Partial Differential Equations - Science topic

usually Floquet is the periodic boundary in solid mechanics module, but I need to use my equation in PDE module to construct the material, however, in pde, there is only periodic boundry but no Floquet, or, is it possible to set the material maually in local variables and put it in equations in the solid mechanics?

thanks for your kind help!

Reduction of one PDE equation by Lie symmetry method and other method is always give the same result ?

I first study finite element method and dealing with matlab program, I can solving PDE by linear trianglution mesh but I can't by quadratic or higher degree for triangulation. My question is: How I can make a matlab code for solving a PDE by Quadratic finite element method ?

We are certainly grateful for your recommendations. Kind regards! ------------------------------------------------------------------------------

Hello everyone,

Some discretizations of pressure-displacement formulation in Biot poroelasticity produce spurious pressure oscillations in some cases, for example, for small time step sizes. If I understood correctly, FVM for flow and P1-FEM for mechanics fall in this category. How does one rigorously prove existence of such instability? Is it possible to obtain any estimates on time step size for instability to occur? I guess this analysis can be carried out in FEM framework in some way.

Thanks in advance!

i have attached some problem i m trying to do with Mathematica HAM for this kind equations

Hello everyone,

I want to solve a elastic wave problem in Comsol using PDE module since my model have a different consititutive relation which is not included in the present modules.

I have difficulty when using the " Coefficient type partial differential equations" because I can't find "Frequency Domain" in research module. "Transient" "Eigenvalue" and "Steady" are available.

How can I calculate a frequancy domain problem with PDE in Comsol?

By the way, if "PML" can be realized in PDE?

I would be really appreciated it if you can help me find out.

Best Regards,

Hao Qiu.

A trace of Ciprofloxacin was measured in the product. Need to set the PDE to justify the measured amount is low in the product.

Is it mandatory to calculate permitted daily exposure from human data as well as animal data and whichever PDE is low should be considered ?

Hello,

Would you have an idea on how to model stable regular networks in a phase field model ?

For instance a square network like in the joined picture (the picture was made artificially) ? I know that the Swift-Hohenberg equation allows to create some regularity but I would like to control more the size of the network and the distance between the branches, and maybe the type of network.

Thank you in advance :)

I want to model a photovoltaic panel that contains sublayers to obtain layers temperature changes over time. I have a bunch of formulas that discretize in terms of time and space. In other words, I want to solve a system of PDEs( each PDE is a discrete formula for each panel layer).

I would like to start a discussion of this specific topic.

Here I would like to discuss the list of possible techniques helpful for performing the simulation of oscillating bodies in quiescent fluid.

This discussion is open to all the students, teachers, and researchers.

I request you to reply here if you are familiar with code development in OpenFoam, IBPM, NEEK1000, lilypad and CFX

1. Bose-Einstein condensation: How do we rigorously prove the existence of Bose-Einstein condensates for general interacting systems? (Schlein, Benjamin. "Graduate Seminar on Partial Differential Equations in the Sciences – Energy, and Dynamics of Boson Systems". Hausdorff Center for Mathematics. Retrieved 23 April 2012.)

2. Scharnhorst effect: Can light signals travel slightly faster than c between two closely spaced conducting plates, exploiting the Casimir effect?(Barton, G.; Scharnhorst, K. (1993). "QED between parallel mirrors: light signals faster than c, or amplified by the vacuum". Journal of Physics A. 26 (8): 2037.)

I am trying to solve a time-dependent diffusion equation with finite-difference discretizations using the Newton–Raphson method. However, I encountered some problems in convergence--the solution result changes when changing time step size. Here I want to find some coding examples to refer to. Thank you.

I am trying to solve the differential equation. I was able to solve it when the function P is constant and independent of r and z. But I am not able to solve it further when P is a function of r and z or function of r only (IMAGE 1).

Any general solution for IMAGE 2?

Kindly help me with this. Thanks

Consider the powerful central role of differential equations in physics and applied mathematics.

In the theory of ordinary differential equations and in dynamical systems we generally consider smooth or C^k class solutions. In partial differential equations we consider far more general solutions, involving distributions and Sobolev spaces.

I was wondering, what are the best examples or arguments that show that restriction to the analytic case is insufficient ?

What if we only consider ODEs with analytic coeficients and only consider analytic solutions. And likewise for PDEs. Here by "analytic" I mean real maps which can be extended to holomorphic ones. How would this affect the practical use of differential equations in physics and science in general ? Is there an example of a differential equation arising in physics (excluding quantum theory !) which only has C^k or smooth solutions and no analytic ones ?

