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Hi,Just had a query on Douglas Rachford in 2 space dmiensions...I understand that the scheme is unconditionally stable for 3-D (no cross terms). Is it also stable under 2-D? Or do I have to resort to Peaceman Rachford for the 2-D case?Are there any benefits of using Douglas-Rachford over Peacman Rachford (other than the former can be extended to 3-D)? Thanks in Advance,

QuoteOriginally posted by: samHi,Just had a query on Douglas Rachford in 2 space dmiensions...I understand that the scheme is unconditionally stable for 3-D (no cross terms). Is it also stable under 2-D? Or do I have to resort to Peaceman Rachford for the 2-D case?Are there any benefits of using Douglas-Rachford over Peacman Rachford (other than the former can be extended to 3-D)? Thanks in Advance,What PDE are you investigating? the heat equation?I'll check but if I can remember offhand, the straightforward DR for 3d is unstable (because we have 3 legs in ADI and each leg is unstable so in 3d there is always one extra leg). That's why we need the PR scheme.

I thought for a parabolic pde like heat equation that PR and DR are stable in 2D with O(delta t^2 and delta x^2) and O(delta t + delta x^2) respectively but that PR cant be applied to 3D thus using DR for higher dimensions. I dont have a book in front of me but ill check and get back to you

QuoteOriginally posted by: desucofI thought for a parabolic pde like heat equation that PR and DR are stable in 2D with O(delta t^2 and delta x^2) and O(delta t + delta x^2) respectively but that PR cant be applied to 3D thus using DR for higher dimensions. I dont have a book in front of me but ill check and get back to youSam, DesI checked in Peaceman's Oil reservoir book1. In 3d PR is unstable if k > 0.5 * h^2 so the 3d version of PR from 2d is NOT stable2. DR was the first ADI in 3d that was unconditionally stableThis is for the heat equation. Use Von Neumann analysis.BTW the ADE (Alternating Direction Explicit) from Saul'yev 1964 is unconditionally stable and explicit for the heat equation. Thus no matrix needed at each time level. Something to ponder on. See Roache CFD 1998.Des BTW you are spot on (without using the book).

yes, ADE is meant to be very useful. I will definitely look at it. I'll look at that Roache CFD article you mentioned and go from there.

Dear Cuchulainn,I have the following request: When I looked at Shaw's presentation (http://www.occf.ox.ac.uk/slides/shaw_slides.pdf), I did try to reproduce the case of "Douglas with special alpha = 1/sqrt(20)" - over the Test Problem (96)-(100) /slide # 26/, but I failed. More precisely I tried with the following settings/parameters of the Douglas scheme: dx dtau Spatial levels Time levels 0.04 0.000357770876399966 101 28 0.0392156862745098 0.000343878197231802 103 29 0.0333333333333333 0.000248451997499976 121 40 , respectively searching the solution at tau = 0.0100176; 0.00997247 and 0.00993808. Do you make a suggestions relative to: 1. The settings/parameters of the Douglas scheme, so that I can produce the case of "Douglas special alpha" - (slide #27)? 2. The reasons of my failure. Thanks in advance. Kpacu.

kpacu,So, you are solving the heat equation (96)-(100) using the Douglas scheme (39)+(45), yes? and alpha = 1/sqrt(20) Quote respectively searching the solution at tau = 0.0100176; 0.00997247 and 0.00993808.This is very early on the calculations. What kind of output do you get? The number of time steps could be increased to as many, at least of the space steps, e.g. NT = 200(BTW is there a specific reason for using Douglas method?)

Cachulainn,I finded implementation error in my source-code. So I produce the wanted result.Yes the number of time-steps increased, but I can reduce that nuber with 22% , there is other theoretical - alpha with similar properties as of the alpha = 1/sqrt(20).However, I understand (I agree with you) and may by these 22% are not enough.Cachulainn, all your comments and suggestions are welcome.Best,kpacu.

Hi kpacu,Another very powerful approach (and it seems not to be well known) is to use fully implicit Euler with Richardson extrapolation. It's discussed on this forum. And I discuss them in my FDM book.D Discussion