2. Discretization and Algorithm

This chapter lays out the numerical schemes that are employed in the core MITgcm algorithm. Whenever possible links are made to actual program code in the MITgcm implementation. The chapter begins with a discussion of the temporal discretization used in MITgcm. This discussion is followed by sections that describe the spatial discretization. The schemes employed for momentum terms are described first, afterwards the schemes that apply to passive and dynamically active tracers are described.

• 2.1 Time-stepping
• 2.2 Pressure method with rigid-lid
• 2.3 Pressure method with implicit linear free-surface
• 2.4 Explicit time-stepping: Adams-Bashforth
• 2.5 Implicit time-stepping: backward method
• 2.6 Synchronous time-stepping: variables co-located in time
• 2.7 Staggered baroclinic time-stepping
• 2.8 Non-hydrostatic formulation
• 2.9 Variants on the Free Surface
  • 2.9.1 Crank-Nickelson barotropic time stepping
  • 2.9.2 Non-linear free-surface
  
  
• 2.10 Spatial discretization of the dynamical equations
  • 2.10.1 Notation
  • 2.10.2 The finite volume method: finite volumes versus finite difference
  • 2.10.3 C grid staggering of variables
  • 2.10.4 Grid initialization and data
  • 2.10.5 Horizontal grid
  • 2.10.6 Vertical grid
  • 2.10.7 Topography: partially filled cells
  
  
• 2.11 Continuity and horizontal pressure gradient terms
• 2.12 Hydrostatic balance
• 2.13 Flux-form momentum equations
  • 2.13.1 Advection of momentum
  • 2.13.2 Coriolis terms
  • 2.13.3 Curvature metric terms
  • 2.13.4 Non-hydrostatic metric terms
  • 2.13.5 Lateral dissipation
  • 2.13.6 Vertical dissipation
  • 2.13.7 Derivation of discrete energy conservation
  
  
• 2.14 Vector invariant momentum equations
  • 2.14.1 Relative vorticity
  • 2.14.2 Kinetic energy
  • 2.14.3 Coriolis terms
  • 2.14.4 Shear terms
  • 2.14.5 Gradient of Bernoulli function
  • 2.14.6 Horizontal dissipation
  • 2.14.7 Horizontal dissipation
  • 2.14.8 Vertical dissipation
  
  
• 2.15 Tracer equations
  • 2.15.1 Time-stepping of tracers: ABII
  
  
• 2.16 Linear advection schemes
  • 2.16.1 Centered second order advection-diffusion
  • 2.16.2 Third order upwind bias advection
  • 2.16.3 Centered fourth order advection
  • 2.16.4 First order upwind advection
  
  
• 2.17 Non-linear advection schemes
  • 2.17.1 Second order flux limiters
  • 2.17.2 Third order direct space time
  • 2.17.3 Third order direct space time with flux limiting
  • 2.17.4 Multi-dimensional advection
  
  
• 2.18 Comparison of advection schemes
• 2.19 Shapiro Filter

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