It seems we could not even have an analytic version of the theory of distributions as there could be no test functions .There are no non-zero analytic functions with compact support.

Is Newtonian physics analytic ? Is Newton's law of gravitation only the first term in a Laurent expansion ? Can we add terms to obtain a better fit to experimental data without going relativistic ?

Maybe we can consider that the smooth category is used as a convenient approximation to the analytic category. The smooth category allows perfect locality. For instance, we can consider that a gravitational field dies off outside a finite radius.

Cosmologists usually consider space-time to be a manifold (although with possible "singularities"). Why a manifold rather than the adequate smooth analogue of an analytic space ?

Space = regular points, Matter and Energy = singular points ?

Suppose one has the following PDE in (x,y):

a(x,y)*(f_xx + f_yy) + b(x,y)*f_x = e(f), (1)

where f_x= partial f/partial x, etc, and

where a(x=1,y) = 0, so that along the line x=1, the PDE becomes

b(1,y)*f_x = e(f). (2)

Does condition (2) effectively serve as a boundary condition on the original PDE (1) to solve in the region x <= 1?

We use transformations to convert the PDE into ODE to find the exact solutions of the PDE. If we use the different definitions of derivative in the transformations, then is it possible to obtain a different ODE of the given same PDE?

I have the following Laplace's equation on rectangle with length a and width b (picture is attached):

ΔU(x,y)=0

Ux(0,y)=0    : Isolated boundary

Ux(a,y)=f(y) : Current source

U(x,b)=0      : Zero potential

The fourth boundary is quite complex :  Mixed boundary condition (isolated except in two points)

if x=a/3 or 2a/3

U(x,0)=0

else

Uy(x,0)=0

Is there an analytical way to solve this kind of mixed boundary problem? can someone point me to the right direction? I'm a bit lost

thanks in advance

Which numerical method is more effective for solving parabolic diffusion PDE in 2D [ with Dirichlet boundary conditions, initial condition at time t =0 is given]? The PDE may represent time dependent heat equation in two space variables, an IBVP.

Thanks!

I want to solve partial differential equations (PDEs), which contains both space (x) and time (t). What is the best software for this purpose?

I also want to know the most appropriate numerical algorithm for this so that I can write a program to solve PDEs.

All types of suggestions are highly appreciable.

Thank You.

Hello!

Could anyone provide and advice regarding if its better to use in your PDE calculation the doses expressed in mg/m2 or if its better to convert them on a mg/kg basis. I saw that is possible to convert the dose expressed in mg/m2 to mg/kg by dividing the Animal Dose by Km value (according to FDA guidance for industry) , is this a correct approach?

I was wondering this because firstly i took in consider to multiply in the PDE formula the animal dose (mg/m2) from preclinical studies with average body surface for humans (1.62m2) before applying the adjustments factor, and i dont know if this or the previous one is the correct approach.

In the Reproductive toxicity test，Maternal general toxicity NOAEL:0.3 mg/m2.Embryofetal developmental toxicity NOAEL:0.45 mg/m2.Can I only choose embryo fetal toxicity NOAEL to calculate PDE？as follows。

Many Monte Carlo methods to solve a given Partial Differential Equation (PDE) are built by sampling the PDE's Green's function. E.g., for heat diffusion, diffusion-convection-reaction type of equations, and so on, have algorithms that can be derived directly from the PDE (i.e., through Ito calculus or stochastic integral). On the other hand, for the Radiative Transfer Equation (RTE), there is an Integral representation. However, the argument for explaining Monte Carlo Radiative Transfer (MCRT) ALWAYS revolves around the physical interpretation.

I even found a review article [1] that states on page 16: "Unlike traditional approaches to RT problems, MCRT calculations do not attempt to solve the RTE directly."

Is there really NO relation (discovered yet) between MCRT and the RTE? or is it just that no one has ever proven this?. I understand the physical interpretation; it is just that having mathematical foundations would also help teach it in class. Can anyone help me by directing me to a reference that derives this?.

[1] Noebauer, U. M., & Sim, S. A. (2019). Monte Carlo radiative transfer. Living Reviews in Computational Astrophysics, 5(1), 1-103.

Hi

I wanna solve partial differential equation in terms of x and t (spatial and time), As I know one of the most useful way for solving pde is variable separation. well explained examples about mentioned way are wave equation, heat equation, diffusion....

wave equation is Utt=C^2 .Uxx

in other word; derivatives of displacement to time, equals to derivatives of displacement to spatial multiplied by constant or vice versa.

however my equation is not like that and derivatives are multiplied to each other.for example : Uxx=(1+Ux)*Utt

Im wondering how to solve this equation.

