This

Article A practical difference scheme for Fokker-Planck equations

is a very classical approach, whereas this

Article A Semi-Lagrangian scheme for the Fokker-Planck equation

is a more recent proposal. Be careful, since the Fokker Planck equation is not hyperbolic, it would rather fall in the cathegory of 'advection-diffusion' equations, since it contains both first and second order terms in space.

HI

Try supportive series solution. If you working on reasonant domain.

Luca is right, the F-P equation is a linear second order PDE that can be classified of parabolic type for a non-vanishing matrix diffusion coefficient.

At very low value you can thing of this equation as perturbed hyperbolic equation (owing to the linear character, provided that the initial condition is regular, you will not get singularity in the solution).

There are no particular problems in solving numerically such a type of equation. A very common method of solution for this type of equation - we use in CFD - is the Crank-Nicolson time discretization for the diffusive terms and the Adams-Bashfort for the convective ones. Generally, this is done along a second order central discretization for the spatial derivative (you can decide if in the FD or FV form). You will get a good range of stability region, a quite fast method and for a linear equation the order of discerization is sufficient to get accurate solutions. But different time integrations will work easily.

Of course, other methods such as FEM or SM are suitable, too.

For details have a look to this book http://sgpwe.izt.uam.mx/files/users/uami/mlss/documentos/Morton.pdf

Dear Ali,

are you sure that your version of the Fokker-Planck equation is hyperbolic? I am asking because, in general, the FK equation can be hyperbolic or parabolic, depending on the contribution of drift and diffusion term.

If it is hyperbolic, i.e., the drift term dominates the diffusion term, then, keeping all else equal, a Finite-Volume method is a good choice.

Dear Dr. Bonaventura , Dr. Denaro and Dr Hesse , you are right, generally FPE is a parabolic PDE.

In fact I need a method that can be implemented in both case; small diffusion part and large diffusion part. in my works, I am utilizing PS method for solving FPE of second order SDE.

because I need to control the PDF of an SDE my method should be fast (in order to be able to apply the control online)

thus among different numerical solutions I choose Pseudo-spectral method, Because I read that it uses fewer nodes than Finite Element method and gives higher accuracy. in my case, I use N1 nodes along time and N2 nodes along SDE states. so N1*N2*N2 nodes is used for discretization. I choose LGL nodes for this purpose. (maximum nodes are N1=4,N2=40). as you know using PS method the solution of FPE transformed to the solution of the corresponding algebraic equation and with a matrix inversion method, the solution can be obtained.

but in my simulations, the negative values appear (it is not correct, because the solution of FPE is a pdf). I want to know if there is a remedy to this problem? should I switch to FE methods , and if I did, does the computation complexity increase?

Thank you professor Marwan Alqura

I apologize for not following up this question, I had missed your final comment on negativity of the numerical solution. This is something that can happen, unfortunately, because of either the time or space discretization. If the solution is very regular, however, starting from a positive initial datum and using reasonably small time steps one should be able to keep positivity, since violations of positivity are usually associated to large gradients in the solution. Furthermore, if parabolic terms are present, they tend to help in maintaining postivity. So one should check for implementation errors first. However, as commented before violations of positivity can arise and need not be the result of an implementation error. If no error were present, then you may consider switching to a more local approach, since guaranteeing positivity with pseudospectral methods is usually harder than doing it with more local FE or FV methods.

Dear Dr. Bonaventura and Dr. Maria Denaro

Can I use the method of lines for FPE? i.e discretizing the time in a way that could be solved by RK4.( reducing the solution to ODEs)

If it is possible, could you please give me a basic and simple reference for learning how to do that?

Yes, what you want is to use a semi-discrete formulation. That means, you discretize in space the PDE, this way reducing to an ODE that can be solved by any method. This is valid for parabolic and hyperbolic equations but not for elliptic equation.

A general references to the method of lines methods could be

Gustafsson, Bertil, Heinz-Otto Kreiss, and Joseph Oliger. Time dependent problems and difference methods. Vol. 24. John Wiley & Sons, 1995.

I am not sure whether FP equation is explicitly discussed there, but in general, as suggested in the previous comments, the idea is to discretize in space by some (reasonable) method and obtain in this way an ODE system. Unfortunately, what is reasonable (in terms of accuracy and computational cost) is very much problem dependent. Furthermore, the choices made for the spatial discretization have consequences on the choice of the time discretization, in the sense that they determine the stability constraints which have to be respected by some time discretization methods and other qualitative properties of the numerical solution (positivity, discrete maximum principle and such).

Dear Dr. Bonaventura Luca Bonaventura

in your answer you mentioned;

"violations of positivity are usually associated to large gradients in the solution"

It seems that if we have more meshes in the area with large gradient we can solve this problem, do you agree?

Is there any mesh generation method that can solve this problem? or because we do not now any information about the solution of the PDE before solving it, we do not know where we have large gradients in order to create more meshes?

Ali, the violiation of the monotonicity of the numerical solution is stated by the Godunov theorem. Only first order linear scheme can retain positive coefficients. That happens indipendently from the mesh size. The way to overcome such limitation is to adopt non-linear schemes (also for solving linear PDE), a hystorical approach that is at the basis of the limiters.

