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

From Interactive System for Ice sheet Simulation
Jump to: navigation, search
Line 12: Line 12:
 
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.
 
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.
  
... unsure on how much more detail to go into on this ...
+
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
  
  
- symmetry in the equations means that the same subroutines can be used for the discretization of both the ''u'' and ''v'' equations. For example,
+
<math>\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 }</math>
 +
 
 +
 
 +
where hatted values refer to the coordinate directions in sigma coordinates. Similarly, the first cross-stress term on the RHS is given by
 +
 
 +
 
 +
<math>\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 }}\,</math>

Revision as of 22:21, 12 August 2009

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 }}\,