wiki:u/johannjc/scratchpad24

Version 3 (modified by Jonathan, 8 years ago) ( diff )

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 for

Now 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

Now if we ignore the source term, we can directly solve for

and under the change of variables we can rewrite the integrals as

which just gives us

where

and use the characteristics (eigenvectors of the Jacobian of the flux) to linearize the flux

and then we can use the solution for the characteristics to turn the time integral into a spatial integral

which we can do a variable substitution

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