ATM S 582 A Sp 21: Advanced Numerical Modeling Of Geophysical Flows

Course Outline

Differential-Difference Equations for the Scalar Transport

• Numerical Dissipation and Numerical Dispersion (pp. 100-110)
  • Dispersion of a spike (mp4)
  • 1st, 2nd, 4th order spike (mp4)
  • 1st, 2nd, 4th order sum of 7.5 and 10 Dx waves (mp4)
  • 1st, 2nd, 3rd order spike (mp4)
  • 1st, 2nd, 3rd order sum of 7.5 and 10 Dx waves (mp4)
• Figures
  • Frequency as a function of numerical resolution
  • Translation of a 2Dx wave mis-interpreted as damping
  • Phase speed as a function of numerical resolution
  • Group velocity as a function of numerical resolution
  • Modified equation for 1st -- 4th order differences
  • Scale dependence of numerical smoothers

Further Considerations in Finite Difference Approximations

• Phase speed as a function of numerical resolution for compact schemes (Fig. 3.10)
• Staggered meshes (pp. 153-157)
  • Phase speed on the staggered mesh as a function of numerical resolution
• Discrete dispersion relation for systems of equations (pp. 167-169)
  • Arakawa C-grid
  • Finite difference approxiation to linearized Boussinesq system
• Stability constraints in simulations of multi-dimensional waves (pp. 157-159)
  • Wavefronts oriented at 45 degrees
• Splitting into fractional steps (pp. 169-176)
  • Swirl test
  • Effect of Strang splitting (from Skamarock, 2006, MWR, p. 2243)
  • Optional background reading on the stability of systems (pp. 148-151)
    

Spectral and Pseudo-Spectral Models

• Basics of series expansion methods (pp. 281-293)
  • Piecewise linear finite element basis
  • Effects of poor resolution
  • Table of coefficients for an equivalent finite difference method
• Conservation and the Galerkin approximation (pp. 298-299)
• Aliasing error (pp. 178-179)
• Misrepresentation of a 4/3 dx wave
• Aliasing when computing a product
• Improving efficiency with the transform method (pp. 292-298)
  • Avoiding aliasing error
• The pseudo-spectral method (pp. 299-303)
  • Spectral, pseudo-spectral comparison for Burgers Eqn
• Spherical Harmonic Functions (pp. 303-308)
• Mercator map of spherical harmonic nodal lines
• Y_{6,8}
• Triangular and rhomboidal truncation
• Elimination of the Pole Problem (pp. 308-310)
• Finite-element method, linear elements (pp. 320-323)
  • Piecewise linear finite element
• Quadratic elements (pp. 327-336)
• Piecewise quadratic finite element
• QFEM, linear FEM and 4th-order FD for 8dx spike
• Phase speed comparison

The Cube Sphere

The Discontinuous Galerkin and Spectral Element Methods

• h-p methods
• Discontinuous Galerkin method (pp. 341-350)
  • Comparison of modal and nodal basis
  • Nodal mass matix for N=4
  • Nodal solutions: 2-wave problem
  • Comparison of h and p refinement
  • Modal solutions: 2-wave problem

Semi-Lagrangian Methods -- or maybe Boundary Conditions depending on student preferences.  The following is for semi-Lagrangian methods.

• The scalar advection equation (pp. 357-360)
  • Backward trajectory
  • CFL condition (pp. 98-100)
  • 250 hPa winds forecast for March 14, 2015; Todays loop
  • Cascade interpolation (pp. 362-365)
• Finite volume integrations with large time steps (pp. 369-372)
  • PPM with selective limiting, CFL=1 & 4, 50x50 mesh
• Forcing in the Lagrangian frame (pp. 372--377)
  • Stability regions
  • Test case
• Comparison with the method of characteristics (pp. 377-379)