# Solution of the Blatter-Pattyn model

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

\begin{align} & x:\quad 4\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial x} \right)+\frac{\partial }{\partial y}\left( \eta \frac{\partial u}{\partial y} \right)+\frac{\partial }{\partial z}\left( \eta \frac{\partial u}{\partial z} \right)=-2\frac{\partial }{\partial x}\left( \eta \frac{\partial v}{\partial y} \right)-\frac{\partial }{\partial y}\left( \eta \frac{\partial v}{\partial x} \right)+\rho g\frac{\partial s}{\partial x} \\ & y:\quad 4\frac{\partial }{\partial y}\left( \eta \frac{\partial v}{\partial y} \right)+\frac{\partial }{\partial x}\left( \eta \frac{\partial v}{\partial x} \right)+\frac{\partial }{\partial z}\left( \eta \frac{\partial v}{\partial z} \right)=-2\frac{\partial }{\partial y}\left( \eta \frac{\partial u}{\partial x} \right)-\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial y} \right)+\rho g\frac{\partial s}{\partial y} \\ \end{align}

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

$\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial x} \right)=\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \hat{x}} \right)+\frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \hat{x}} \right)+\frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \sigma } \right)+\left( \frac{\partial \sigma }{\partial \hat{x}} \right)^{2}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \sigma } \right)+\left( \frac{\partial _{{}}^{2}\sigma }{\partial \hat{x}_{{}}^{2}} \right)\eta \frac{\partial u}{\partial \sigma }$

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

$\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial y} \right)=\underset{{}}{\mathop{\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \hat{y}} \right)}}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \hat{y}} \right)}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{y}}\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \sigma } \right)}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{x}}\frac{\partial \sigma }{\partial \hat{y}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \sigma } \right)}\,+\underset{{}}{\mathop \frac{\partial _{{}}^{2}\sigma }{\partial \hat{x}\partial \hat{y}}\eta \frac{\partial u}{\partial \sigma }}\,$