Open Access

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

Communications in Applied and Industrial Mathematics's Cover Image
Communications in Applied and Industrial Mathematics
"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

Cite

1. G. Tumolo and L. Bonaventura, A semi-implicit, semi-Lagrangian discontinuous Galerkin framework for adaptive numerical weather prediction., Quarterly Journal of the Royal Meteorological Society, vol. 141, pp. 2582-2601, October 2015.Search in Google Scholar

2. J. Thuburn, Some conservation issues for the dynamical cores of NWP and climate models., Journal of Computational Physics, vol. 227, pp. 3715-3730, 2008.Search in Google Scholar

3. J. Thuburn, Conservation in Dynamical Cores: What, How and Why?, in Numerical Techniques for Global Atmospheric Models (P. Lauritzen, C. Jablonowski, M. Taylor, and R. Nair, eds.), vol. 80, pp. 345-356, Lecture Notes in Computational Science and Engineering, Springer, 2010.10.1007/978-3-642-11640-7_11Search in Google Scholar

4. M. A. Taylor, Conservation of Mass and Energy for the Moist Atmospheric Primitive Equations on Unstructured Grids, in Numerical Techniques for Global Atmospheric Models (P. Lauritzen, C. Jablonowski, M. Taylor, and R. Nair, eds.), vol. 80, pp. 357-380, Lecture Notes in Computational Science and Engineering, Springer, 2010.10.1007/978-3-642-11640-7_12Search in Google Scholar

5. L. Bonaventura, R. Redler, and R. Budich, Earth System Modelling 2: Algorithms, Code Infrastructure and Optimisation. New York: Springer Verlag, 2012.Search in Google Scholar

6. V. T. Chow, Open Channel Hydraulics. New York: McGraw-Hill, 1959.Search in Google Scholar

7. V. Casulli, Semi-implicit finite difference methods for the two dimensional shallow water equations., Journal of Computational Physics, vol. 86, pp. 56-74, 1990.10.1016/0021-9991(90)90091-ESearch in Google Scholar

8. V. Casulli and E. Cattani, Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water ow., Computational Mathematics and Applications, vol. 27, pp. 99-112, 1994.10.1016/0898-1221(94)90059-0Search in Google Scholar

9. T. Davies, M. Cullen, A. Malcolm, M. Mawson, A. Staniforth, A. White, and N. Wood, A new dynamical core for the Met Ofice's global and regional modelling of the atmosphere., Quarterly Journal of the Royal Meteorological Society, vol. 131, pp. 1759-1782, 2005.Search in Google Scholar

10. G. Tumolo, L. Bonaventura, and M. Restelli, A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations., Journal of Computational Physics, vol. 232, pp. 46-67, January 2013.10.1016/j.jcp.2012.06.006Search in Google Scholar

11. M. Restelli, L. Bonaventura, and R. Sacco, A semi-Lagrangian Discontinuous Galerkin method for scalar advection by incompressible flows., Journal of Computational Physics, vol. 216, pp. 195-215, 2006.10.1016/j.jcp.2005.11.030Search in Google Scholar

12. D. Williamson, J. Drake, J. Hack, R. Jacob, and P. Swarztrauber, A Standard Test Set for the Numerical Approximations to the Shallow Water Equations in Spherical Geometry., Journal of Computational Physics, vol. 102, pp. 211-224, 1992.10.1016/S0021-9991(05)80016-6Search in Google Scholar

13. R. Bank, W. Coughran, W. Fichtner, E. Grosse, D. Rose, and R. Smith, Transient Simulation of Silicon Devices and Circuits, IEEE Transactions on Electron Devices., vol. 32, pp. 1992-2007, 1985.Search in Google Scholar

14. M. Hosea and L. Shampine, Analysis and implementation of TR-BDF2., Applied Numerical Mathematics, vol. 20, pp. 21-37, 1996.10.1016/0168-9274(95)00115-8Search in Google Scholar

15. A. Staniforth and J. Côté, Semi-lagrangian integration schemes for atmospheric models-a review., Monthly Weather Review, vol. 119, pp. 2206-2223, 1991.Search in Google Scholar

16. K. W. Morton and E. Süli, Evolution-Galerkin methods and their supraconvergence.,Numerische Mathematik, vol. 71, pp. 331-355, 1995.10.1007/s002110050148Search in Google Scholar

17. K. W. Morton, On the analysis of finite volume methods for evolutionary problems., SIAM Journal of Numerical Analysis, vol. 35, pp. 2195-2222, 1998.Search in Google Scholar

18. F. Giraldo, The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids., Journal of Computational Physics, vol. 147, pp. 114-146, 1998.10.1006/jcph.1998.6078Search in Google Scholar

