PDE U_t = aU_xy (mixed derivatives)

[RedBranchKnight]

I am trying to solve

(1) U_t = 2bU_xy (as part of U_t = aU_xx + 2bU_xy + cU_yy)

using centred finite difference method. When a > 0 everyhing is OK but when a < 0 I get some oscillation problems.

My questions are:

1. is there a pde theory for (1)?
2. what is the 'motivation' for (1) in terms of information flow to and from boundaries/
3. maybe my fdm schemes destroy ellipticity which is the reason why I get oscillations.
4. These oscillations seem to be related to the initial condition in some way but I have not figured it out

thanks for any help!
RBK

 

[gtoguy97]

1. Yes, the PDE theory for (1) is called parabolic partial differential equations. 2. The motivation for (1) is that it is a two-dimensional diffusion equation which describes the flow of information or energy from one point to another over time. The 2bU_xy term in the equation represents the diffusion coefficient across the x and y directions. 3. It is possible that your FDM schemes are destroying the ellipticity of the equation, which can lead to oscillations. To check this, you could examine the eigenvalues of the elliptic operator associated with the equation. 4. Oscillations due to the initial condition can be avoided by using a better numerical scheme such as implicit methods or high order explicit methods.