I will be thankful to hear any idea.

Hi everyone!

I want to optimize parameters (K, C, rho) in convection-diffusion equation for heat. I want to minimize the difference between the final simulation temperature and actual process temperature by changing the parameters in PDE equation. Please let me know about MATLAB tools and if there is any solved example etc.

Your guidance and opinion will be highly appreciated.

Thanks and Regards

I want to solve analytically a coupled 2nd order space-time problem, originated from an optimal control problem. One of the problems is forward, another is backward in time. For example, (i) $y_t-y_{xx}=u, y(x,0)=0, y(0,t)=0, y(1,t)=g(t)$ (ii) $-p_t-p_{xx}=y, p(x,T)=0,p(0,t)=0, p(1,t)=h(t)$ with the coupling condition $p(x,t)+c*u(x,t)=0$ in $(0,1)\times (0,T)$. I have tried separation of variables, but it is getting complicated, any suggestions?

Definitely, I ask myself whether it was useful to introduce fractional partial differential equations.

As based on fractional differential operators this class of equations also surfers from the same drawbacks as the one described in:

Fractional partial differential equations were introduced to model anomalous diffusion, i.e. phenomena that exhibits power law behaviours other than 1/2. But it was shown recently that these kinds of behaviours can also be obtained with classical partial differential equations with spatially variables coefficients:

Where it is difficult to propose a physical interpretation with the fractional partial differential equations, classical partial differential equations with spatially variables coefficients allow interpretations relating to the geometry of the systems studied.

What is your opinion ?

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I work on spectral method for solving ODE and PDE. If you have any equation to solve, send me the govening equation, the boundary conditions and the initial conditions and I can propose a matlab code to solve these equations.

Best regards.

Pleas see the attached file.

Hello.

Any idea to choose a suitable journal for a manuscript, whose abstract is given below? More specifically, a high-quality journal that makes a final decision within three months or so.

Thank you so much.

In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions. For instance, to produce such functions in the mixed Generalized Multiscale Finite Element Method (mixed GMsFEM), a number of Partial Differential Equations (PDEs) must be solved. Thus, it makes sense to replace PDE solvers with data-driven methods given their great capabilities and general acceptance in recent decades. The main purpose of the present study was to develop four distinct Convolutional Neural Network (CNN) models to predict four different multiscale basis functions for the mixed GMsFEM. These models were applied to 249,375 samples, with the permeability field as the only input. The statistical results indicate that the AMSGrad optimization algorithm with a coefficient of determination (R2) of 0.8434 - 0.9165 and Mean Squared Error (MSE) of 0.0078 - 0.0206 performs slightly better than Adam with an R2 of 0.8328 - 0.9049 and MSE of 0.0109 - 0.0261. Graphically, all models precisely follow the observed trend in each coarse block. This work could contribute to many domains, especially the determination of pressure, velocity, and saturation in the development of oil/gas fields. Looking at this work as an image (matrix)-to-image (matrix) regression problem, the constructed data-driven-based models may have applications beyond reservoir engineering, such as hydrogeology and rock mechanics.

I am new to MATLAB and want to solve the following equation (in image).

Here c1 and ck are functions of the variable z. v1, D1, vk, Dk, k, Kd and c_cmc are all constants.

Boundary conditions are-> c1(0)=0, c1(0.1)=0.025, ck(0)=0.125, ck(0.1)=0

How do I solve this equation? Please help.

Even with other values of constants, how do I solve such type of equations?

In science and engineering, one often solves a perturbed model representing; an ODE, a PDE, an algebraic equation...

Are we deliberately perturbing the equations and solve or what is the physical meaning?

It may also occur during Lab based practical experiments or what?

How does it affect the model or equation?

So many thanks!

PDE is

(1/r)d/dr(rdT/dr) = C*exp(E*(1-(1/T))

Boundary conditions are: At r = 0; dT/dr = 0 and at r = R0 T = Tw

where C, E, and Tw are constants.

In 2010, Dr. Khmelnik has found the suitable method of resolving of the Navier-Stokes equations and published his results in a book. In 2021, already the sixth edition of his book was released that is attached to this question for downloading. Here it is worce to mention that the Clay Mathematics Institute has included this problem of resolving of the Navier-Stokes equations in the list of seven important millennium problems. Why the Navier-Stokes equations are very important?

Dear RG members, I come across an extension of Green's function technique to nonlinear differential operators that can treat inhomogeneous terms present in the nonlinear ODE cases from Ref. and few following works. But I am unable to find any extensions to treat the nonlinear PDE cases with inhomogeneous terms using green's function techniques. Are they developed? And where I can find more information. Kindly suggest if known.

