Convergence study with SCHISM

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SCHISM is an implicit model using ELM. This means that large time step is not only allowed but also encouraged! In fact the numerical diffusion in ELM of SCHISM will increase when the CFL number is below ~0.4, which may lead to undesirable results. So estimate the CFL number in your application when doing grid generation Mesh_generation and start from a large time step (e.g., 100-400s step for barotropic applications; for baroclinic applications, the time step is constrained by the internal Courant number and so may need to be slightly smaller. For field-scale applications we typically use 100-200s step).

If you are doing a convergence study, you need to keep the CFL number fixed while reducing the time step (which means you have to reduce the grid spacing).

For a given grid, the errors changes with dt in a nonlinear manner, as shown in the plot below:

Fig. 1: Model errors as a function of time step, on a fixed grid

This is due to the use of ELM (Baptista 1987; Zhang and Baptista 2008). So beware of this behavior when you reduce the time step. If you have to reduce the time step for some reason (e.g., extremely fine grid and strong wet/dry), you

have to refine the grid to satisfy the condition CFL>0.4; see [1] for more info.

The 'peculiar' behavior is often the most misunderstood part of SCHISM model. No convergence is expected with dt-->0 when dx is fixed. An analogy is that in explicit models, no convergence is expected with dx-->0 when dt is fixed (due to stability condition). Both types of models converge (and are consistent) with dx,dt-->0 and dx/dt=constant. SCHISM is a consistent and convergent model.


References

  • Baptista, A.M. (1987) Solution of advection-dominated transport by Eulerian-Lgrangian Methods using the backwards method of characteristics. Ph.D. Dissertation, MIT, Cambridge.
  • Zhang, Y.-L. and Baptista, A.M. (2008) "SELFE: A semi-implicit Eulerian-Lagrangian finite-element model for cross-scale ocean circulation", Ocean Modelling, 21(3-4), 71-96.
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