# Kees' assignment

## Model equation

$\frac{\partial H}{\partial t} = - \frac{\partial}{\partial x}\left(-D(x) \frac{\partial h}{\partial x}\right) + M$

where

$D(x) = C H^{n+2}\left|\frac{\partial h} {\partial x}\right| ^{n-1},$

and

$C = \frac{2 A}{n+2} \left(\rho g\right)^n$

## Model parameters

• $\frac{\partial b}{\partial x} = -0.1$
• $M(x) = M_0 - x M_1 = 4.0 - (0.2\times 10^{-3}) x$ km/yr
• $\rho$ = 920 $kg/m^3$
• g=9.8 $m/s^2$
• A = 1e-16 $Pa^{-3} a^{-1}$
• n=3
• dx=1.0 km

## Boundary conditions

• $H_l = 0$ (left boundary)
• $H_r>0$ (right boundary)

## Numerical tips

Use a staggered grid such that the $D(x_{j+1/2})$ are computed at the centers of the grid (as opposed to the vertices, as we have been doing), so

$D(x_{j+1/2}) = C \left(\frac{H_j + H_{j+1}}{2}\right)^{n+2} \left(\frac{h_{j+1} - h_j}{\Delta x}\right)^{n-1}.$

From the diffusivity, the flux is computed

$\phi_{i+1/2} = -D(x_{i+1/2}) \frac{\partial h}{\partial x}$,

where

$\frac{\partial h}{\partial x} = \frac{h_{i+1}-h_{i}}{\Delta x}$

and then the flux ($\phi_i$) can be used to compute the rate of change of the surface from

$\frac{\partial H}{\partial t} = -\frac{\partial }{\partial x} H\bar u + M = - \frac{\phi_{i+1/2} - \phi_{i-1/2} }{\Delta x} + M$