From a different perspective, when a physical diffusion is present, the equation is no longer hyperbolic but parabolic. For small values of the diffusion you can, however, encounter the same difficulties in preventing wiggles (that is violation of the monotonicity). But now you can exploit the effect of the grid size, indeed a refining of the mesh can be useful to get a monotone solution. This is at the basis of the cell Reynolds (or Peclet) number constraint.

@ Filippo Maria Denaro

Dear Dr. Filipo Maria Denaro

could you please introduce me some of the best non-linear schemes for solving advection-diffusion PDEs?

Have a look here

http://www.tevza.org/home/course/modelling-II_2016/books/Leveque%20-%20Finite%20Volume%20Methods%20for%20Hyperbolic%20Equations.pdf

Dear Filippo,

Numerical solving of Fokker-Planck equation in 3D becomes very expensive with conventional methods (like what you find in FVM of Randy Leveque book) and high dimensions (Just write the matrix with ADI you see that). One cutting edge very numerically efficient method for that target (and SPDEs and high dimensional deterministic PDEs in general) is Proper Generalized Decomposition (PGD).

Metthod is new and only one book on that (to the best of my knowledge) but many papers published on that.

See:

https://www.springer.com/us/book/9783319028644

In fundamental philosophy the method is extension of FEM based on the basic functions which came from the truncated previous solution of the system so they converge faster and numerically efficient.

I have some easy to understand Matlab codes on that if you want.

Hope it helps,

Kaveh

@ Ali Namadchian

If you are looking for FDM and FVM look at the book by:

Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection–diffusion-reaction equation. Springer Series in Computational Mathematics 33. Springer, Berlin

I prefer the old good book of Leveque more than his new book, it is readable and simple and give you the main synapses:

Numerical Methods for Conservation Laws, by Randall J. LeVeque, Lectures in Mathematics, ETH-Zurich Birkhauser-Verlag, Basel, 1990. ISBN 978-3-7643-2464-3 (Print), 978-3-0348-5116-9 (Online)

There are nice books on FEM solution of ADR (or ADE) equation as well like :

The Finite Element Method for Fluid Dynamics, BY Olgierd Zienkiewicz

Also see this notes:

http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf

Hope it helps,

Kaveh

Dear Dr. Zamani @ Kaveh Zamani

as I mentioned in my question I really need a very fast method for solving the FPE which maintain the positivity of the solution and conservation law, and it is really desirable for me to have positive solution even with very singular initial condition, such as delta Dirac initial condition.

you believe PGD outperforms Finite difference, Finite elements, Finite Volume and Spectral methods for FPE?

" really desirable for me to have positive solution even with very singular initial condition, such as delta Dirac initial condition "

This is a hard goal! what is your ultimate purpose of modeling, what physical system you are going to study? what is your computational power? only a PC or you have access to HPC facilities? you may over kill with very dense mesh and conventional 2nd order FVM, or local refinements. If you tell me more (dimensions, length of simulation and ...) I may be able to give you a better solution.

PGD is faster but I do not know if it can handles sharp discontinuity, also very loosely speaking your method must fall under the arch of FVM as FDM is not mass conservative also FEM does not preserve shape as good as FVM (this is loosely speaking and most conventional methods)

Dear Dr. Zamani @ Kaveh Zamani

I am not dealing with a specific physical system. I look at FPE as Kolmogorov forward equation that can characterize the pdf of the states of stochastic differential equation.

so I form my FPE based on some engineering systems SDE (for example SDE for continuous stirred-tank reactor )

I want to use the FPE as the model of the system in order to control the pdf of the state of the system.

(as you know in many stochastic optimal control schemes we just try to control the mean of the states of the system or some moments of the states but here with FPE we can control the pdf of the system)

as a practical scheme that can be implemented online (for example with receding horizon approach) at each instance of time I need to solve the system of optimality which contains the FPE as a constraint on it.

so I need a very fast and general method for solving FPE.

Dear Ali,

I am a geo-physical and Environmental Fluid Mechanic person, I do not have any clue about control

Dear Dr. Zamani

I just wanted to point out that I need a method that can solve FPE fast while preserving the positivity and conservation of the solution.

if you think PGD might help me, could you please introduce me a paper with very brief introduction to the PGD and send me your Matlab codes

Please see the resources I introduced to Filippo above.

Best of luck,

Kaveh

You cannot define PDE equations for non regular initial data simply because the derivatives do not exist. COnversely, FVM can handle singularity as they start from the weak form.

Thank you Dr. Filippo Maria Denaro Filippo Maria Denaro

I am not familiar with FVM.

if I used FVM, for FPE I should not be worried about psoitivity and conservation of the solution for even singular initial condition?

If FVM can solve my problem dealing with singular initial conditions, what about computational speed? is FVM faster than FD for a specific PDE such as FPE?

Conservation is ensured in FVM by the unicity of the numerical flux function but not the positivity. For that you have to consider TVD formulation.

Computational speed depends on the algorithm has nothing to do with FD or FV formulation