Solution of the Blatter-Pattyn model
The final form of the equations we'd like to solve is:
For ice sheet modeling, it is convenient to recast the governing equations using a dimensionless, stretched vertical coordinate (often called a sigma coordinates). The stretched vertical coordinate is defined as:
This means that at the surface of the ice sheet , and at the base regardless of the ice thickness (Pattyn 2003). As a result of this transformation, a coordinate is mapped to (Pattyn 2003). This means that function derivatives must be re-written (using as an example) as:
Similarly for and . Pattyn simplifies this by assuming that
This assumption is valid if the bed and surface gradients are not too large (Pattyn 2003). This simplifies the above to:
Rescaling parameters , , , , and are defined. Presenting the x derivative case, as the y derivative case is analogous,
Using these, expressions for the x derivatives become:
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 ...
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.
General Matrix Form
A general matrix form of the equations, where coefficients on the u and v velocity components (i.e. viscosity, grid spacing, scalars) are contained in the block matrices A, is given by
where the uu subscript denotes block matrices containing coefficients for gradients on u in the equation for the x component of velocity (i.e. u). The subscript uv denotes block matrices containing coefficients for gradients on v in the equation for the x component of velocity (and similarly for the vv and vu subscripts). On the right-hand side, the subscripts denote the geometric source terms for the x and y components of velocity (subscripts u and v, respectively).
Solution of the Non-linear System Through a Fixed Point Iteration
The non-linearity in the equations - the fact that the coefficients on the velocity components (the viscosity) are dependent on the velocity (or more specifically, the velocity gradients) - is handled through a fixed-point iteration.
Go to Strang for this ..
Final Matrix Form
The final matrix form of the equations, accounting for the Picard iteration on the viscosity, is given by
where the index k denotes an unknown value being solved for during the current non-linear iteration and the index k-1 denotes a lagged value taken from solution at the end of the previous non-linear iteration.
Solution of the Linear System
Newton-based Methods for Solutions of the Non-linear System
Go to Blatter-Pattyn model.