I don't recognize that particular scaling. But the most useful scaling I know is translational invariance in x(t) = log S(t). This is level or multiplicative scaling in term of S(t).For example, consider bivariate stochastic volatility models where y(t) is the volatility.The transition density, in general, looks like p(x1, y1 | x0, y0; t).With level independence, it reduces to p(x1-x0, y1 | 0, y0; t). A brute force pde solver uses 2 spatial dimensions plus 1 time. With a Fourier transform on the x-dependence, youonly have to solve a pde with one spatial and one time dimension. This kind of reduction is a big helpas you noted. (Then you have to invert a transform, but that is just an integration).regards,

rogers-shibenhamou-duguetuse homgeneity to reduce dimensionality

Hi ,in general you can reduce the dimension by means of symmetry analysis.To do so, you have to determine the Lie algebra. For example, if you consider the heat equationyou can find the following 6D Lie algebra : With the following coefficients :you can determine an 1 parameter lie group, which can be used to reduce the dimensionality of your PDE.If you determine the invariants ( y and v(y) ) you get the following ODE :with the two independent solutions being the Whittaker functions. The invariants can be determined by means of the usual manner you are using for the method of characteristics. No wonder : you are reciving a linear, first order PDE, whose general solution can be determined like this. This will represent the 1D flow, you are searching for.Note that the above mapping C is a diffeomorphism.YOu can play with the Abramowitz Stegun book to convert one set of special functions into another. For example, the " similarity transform" ( Galilei boost ) is obtained with C4 = 1, all others being 0, the travelling wave solutions are determined with ( C1, C2 ) = (1, 1) all others are zero, etc.For a higher dimensional PDE you have to determine the Lie algebra. For 2+1 the corresponding jet space will be 10D ( if I count well ... ). Thus the Lie algebra is ook 10 dimensional. For this you may use e.g. the Desolve, avaliable at RWTH Aachen. ( This program is able to calculate the commutation table, which is a great help if you want to calculate the adjoint representation to determine a linear space I did above ) .

Anita,Interesting. I have always wondered about the following extension:How do you do this analysis for a convection-diffusion equation (forget about mixed derivatives)?U_t = aU_xx + bU_xAll the examples seem to be for the heat equation. And Olver does not put in boundary conditions as far as I can see.For the given example, my first shot would be separation of variables...

Hi!Indeed it is from Olver. according to me you should see it as follows : the boundary conditions make the symmetry deviate from the v_{4} in my notation. ( The fundamental solution of the heat equation has that solution. ) I am sure, others have already explored this fact : it seems to be so obvious, that even I could find it out.What I would do :a. I would calculate the Lie algebra for the convection dominated eqn. And of course, calculate its commutation table.b. and then begins the thinking ... I am not sure, is there any transform, which maps the convection-diffusion eqn into heat eqn ? But this is precisley the question which should turn out from the group analysis. ( If you can transform the KdV into heat equation, then you may be able to do this, too ... ) I am sure, that this question comes out from the paper I was requesting. During the next wilmott- " bijeenkomst" in A'dam we may discuss it in person, if you are still interested.( A personal remark. The papers of the numerical analysts on the FDM, splitting techniques, etc. seem to lack the insight. All right, it is nice to analyse a method, BUT in many cases these analyses do not make you really understand WHAT and WHY do work. And it is a pity, since with this info you could be constructing your other schemes much more quickly. )

Dear Mr. Sitmo,just come to me on Monday in the office, and I can share you some more ideas on this. ( Maybe you could help me with some implementation ? Or the other way around ? ) Note that on Monday I come from the airport, so I will be in the office @~ 1100.

This might be of use. It uses information on exact solutions to improve your grid.Tradable scheme

QuoteI am not sure, is there any transform, which maps the convection-diffusion eqn into heat eqn ? But this is precisley the question which should turn out from the group analysis. ( If you can transform the KdV into heat equation, then you may be able to do this, too ... ) I am sure, that this question comes out from the paper I was requesting. This is repeat of the question I asked, more or less... Might be possible sometimes but I have not seen any.Unfortunately, we are kind of back to square 1...QuoteThe papers of the numerical analysts on the FDM, splitting techniques, etc. seem to lack the insight. All right, it is nice to analyse a method, BUT in many cases these analyses do not make you really understand WHAT and WHY do work. And it is a pity, since with this info you could be constructing your other schemes much more quickly. ) Which articles have you been reading?FDM was used by Euler, Gauss to prove existence of PDEs long before Richardson used them in his factory.

QuoteOriginally posted by: ElmerTradable schemeIt's nice to see that the paper uses a variation of my exponentially fitted scheme to get the FDM. Not having a drift term in the PDE simplifies things indeed.Typo: It mentions Daniel Duffie, but it should be Daniel Duffy, never mind

I completely agree with your remark ( even more, than you think ) .Indeed, DD may have advocated the exp. smoothing scheme, which may work.( It is quasi " philosophical", and I may have read his book very superflously, but it seemed to be a kind of " descriptive approach " . I.e., it gave a solution, but did not provide deeper insight in the solution. This method MAY help you provide this insight. And why is it good ? Firstly : it may be useful to apply it for problems, where without this insight we could not make a progress. Secondly, it gives an intellectual exctiement. Maybe it is a kind of luxury. Other people buy a Merci CLS to satisfy their need for luxury, I keep on investigating these things But this remark of mine may be off-topic, since it adds nothng on Maths and the problem posed in the title of the topic. It may, however, add some extra on my personal viewpoint, but it is not that interesting at all.)

QuoteOriginally posted by: GogolaAnitaI completely agree with your remark ( even more, than you think ) .Indeed, DD may have advocated the exp. smoothing scheme, which may work.( It is quasi " philosophical", and I may have read his book very superflously, but it seemed to be a kind of " descriptive approach " . I.e., it gave a solution, but did not provide deeper insight in the solution. This method MAY help you provide this insight. And why is it good ? Firstly : it may be useful to apply it for problems, where without this insight we could not make a progress. Secondly, it gives an intellectual exctiement. Maybe it is a kind of luxury. Other people buy a Merci CLS to satisfy their need for luxury, I keep on investigating these things But this remark of mine may be off-topic, since it adds nothng on Maths and the problem posed in the title of the topic. It may, however, add some extra on my personal viewpoint, but it is not that interesting at all.)Ok, I see. It's a "how to" book I suppose. It is not a book on theoretical numerical analysis but this is also needed, but is outside the scope. The focus is on schemes that one can program in C++ without too much hassle.