The Alternating Direction Explicit (ADE) Thread

The Alternating Direction Explicit (ADE) Thread

Postby Cuchulainn » Tue Mar 23, 2010 5:22 pm

This thread is for a discussion of your feedback, queries and developments concerning 'new' FDM methods, for example ADE.
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Postby Cuchulainn » Tue Apr 06, 2010 6:49 am

From SSRN

///



"Dear Daniel John Duffy:



Your paper, "Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems", was recently listed on SSRN's Top Ten download list for Derivatives eJournal and Econometrics eJournals. As of 04/05/2010, your paper has been downloaded 88 times."
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Postby Cuchulainn » Tue Apr 27, 2010 5:29 pm

The normal ADE method is for ODE of the form





dU/dt + AU = F



Sometimes the ODE becomes (e.g. from a FEM semi-discretisation)



MdU/dt + AU = F



where M is a (banded) matrix.



Q: how does ADE appear in this case?
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Postby Cuchulainn » Tue Feb 01, 2011 4:36 pm

Snippet (algo) in C# of ADE_BC
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ADE_Backup.cs
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Postby Cuchulainn » Wed Mar 02, 2011 10:37 pm

MSc Thesis Finance using a variation of ADE and other schemes



http://upetd.up.ac.za/thesis/available/ ... tation.pdf







The Larkin variation



http://www.ams.org/journals/mcom/1964-1 ... 4450-X.pdf
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Postby Cuchulainn » Fri Apr 08, 2011 11:58 am

ADE for 3+1 dimensional PDE
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PDEADE.pdf
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Postby Cuchulainn » Wed May 18, 2011 8:48 pm

The 3 variants of ADE (in our experience Barakhat Clark is the best).
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Postby Cuchulainn » Sat Jun 18, 2011 12:55 pm

Some results on Fichera for SABR
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source code location?

Postby cppljevans » Fri Jul 01, 2011 5:48 pm

Cuchulainn wrote:ADE for 3+1 dimensional PDE




Could you make the source code in the article available?

I'd like to try using the array_recur in the attachment to:



http://lists.boost.org/Archives/boost/2 ... 183118.php



which, IIUC, would be applicable if NX, NY, and NZ were

compile-time constants. IOW, instead of:



multi_array<double, Dim>



you could use:



array_recur<double, NX, NY, NZ>



Also, since V and U have the same shape, you could do:



enum{U_ndx, V_ndx};

array_recur<double, V_ndx+1, NX,NY,NZ>



and since U and UOld have the same shape, you could do:



array_recur<double, 2, V_ndx+1,NX,NY,NZ> uv;



and instead of copying UOld to U, you could just

change the outer most indices:



old_ndx=0;

new_ndx=old_ndx+1;



while(/*not converted*/)

{

//calculate uv[new_ndx,...] from uv[old_ndx,...];

...

old_ndx = (old_ndx+1)%2;

new_ndx = (new_ndx+1)%2;

};



Does that make sense?
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Postby Cuchulainn » Tue Jul 12, 2011 2:01 pm

The source code will be providesd in the forthcoming Boost II book by Demming and Duffy. HTH
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Postby cppljevans » Sun Aug 14, 2011 7:15 pm

Cuchulainn wrote:ADE for 3+1 dimensional PDE


On page 8 of the pdeade.pdf file, there's the ExactSolution

function which takes the x,y,z space coordinates and the

time value, t; however, the time value is ignored in return

expression:



exp(-x*x) * exp(-y*y) * exp(-z*z)



was that intentional? I would guess that the returned value

is time dependent, unless I'm missing something.
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eact solution is fundamental solution

Postby cppljevans » Wed Sep 21, 2011 6:00 pm

As mentioned previously, the pdeade.pdf file contains, in the code

for ExactSolution:



exp(-x*x) * exp(-y*y) * exp(-z*z)



however, the expression here:



http://en.wikipedia.org/wiki/Heat_equat ... _solutions



contains the following:



phi(x,t)=exp(-x^2/4*k*t)/sqrt(4*Pi*k*t)



Using http://linux.die.net/man/1/ginsh to check this showed it

satisfies the differential equation:



u_t(x,t) -k*u_x_x(x,t)=0



from the Heat_equation#Fundamental_solutions page.

I'd guess it could be extended to n dimensions by

simply summing the squares of the space values. IOW,

instead of -x^2, there would be -(x^2+y^2+z^2) for

the case when n=3.



Was that the intended expression in ExactSolution?



TIA.



-Larry
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wikipedia fundamental soln wrong?

Postby cppljevans » Wed Sep 21, 2011 8:56 pm

When tested with ginsh, the wikipedia solution fails to solve the

equation. Instead, the argument of the sqrt function should be

just t instead of 4*Pi*k*t.



Attached in the gensh script used to test this and the resulting

output. The last line shows 1, indicating the ratio of the

time derivative and the 2nd space derivative is the same.
Attachments
diff_eq_fundamental.txt
output of the ginsh script. Trailing 1 indicates
ration of time derivative and 2nd space
derivative are indentical.
(612 Bytes) Downloaded 1102 times
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ginsh script

Postby cppljevans » Wed Sep 21, 2011 9:01 pm

For some strange reason, this forum will not allow attaching 2 files

in same reply. This message is just to provide the ginsh script attachment

missing from my last post.
Attachments
diff_eq_fundamental_sh.txt
ginsh demonstrating exact solution.
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Postby cppljevans » Thu Sep 22, 2011 7:35 pm

Running the python script shown here:



http://en.wikipedia.org/wiki/Talk:Heat_ ... _just_t.3F



Indicates that ginsh simply could not simplify the expression with

4*Pi*k*t but that expression also satisfies the heat equation.



Would someone care to explain why both expressions solve the

same equation?
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