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Discretization of advection terms in the momentum equations

We make a distinction between horizontal and vertical advection terms of the momentum equations. Moreover, we consider the momentum equations separately, i.e. the u -momentum equation and the w -momentum equation. Note that the v -momentum equation will be treated as the u -momentum equation in exactly the same way. So, we have four different commands:

Below, they will be outlined, respectively.


Horizontal advection terms of u -momentum equation


We consider terms like

u$\displaystyle {\frac{{\partial u}}{{\partial x}}}$    and    v$\displaystyle {\frac{{\partial u}}{{\partial y}}}$

There are many schemes to approximate these terms. Some of these schemes are accurate but are prone to generate wiggles - typically central schemes. Other schemes generate a certain amount of numerical diffusion and thus may affect the wave amplitude or wave energy of particularly short waves - typically upwind schemes. Higher order upwind schemes still generate small wiggles. If this is not desired, a flux-limiting scheme may be employed instead. Upwind schemes are known to be more stable than central schemes.


The default scheme for the considered terms is the well-known second order BDF scheme (or sometimes called the LUDS scheme). For many applications this is a good choice. However, in some cases central differences (CDS) are preferred. This is especially the case when the higher harmonics are involved or when wave breaking is present (the amount of dissipation of higher harmonics is then important). Note that when the command BREAK is employed, SWASH will automatically apply central differences to the horizontal advection terms. If, for some reason, SWASH becomes unstable, possibly due to the growth of wiggles, the user is then advised to use the BDF scheme.


Other higher upwind schemes (e.g. QUICK) may be used as well, but we did not experience much differences compared to the BDF scheme. In any case, never apply the first order upwind scheme to any horizontal advection term, which is usually too numerically diffusive.


Horizontal advection terms of w -momentum equation


The horizontal advection terms of the w -momentum equation are given by

u$\displaystyle {\frac{{\partial w}}{{\partial x}}}$ + v$\displaystyle {\frac{{\partial w}}{{\partial y}}}$

These terms are usually ignored. For some applications they are negligible small compared to the vertical pressure gradient. However, they will be automatically taken into account in the simulation for If they are included, then the second order BDF scheme will be employed. Sometimes, central differences may be preferred, for instance, when the higher harmonics are involved.


Vertical advection term of u -momentum equation


The vertical advection term of the u -momentum equation reads

w$\displaystyle {\frac{{\partial u}}{{\partial z}}}$

This term is only included in the computation when more than one layer is chosen (K > 1). The default scheme for this advection term is the first order upwind scheme. In most cases, it will not affect the accuracy and it is robust. Especially when there are not so many vertical layers (2 or 3), this scheme is preferable. If, for instance, many layers are involved or the higher harmonics are present, then central differences or a higher order upwind scheme might be a better choice.


Vertical advection term of w -momentum equation


This term is given by

w$\displaystyle {\frac{{\partial w}}{{\partial z}}}$

and is usually ignored (even when K > 1). It will be included automatically if This term is by default approximated with the first order upwind scheme. The user is advised not to alter this.


next up previous index
Next: Moving shorelines Up: Numerical parameters Previous: Momentum conservation   Index
The SWASH team 2017-04-06