| Author | Message | ||
     Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 45 Registered: 03-2004 |
In an earlier post you explained how to use artificial diffusion to smooth a value for display purposes - my post #22 Spatial Filtering on 7/31/06. In the "Help", though, you mention its use to stabilize solutions to nonlinear PDEs. Do you just stick eps*Div(Grad(something)) into the equations? If so, what do you do about boundary conditions? Could you use anisotropic1,pde to illustrate the basic technique? | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1121 Registered: 06-2003 |
There are several different cases in which this might be done: 1) If your equation already contains a diffusion term, then the smoothing mentioned in our "Smoothing Operators" Tech note reduces to nothing more than altering the diffusivity. This may or may not be desirable. If the diffusivity is widely variable, you may want to use a MAX() instead of an addition, merely to prevent pathologically small diffusivities. 2) In time-dependent equations that lack a diffusion term, you can replace dt(u) by dt(u)-div(eps*grad(u)) as described in our Tech note. 3) In steady-state equations that lack a diffusion term, it is best to look at the physics and see if there is some natural diffusive term that has been removed for "simplicity" and add it back in with a coefficient as discussed in our Tech Note. Fluid flow equations, for example, can be written with a viscous term. 4) If you are ambitious, you can form the Fourier transform of the full equation, and follow the procedures described in our Tech note to damp the high spatial-frequency components. In all cases, the addition of a diffusion term div(eps*grad(u)) contributes a boundary term normal(eps*grad(U)) to the Natural BC. Ignoring it implies a reflective boundary. Alternatively, the tautology Natural(U)=normal(eps*grad(u)) provides the boundary terms needed by the Divergence Theorem to allow arbitrary shapes. If you have a specific case in mind, send us a script and we will take a look at it. PS Our Tech Note "Smoothing Operators in PDE's" has font troubles. You need to install the "Lucida Bright Math Italic" font to see the report correctly. (LUCI000.TTF). You can download this font from www.pdesolutions.com/download/Luci000.ttf | ||
| Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 46 Registered: 03-2004 |
Thanks. Viscous? For a steady state problem, I suppose that means I would be adding a time-dependent term that would decay away. I'll put together a case to look at. Thanks also for correcting the reference to my earlier post. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1122 Registered: 06-2003 |
A typical statement of the Navier-Stokes momemtum equation for vector U is r(dt(U)+U•ÑU) = -Ñp + mÑ2U Here mÑ2U is the viscous dissipation term, and has the same form as the smoothing term of our Tech note. It is present even when dt(U) vanishes, so is a component of a steady-state solution and does not decay away. The steady-state form is then rU•ÑU = -Ñp + mÑ2U The viscous term is then a low-pass filter on spatial velocity oscillations. If you try to solve it without m, it may be unstable. Adding a fictitious m introduces an artificial viscosity. This is the analogy I meant to indicate in my remark (3). |