Hello everyone,
I am new to Deal.ii and I am having an issue implementing a problem, which is
an extension of step-21. To iterate through time I am solving the equation
(which is analogous to the saturation equation in step-21):
$\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = F(C, \mathbf{u}) +
\nabla^2 C$
given $\mathbf{u}$ and $C(0)$. For stability purposes, we solve this using the
method of characteristics i.e.
$\frac{\mathrm{d} C}{\mathrm{d} t} = F(C, \mathbf{u} ) + \nabla^2 C $
on the line
$\frac{\mathrm{d} \mathbf{x} }{\mathrm{d} t} = \mathbf{u}$
and discretising time (with timestep \delta t) gives:
$\frac{C^{j+1} - C^j}{\delta t} = F(C^{j+1}, \mathbf{u}^{j+1} ) + \nabla^2
C^{j+1} $
on
$\frac{ \mathbf{x}^{j+1} - \mathbf{x}^j }{\delta t} = \mathbf{u}^{j+1}$
So,
$\frac{C^{j+1}(\mathbf{x}^{j+1} ) - C^j ((\delta t) \mathbf{u}^{j+1} +
\mathbf{x}^{j+1}) }{\delta t} = F(C^{j+1}(\mathbf{x}^{j+1} ) ,
\mathbf{u}^{j+1}(\mathbf{x}^{j+1} ) ) + \nabla^2 C^{j+1} (\mathbf{x}^{j+1} ) $
My problem is that, at each timestep I need to be able to find out the previous
solution for C at the points on the domain $((\delta t) \mathbf{u}^{j+1} +
\mathbf{x}^{j+1})$ to give the solution C^{j+1}(\mathbf{x}^{j+1} )
Many thanks,
K L
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