# Difference between revisions of "Solution of the Blatter-Pattyn model"

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− | + | One term has become five terms and each one of those is pretty scary looking on its own. Luckily, there is also a lot of symmetry here. Notice that if we wanted to design subroutines to discretize the terms on the RHS, we could re-use a lot of them by either applying them to the correct velocity component (either to the ''u'' OR the ''v'' discretization) or by passing the appropriate arguments (pass either the grid spacing in the ''x'' direction OR the ''y'' direction, where appropriate). There is still a lot of work here ... |

## Revision as of 08:26, 13 August 2009

The final form of the equations we'd like to solve is:

Again, note that for the *x* equation we've moved all the terms containing gradients in *v* to the right-hand side (RHS).

We've set it up this way in order to solve the equations using an **operator splitting** approach; for the *x* equation, we treat *v* as known (where we take the values of *v* from the previous iteration) and solve for *u*, and vice versa when we solve they *y* equation for *v*. The "splitting" refers to the fact that we are breaking the multi-dimensional divergence operation into multiple steps. Rather than solving one big matrix equation for *u* and *v* simultaneously we solve two smaller matrix equations in sequence with one of the unknowns treated as a known "source" term.

As with the 0-order model, we need to change from Cartesian to sigma coordinates. The first normal-stress term first term on the left-hand side becomes

where hatted values refer to the coordinate directions in sigma coordinates. Similarly, the first cross-stress term on the RHS is given by

One term has become five terms and each one of those is pretty scary looking on its own. Luckily, there is also a lot of symmetry here. Notice that if we wanted to design subroutines to discretize the terms on the RHS, we could re-use a lot of them by either applying them to the correct velocity component (either to the *u* OR the *v* discretization) or by passing the appropriate arguments (pass either the grid spacing in the *x* direction OR the *y* direction, where appropriate). There is still a lot of work here ...