upwind discretization second mixed derivatives

seb82 · October 8th, 2009, 8:27 am

I am working on the discretization of a PDE which involves second mixed derivatives. I found the standard finite difference discretization:[f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)]/4hkHowever, this is said to be instable. Therefore, I am looking for an upwind scheme similar to the ones used for the first derivatives which depend on the sign of the coefficient function. Can anyone help me with this matter or tell me where to find suitable schemes?I would really appreciate any sort of help!

Cuchulainn · October 8th, 2009, 12:42 pm

double

Cuchulainn · October 8th, 2009, 12:42 pm

Can you tell me which pde and which scheme you are using? Then it is possible to say why the scheme is unstable. e.g. Heston, ADI etc.

seb82 · October 8th, 2009, 1:37 pm

It is the Heston PDE with an implicit Euler discretization in time.

Cuchulainn · October 8th, 2009, 6:07 pm

QuoteOriginally posted by: seb82It is the Heston PDE with an implicit Euler discretization in time.Ah, Ok Much has been written on the mixed derivatiaves (dating from 1964) on the different ways to do it, but Janenko's solution is very robust. Roeleof Sheppard has done thesis on this subject using IE as well with extrapolation. You can find Janenko in his monograph, the thesis and the chapter on Heston in my book.See the pdf here in the threadGood luckhth