19. D. Xiu and G. E. Karniadakis, A semi-Lagrangian High-Order Method for Navier-Stokes Equations., Journal of Computational Physics, vol. 172, pp. 658-684, 2001.10.1006/jcph.2001.6847Search in Google Scholar

20. C. Zhao, B. Hobbes, H. Mühlhaus, and A. Ord, A consistent pointsearching algorithm for solution interpolation in unstructured meshesconsisting of 4-node bilinear quadrilateral elements., International Journal of Numerical Methods in Engineering, vol. 45, pp. 1509-1526, 1999.Search in Google Scholar

21. A. Allievi and R. Bermejo, A Generalized Particle Search-Locate Algorithm for Arbitrary Grids, Journal of Computational Physics, vol. 132, pp. 157-166, 1997.10.1006/jcph.1996.5604Search in Google Scholar

22. Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems., SIAM Journal on Scientific and Statistical Computing, vol. 7, pp. 856-869, 1986.10.1137/0907058Search in Google Scholar

23. B. Cockburn and C. Shu, The Runge-Kutta local projection P1 Discontinuous Galerkin method for scalar conservation laws., Mathematical Modelling and Numerical Analysis, vol. 25, pp. 337-361, 1991.10.1051/m2an/1991250303371Search in Google Scholar

24. G. Karniadakis and S. Sherwin, Spectral hp-Element Methods for Computational Fluid Dynamics. Oxford University Press, 2005.10.1093/acprof:oso/9780198528692.001.0001Search in Google Scholar

25. D. Le Roux and G. Carey, Stability-dispersion analysis of the discontinuous Galerkin linearized shallow-water system., International Journal of Numerical Methods in Fluids, vol. 48, pp. 325-347, 2005.10.1002/fld.893Search in Google Scholar

26. D. Le Roux, Spurious inertial oscillations in shallow-water models., Journal of Computational Physics, vol. 231, pp. 7959-7987, 2013.Search in Google Scholar

27. Y. Hasbani, E.Livne, and M.Bercovier, Finite elements and characteristics applied to advection-diffusion equations., Computers and Fluids, vol. 11, pp. 71-83, 1982.10.1016/0045-7930(83)90002-6Search in Google Scholar

28. K. W. Morton, A. Priestley, and E. Süli, Stability of the Lagrange-Galerkin scheme with inexact integration., RAIRO Modellisation Matemathique et Analyse Numerique, vol. 22, pp. 625-653, 1988.10.1051/m2an/1988220406251Search in Google Scholar

29. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for numerical solution of the compressible Navier-Stokes equations., Journal of Computational Physics, vol. 131, pp. 267-279, 1997.10.1006/jcph.1996.5572Search in Google Scholar

30. B. Cockburn, S. Hou, and C. Shu, The Runge-Kutta Local Projection Galerkin Finite Element Method for conservation laws IV: the multidimensional case., Mathematics of Computation, vol. 54 (190), pp. 545- 581, 1990.10.1090/S0025-5718-1990-1010597-0Search in Google Scholar

31. B. Cockburn and C. Shu, The Runge-Kutta Discontinuous Galerkin method for conservation laws, V., Journal of Computational Physics, vol. 141, pp. 198-224, 1998.10.1006/jcph.1998.5892Search in Google Scholar

32. M. Restelli and F. Giraldo, A conservative Discontinuous Galerkin semiimplicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling., SIAM Journal of Scientific Computing, vol. 31, 10.1137/070708470Search in Google Scholar

33. C. Temperton and A. Staniforth, An efficient two-time-level semi-Lagrangian semi-implicit integration scheme., Quarterly Journal of the Royal Meteorological Society, vol. 113, pp. 1025-1039, 1987.Search in Google Scholar

34. A. Priestley, Exact Projections and the Lagrange-Galerkin Method: A Realistic Alternative to Quadrature., Journal of Computational Physics, vol. 112, pp. 316-333, 1994.10.1006/jcph.1994.1104Search in Google Scholar

35. C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations. Philadelphia: SIAM, 1995.10.1137/1.9781611970944Search in Google Scholar

36. H. Weller, Shallow Water Test Case on a Beta Plane with Dynamic Adaptivity., http://www.met.reading.ac.uk/sws02hs/inprogress/newtonadaptivitytests.pdf, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK., October 2012. Multiscale Numerics for the Atmosphere and Ocean program.Search in Google Scholar

37. M. Läuter, D. Handorf, and K. Dethloff, Unsteady analytical solutions of the spherical shallow water equations., Journal of Computational Physics, vol. 210, pp. 535-553, 2005. pp. 2231-2257, 2009.Search in Google Scholar

eISSN:
2038-0909
Language:
English
Publication timeframe:
Volume Open
Journal Subjects:
Mathematics, Numerical and Computational Mathematics, Applied Mathematics