ch²
11-20-2006, 08:36 AM
Hi,
I cannot figure out how to handle non rectangular domain/boundaries (I mean a not a simple rectangle for the domain) when solving pdes with Crank-Nicolson (or with other implicit methods). I've read the part about using the boundary values as additionnal source terms, but I don't know how to implement that in a clear/economic fashion when the boundary is not the including rectangle.
In my precise example I would like to solve a 2d or 3d Stokes equation for a incompressible fluid (-grad(P)_i+laplacian(v_i)=0 and laplacian(P)=0 for i=1,2,3) in a cross geometry that is :
00001--10000
00001--10000
11111--11111
----------------
11111--11111
00001--10000
00001--10000
Where (0) are outside o f the domain, (1) are the walls and (-) are the inner channels where is the flow. I impose pressure at the channel exits and entrances, plus normal gradient of pressure = 0 on the walls, and zero velocity on the walls.
With an explicit FTCS I get a solution (first solving laplacian P =0 with dP/dt=laplacian P, then integrating the velocity equation), but it's really slow of course.
To use an implicit method with the boundaries in the rhs, I've the feeling that I would have to track the values of the fields outside my domain of interest, which could be quite annoying because the interesting part is often quite smaller than the rectangle needed to enclose it.
Has anyone got some hints to handle these kind of boundaries?
I cannot figure out how to handle non rectangular domain/boundaries (I mean a not a simple rectangle for the domain) when solving pdes with Crank-Nicolson (or with other implicit methods). I've read the part about using the boundary values as additionnal source terms, but I don't know how to implement that in a clear/economic fashion when the boundary is not the including rectangle.
In my precise example I would like to solve a 2d or 3d Stokes equation for a incompressible fluid (-grad(P)_i+laplacian(v_i)=0 and laplacian(P)=0 for i=1,2,3) in a cross geometry that is :
00001--10000
00001--10000
11111--11111
----------------
11111--11111
00001--10000
00001--10000
Where (0) are outside o f the domain, (1) are the walls and (-) are the inner channels where is the flow. I impose pressure at the channel exits and entrances, plus normal gradient of pressure = 0 on the walls, and zero velocity on the walls.
With an explicit FTCS I get a solution (first solving laplacian P =0 with dP/dt=laplacian P, then integrating the velocity equation), but it's really slow of course.
To use an implicit method with the boundaries in the rhs, I've the feeling that I would have to track the values of the fields outside my domain of interest, which could be quite annoying because the interesting part is often quite smaller than the rectangle needed to enclose it.
Has anyone got some hints to handle these kind of boundaries?