Version 2 (modified by 8 years ago) ( diff ) | ,
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Tighter coupling of source terms with Corner Transport Upwind Scheme
Consider wave equation with a source term
Exact integral finite volume solution to this problem is
Where
- is the volume average of over cell at time
- is the time and area averaged flux at the boundary between cell and cell
- is the time and volume averaged source term over cell
In Riemann methods, in 1D, we can write
where
is the time averaged area average of on the left side of the interface and is the Riemann solver forNow the trick is how to calculate
We can use the midpoint rule to estimate
Usually some form of spatial reconstruction is used to calculate
and then the evolution equation for q is used to approximate time derivatives using spatial derivatives.
which we then use the evolution equation to replace time derivatives with spatial ones
and then we can linearize the flux function about
and use the characteristics (eigenvectors of the Jacobian of the flux) to linearize the flux
and then we can use the
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