant, R., Friedrichs, K. and Lewy, H., 1928, Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32–74.
Denner, F. and van Wachem, B. G. M., 2015, TVD differencing on three-dimensional unstructured meshes with monotonicity-preserving correction of mesh skewness. Journal of Computational Physics, 298, 466–479.
Diamantakis, M., 2013, The semi-Lagrangian technique in atmospheric modelling: current status and future challenges. ECMWF Seminar in numerical methods for atmosphere and ocean modelling, pp. 183–200.
Dougherty, R. L., Edelman, A. S. and Hyman, J. M., 1989, Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation. Mathematics of Computation, 52(186), 471–494.
Dubey, R. K., 2013, Flux limited schemes: Their classification and accuracy based on total variation stability regions. Applied Mathematics and Computation, 224, 325–336.
Durran, D. R., 2010, Numerical Methods for Fluid Dynamics with Applications to Geophysics. Second Edition, Springer, 516 pp.
Fringer, O., Armfield, S. and Street, R., 2005, Reducing numerical diffusion in interfacial gravity wave simulations. International Journal for Numerical Methods in Fluids., 49(3), 301-329.
Fritsch, F. N. and Carlson, R. E., 1980, Monotone piecewise cubic interpolation. SIAM Journal on Numerical Analysis, 17(2), 238–246.
Germaine, E., Mydlarski, L. and Cortelezzi, L., 2013, 3DFLUX: A high-order fully three-dimensional flux integral solver for the scalar transport equation. Journal of Computational Physics, 240: 121–144.
Harris, L. M., Lauritzen, P. H. and Mittal, R., 2011, A flux-form version of the conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid. Journal of Computational Physics, 230(4), 1215–1237.
Harten, A., 1983, High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49(3), 357–393.
Hortal, M., 2004, Overview of the numerics of the ECMWF atmospheric forecast model. Published in the Proceedings of the ECMWF Seminar on “Recent developments in numerical methods for atmospheric and ocean modelling’’, pp. 6–10.
Hundsdorfer, W. and Trompert, R., 1994, Method of lines and direct discretization: a comparison for linear advection. Applied Numerical Mathematics, 13(6), 469–490.
Jiang, G. S. and Shu, C. W., 1996, Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1), 202–228.
Kadalbajoo, M. K. and Kumar, R., 2006, A high resolution total variation diminishing scheme for hyperbolic conservation law and related problems. Applied Mathematics and Computation, 175(2), 1556–1573.
Kahaner, D., Moler, C, and Nash, S., 1989, Numerical Methods and Software. Prentice-Hall, Inc., 495 pp.
Lax, P. and Wendroff, B., 1960, Systems of conservation laws. Communications on Pure and Applied Mathematics, 13(2), 217–237.
Lee, J. L., Bleck, R. and MacDonald, A. E., 2010, A multistep flux-corrected transport scheme. Journal of Computational Physics, 229(24), 9284–9298.
Machenhauer, B., Kaas, E. and Lauritzen, P. H., 2009, Finite-volume methods in meteorology. Handbook of Numerical Analysis, 14, 3–120.
Mehrenberger, M. and Violard, E., 2007, A Hermite type adaptive semi-Lagrangian scheme. Applied Mathematics and Computer Science,‏ 17, 329–334
Mohebalhojeh, A. R. and Dritschel, D. G., 2007, Assessing the numerical accuracy of complex spherical shallow-water flows. Monthly Weather Review, 135(11), 3876–3894.
Mohebalhojeh, A. R. and Dritschel, D. G., 2009, The diabatic contour-advective semi-Lagrangian algorithms for the spherical shallow water equations. Monthly Weather Review, 137(9), 2979–2994.
Nair, R. D., Scroggs, J. S. and Semazzi, F. H., 2003, A forward-trajectory global semi-Lagrangian transport scheme. Journal of Computational Physics, 190(1), 275–294.
Nair, R. D. and Lauritzen, P. H., 2010, A class of deformational flow test cases for linear transport problems on the sphere. Journal of Computational Physics, 229(23), 8868–8887.
Priestley, A., 1993, A quasi-conservative version of the semi-Lagrangian advection scheme. Monthly Weather Review, 121(2), 621–629.
Qian, J. H., Semazzi, F. H. and Scroggs, J. S., 1998, A global nonhydrostatic semi-Lagrangian atmospheric model with orography. Monthly Weather Review, 126(3), 747–771.
Rasch, P. J. and Williamson, D. L., 1990, On shape-preserving interpolation and semi-Lagrangian transport. SIAM Journal on Scientific and Statistical Computing, 11(4), 656–687.
Ritchie, H., 1987, Semi-Lagrangian advection on a Gaussian grid. Monthly Weather Review, 115(2), 608–619.
Robert, A., 1981, A stable numerical integration scheme for the primitive meteorological equations. Atmosphere–Ocean, 19(1), 35–46.
Robert, A., 1982, A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations. J. Meteor. Soc. Japan, 60(1), 319–325.
Rõõm, R., Männik, A. and Luhamaa, A., 2007, Non-hydrostatic semi-elastic hybrid-coordinate SISL extension of HIRLAM. Part I: numerical scheme. Tellus A, 59(5), 650–660.
Skamarock, W. C., Kelmp, J. B., Dudhia, J., Gill, D. O., Barker, D. M., Wang, W. and Powers, J. G., 2005, A description of the advanced research WRF version 2, DTIC Document, NCAR Technical Note NCAR/TN-468+STR.
Smolarkiewicz, P. K. and Pudykiewicz, J. A., 1992, A class of semi-Lagrangian approximation for fluids. J. Atmos. Sci., 49, 2082–2096.
Staniforth, A. and Côté, J., 1991, Semi-Lagrangian integration schemes for atmospheric models—A review. Monthly Weather Review, 119(9), 2206–2223.
Sweby, P. K., 1984, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 21(5), 995–1011.
Temperton, C. and Staniforth, A., 1987, An efficient two-time-level semi-Lagrangian semi-implicit integration scheme. Quarterly Journal of the Royal Meteorological Society, 113(477), 1025–1039.
Temperton, C., Hortal, M. and Simmons, A., 2001, A two-time-level semi-Lagrangian global spectral model. Quarterly Journal of the Royal Meteorological Society, 127(571), 111–127.
Verma, S., Xuan, Y. and Blanquart, G., 2014, An improved bounded semi-Lagrangian scheme for the turbulent transport of passive scalars. Journal of Computational Physics, 272, 1–22.
Waterson, N. P. and Deconinck, H., 2007, Design principles for bounded higher-order convection schemes––A unified approach. Journal of Computational Physics, 224(1), 182–207.
Williamson, D. L., and Olson, J. G., 1998, A comparison of semi-Lagrangian and Eulerian polar climate simulations. Monthly weather review, 126(4), 991–1000.
Wolberg, G. and Alfy, I., 2002, An energy-minimization framework for monotonic cubic spline interpolation. Journal of Computational and Applied Mathematics, 143, 145-188.
Zalesak, S. T., 1979, Fully multidimensional flux-corrected transport algorithms for fluids. Journal of Computational Physics, 31(3), 335–362.
Zhang, D., Jiang, C., Liang, D. and Cheng, L., 2015, A review on TVD schemes and a refined flux-limiter for steady-state calculations. Journal of Computational Physics, 302, 114–154.