wiki:u/johannjc/scratchpad24

Version 4 (modified by Jonathan, 8 years ago) ( diff )

Tighter coupling of source terms with Corner Transport Upwind Scheme

1D wave equation with 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

Method of Characteristics

If we operator split the source term from the hyperbolic term, and if the hyperbolic term is trivial

where is a constant and via the method of characteristics

which gives

and

and the integral for becomes

Source term contribution

For the source term contribution to the time averaged interface state, things are a little tricky. Imaging a constant source term

Then the contribution to and the time averaged value for would be

Now if we include the solution from characteristic tracing in our source term calculation, the first term just gives us the same spatial integration, however the second term is a double integral.

There are lots of ways to reduce this double integral into a single (or finite number) of source evaluations using any combination of midpoint rules or trapezoid rules. Ideally we would like to use our already calculated spatial integral which we can do by using the trapezoid rule on the outer integral and exchanging the source evaluation with the integral average. This gives

where

Summary of 1D

  1. Spatial reconstruction and averaging of solution over domain to get interface states
  2. Application of source term to interface states over
  3. Calculate Fluxes from interface states
  4. Update Cells using those fluxes
  5. Calculate source term using time averaged cell center
  6. Update final states using source term for

CTU

PPM details

In more general terms, we can write

and we can estimate

Now and we can also add and subtract so that

and try to use characteristic tracing with

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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