The Alternating Direction Explicit (ADE) Thread

[Cuchulainn]

This thread is for a discussion of your feedback, queries and developments concerning 'new' FDM methods, for example ADE.

[Cuchulainn]

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."

[Cuchulainn]

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?

[Cuchulainn]

Snippet (algo) in C# of ADE_BC

[Cuchulainn]

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

[Cuchulainn]

ADE for 3+1 dimensional PDE

[Cuchulainn]

The 3 variants of ADE (in our experience Barakhat Clark is the best).

[Cuchulainn]

Some results on Fichera for SABR

[cppljevans]

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?

[Cuchulainn]

The source code will be providesd in the forthcoming Boost II book by Demming and Duffy. HTH

[cppljevans]

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.

[cppljevans]

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

[cppljevans]

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.

[cppljevans]

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.

[cppljevans]

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?