Kees' assignment

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Contents

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

Solutions