Solving Advection/Diffusion like equation on non-rectangular domain.

[Cyril Gadal]

Dear all,

I am trying to solve the following equation:

 d_phi/d_t + nabla . (phi*vec{u}) -  d(Fphi*(1-phi))/d_z = nabla . (D grad{phi}),

where:

- phi is a scalar, vec{u} and vec{F}  are vector fields
- D and F are coefficients that can be taken constant, but that will probably become functions of phi but also x and z.

The domain is bounded by two parabolas  (red and brown) :

 and the black lines represent streamlines of the advection field vec{u}.

The boundary conditions are non-flux through the boundaries.

Is ClawPack the right tool for this? I am especially interested in solving the equation from D = 0 (purely hyperbolic) to very large values of D.

Also, as the domain is not rectangular, I was planning on using a triangular mesh, but I could not find any mention of non-cartesian grids for ClawPack ?

Thanks a lot !

[Kyle Mandli]

Hi Cyril,

Clawpack can be used to solve advection-diffusion problems, usually by using operator splitting.  That being said, this is generally ok for small values of D but larger values you almost certainly would be better off using a finite difference approach like Crank-Nicholson.  On top of that, Clawpack only can do logically rectangular grids, not fully unstructured ones.  The domain you showed may be able to be mapped but it may produce a grid that is not ideal.

If you can get the domain to map, it may be worthwhile setting up the D=0 case in Clawpack and use a Crank-NIcholson or other, implicit method for the rest as a good comparison.

Kyle

On Jan 16, 2025 at 12:53 PM -0500, Cyril Gadal <cyril...@gmail.com>, wrote:

Dear all,

I am trying to solve the following equation:

 d_phi/d_t + nabla . (phi*vec{u}) -  d(Fphi*(1-phi))/d_z = nabla . (D grad{phi}),

where:

- phi is a scalar, vec{u} and vec{F}  are vector fields
- D and F are coefficients that can be taken constant, but that will probably become functions of phi but also x and z.

The domain is bounded by two parabolas  (red and brown) :

<domain.png>
 and the black lines represent streamlines of the advection field vec{u}.

The boundary conditions are non-flux through the boundaries.

Is ClawPack the right tool for this? I am especially interested in solving the equation from D = 0 (purely hyperbolic) to very large values of D.

Also, as the domain is not rectangular, I was planning on using a triangular mesh, but I could not find any mention of non-cartesian grids for ClawPack ?

Thanks a lot !

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