MPO662 Course Material

• Here are my course notes for MPO662. You will also find the code and description of the Shallow Water Equations homework problem. There is matlab script that will read a single snapshot of a fortran unformatted binary file.
• Additional section added to chapter 3 concerning compact differencing .
• Additional lecture added concerning Numerical Dispersion for Shallow Water Equations , see also Dukowicz 1995 below.

2012 Homework Problems

1. Finite Difference Approximations
   The matlab script used to produce the finite difference approximation shown in class can be downloaded for inspiration.
   For those of you trying to brush up on Fortran sample code, and Makefile files are available to get you started.
2. Compact Differencing and FDA of Heat
3. Consistency and Stability of model PDEs
   
4. Time Stability, Dispersion and SWE
   the matlab scripts and the fortran codesare available.
5. Nonlinear SWE
   

2009 Homework Problems

2010 Homework Problems

2011 Homework Problems

Final Projects

• shinit.f90 Initiatlization for advection of a passive tracer in a Stommel Gyre.
• shinit.f90 Initiatlization for monopole vortex propagation.
• shinit.f90 Initiatlization for vortex merger.
• shinit.f90 Initiatlization for double gyre circulation.

Tutorials and Manuals

• Skeleton script for symbolic manipulation in matlab.
• make
• sfmakedepend is an extremely useful utility to sort out automatically the object files dependency order for fortran. To use it you need to:
  1. have perl installed,
  2. to download the sfmakedepend file,
  3. and to type "make depend" on the command line.
  Finally, we should all thank Kate Hedstrom, who wrote sfmakedepend, for making our life easier.

Programming

• Fortran Programming
  There are lots of resources available and the best way to learn a language is to practice it with a web browser pointed to a tutorial or documentation page. Here are some links to get you going.
  • You can visit the MSC321 web site for notes and sample code.
  • Walter S. Brainerd, Guide to Fortran 2003 Programming, Springer, 2009.
  • Ian D. Chivers and Jane Sleightholme, Introduction to Programming with Fortran, With Coverage of Fortran 90, 95, 2003, and 77 Springer, 2012.
• Parallel Programming
  • mpi.pdf

Reading Material

• Lax-Richtmeyer Equivalence Theorem
  1. Peter I. Lax and R. D. Richtmyer Survey of the Stability of Linear Finite Difference Equations Communications on pure and applied mathematics, 9 pp 267-293, 1956.

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• Compact Differencing Schemes
  1. Sanjiva K. Lele Compact finite difference schemes with spectral-like resolution Journal of Computational Physics 103, Issue 1, pp 16-42, 1992.
  2. Peter C. Chu and Chenwu Fan A Three-Point Combined Compact Difference Scheme Journal of Computational Physics, 140, pp370-399 (1998).

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• Advection Schemes
  1. B. P. Leonard, M. K. MacVean and A. P. Lock, The flux integral method for multidimensional convection and diffusion Applied Mathematical Modelling, 19, pp 333-342, (1995).
     ( Presentation of The UTOPIA scheme, but has a typo in formula)
  2. B. P. Leonard, Order of accuracy of QUICK and related convection-diffusion schemes Applied Mathematical Modelling, 19, pp 640-653, (1995).
     (Finite difference/Finite Volume schemes discussion)
  3. B. P. Leonard, A. P. Lock and M. K. MacVean Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes Monthly Weather Review, 124, pp 2588-2606, (1996).
  4. Shian-Jiann Lin and Richard B. Rood, Multidimensional flux-form semi-Lagrangian transport schemes Monthly Weather Review, 124, pp 2046-2070, (1996).
  5. Piotr K. Smolarkiewicz, A fully multidimensional positive definite advection transport algorithm with small implicit diffusion Journal of Computational Physics, 54, pp 325-362, 1984.
  6. Piotr K. Smolarkiewicz and Len G. Margolin MPDATA: A finite-difference solver for geophysical flows Journal of Computational Physics>, 140, pp 459-480, 1998.
  7. John K. Dukowicz and John R. Baumgardner, Incremental Remapping as a Transport/Advection Algorithm Journal of Computational Physics, 160, pp 318-335, 2000.

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• Weighted Essentially Non-Oscillatory Schemes (WENO)
  1. Chi-Wang Shu Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws ICASE report 97-65, 1997.
  2. Guang-Shan Jiang and Chi-Wang Shu, Efficient Implementation of Weighted ENO Schemes Journal of Computational Physics, 126, Issue 1, pp 202-228, 1996.
  3. G.A. Gerolymos, D. Sénéchala and I. Vallet Very-high-order weno schemes Journal of Computational Physics, 289, pp 8481-8524, 2009.

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• Flux Corrected Transport and Limiters
  1. Steven T. Zalesak Fully Multidimensional Flux-Corrected Transport Algorithms for Fluids Journal of Computational Physics, 31, pp 335-362 (1979).
  2. Peter K. Sweby High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM Journal on Numerical Analysis, 21, Issue 5, pp. 995-1011, (1984).

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• Finite Difference Staggering
  1. John K. Dukowicz, Mesh Effects for Rossby Waves Journal of Computational Physics, 119, pp 188-194, 1995.
  2. Eric Blayo Compact Finite DifferenceSchemes for Ocean Models: 1. Ocean Waves Journal of Computational Physics 164, Issue 2, pp 241-257 (2000).

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• Energy and (Potential) Enstrophy Conserving Schemes
  1. Akio Arakawa, 1997, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I Journal of Computational Physics, 135, pp 103-114.
  2. Akio Arakawa and Vivian R. Lamb, 1981, A potential enstrophy and energy conserving scheme for the shallow water equations. Monthly Weather Review, 109, pp 18-36.
  3. Akio Arakawa and Yueh-Jiuan G. Hsu, 1990, Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations. Monthly Weather Review, 118, pp 1960-1969.
  4. Robert Sadourny, 1975, The Dynamics of Finite-Difference Models of the Shallow-Water Equations Journal of the Atmospheric Sciences , 32, pp 680-689.
  5. Rick Salmon, 2004, Poisson-Bracket Approach to the Construction of Energy- and Potential-Enstrophy- Conserving Algorithms for the Shallow-Water Equations Journal of the Atmospheric Sciences , 61 , pp 2016-2036.
  6. Rick Salmon, 2005, A general method for conserving quantities related to potential vorticity in numerical models Nonlinearity, 18, pp 1-16.
  7. G.S. Ketefian, M.Z. Jacobson,(2008), A Mass, Energy, Vorticity, and Potential Enstrophy Conserving Lateral Fluid-Land Boundary Scheme for the Shallow Water Equations, Journal of Computational Physics , 228, Issue 1, pp 1-32.
  8. T.D. Ringler, J. Thuburn, J.B. Klemp, W.C. Skamarock, (2010), , A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids Journal of Computational Physics,

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• Open Boundary Conditions for the Shallow Water Equations
  1. E. Blayo and L. Debreu, 2005, Revisiting open boundary conditions from the point of view of characteristic variables, Ocean Modelling, 9, Issue 3, pp 231-252.
  2. J. Nycander and K. Döös, 2003, Open boundary conditions for barotropic waves Journal of Geophysical Research, 108, NO C5, 3168.

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• A History of Numerical Weather Forecasting
  1. Peter Lynch, 2008, The origins of computer weather prediction and climate modeling, Journal of Computational Physics, 227,pp 3431-3444.
  2. Peter Lynch, 2008, Margules' tendency equation and Richardson's Forecast, Weather, 58.
  3. Peter Lynch, 1999, Richardson's Marvellous Forecast.