The following article also discusses a related issue: https://downloads.hindawi.com/journals/amp/2018/7179160.pdf

Sometimes to solve a PDE, the PDE is converted into an ODE of abstract form and then the solution of the abstract form is said to be the solution of the given PDE. What are the methods available in the literature to convert a PDE into an abstract form ODE?

There are many references in math and physics that describe how to write and solve the Diffusion Equation in different coordinate systems when the spatial dimension (d) of the problem is an integer, and ordinary diffusion is taking place (with no heavy tails).

If the spatial dimension of the region in which the diffusion is occurring is a non-integer,

and ordinary diffusion is taking place,

how do you write the Diffusion Equation for this type of problem,

and what are the methods for solving it for a specified initial condition and boundary conditions.

Any information about solving problems like this

or suggestions for references related to this topic will be greatly appreciated.

Either an elliptical plate solicits in biaxial traction by 4 concentric nodes, only the surface forces that the Matlab Pdetool software accepts in the boundary conditions window, how can I apply a concentric force.

Hello, I went through the Fluent Theory Guide and other Ansys learning resources, however I couldn't find the answers to the following issues.

Let's discuss the problem on the example of a general steady RANS k-omega case.

No slip Wall

U is set to zero, k is set to zero, omega is given by a specific equation.

1. What about pressure? Does Fluent use the zeroGradient condition for pressure on a wall? It's done by default in OpenFOAM.

2. NS momentum equations are 2nd order in space, so the boundary conditions should involve the normal derivative (e.g. dUn/dn=0), if I am not mistaken ?

3. Consequently, the turbulence transport equations are 2nd order in space as well, so should their respective derivatives be specified on a wall?

Velocity Inlet

U, k and omega are specified.

4. What about pressure? zeroGradient?

5. Refering to question 2,3 - should a Neumann BC be specified for U, k, omega?

Pressure outlet

Fluent Theory guide says that all variables (apart from pressure) are extrapolated from the interior. However, that does not sound right. Imagine a highly diffusive system, wherein turbulence (or any other scalar) is diffused upstream, faster than the downstream convection. In such a case, turbulence properties at the outlet would affect the solution in the domain.

6. As a consequence, what BC are imposed for U, k, omega? Mentzer suggests "intervals" which freestream turbulence properties should be taken from. https://turbmodels.larc.nasa.gov/sst.html

Symmetry

Fluent specifies zero convective/diffusive flux across the symmetry plane.

7. We should specify a value of k and omega at the symmetry anyway... right? It has to be done in OpenFOAM...

Thank you very much for your time and effort, feel free to make a comment on that.

Could someone provide the description/representation of nonlinear subdiffusion equation?

I also like to know the physical significance and the difference between diffusive and subdiffusive PDE.

Thanks in advance

Amitha Rao

I am looking for the Matlab code to solve PDE using Deep neural networks. If anyone could help me. Thanks in advance.

In Ansys dynamics explicit module, I have modeled a water tank for an earthquake analysis to observe sloshing behavior. I am interested in how Ansys solves the problem using conservation of mass, momentum, and Energy. I have used the eulerian body for water and the lagrangian domain for the water tank. My question Is it using the Finite Element Method (FEM) or Finite Volume Method (FVM) or Finite Difference Method (FDM)? to solve the PDE's

How to find the exact solution of this PDE?

give me roughly idea or method please.

thank you.

Question about the loss of hyperbolicity in nonlinear PDE: when complex eigenvalues appear, what is the effect on flow? I understand that we do not have general results on existence in this case, but is it only the mathematical tools that are lacking where can we show physical phenomena of instability?

VOF is generally used for interface reconstruction. However, VOF discretization scheme with a special interface treatment is used to reduce the numerical diffusion and maintain the interface sharpness. Does the VOF discretization scheme use to discretize PDEs?

I'm solving an eigenvalue problem in COMSOL. In the coefficient-form PDE interface, the coefficients of my partial differential equation contain eigenvalue terms (λ). However, if I enter "lambda" as the eigenvalue in the coefficient field as described in the COMSOL tutorial (https://doc.comsol.com/5.5/doc/com.comsol.help.comsol/comsol_ref_equationbased.23.011.html), the software will show it as an undefined variable. Then I can't solve the problem.

So, What is the expression for the eigenvalues in coefficient form PDE interface (eigenvalue study) in COMSOL? How can I enter the coefficients with eigenvalue terms (for example, as shown in the picture)? Thanks very much for your help.

I am using coefficient form PDE and the coefficients in the equation are in the matrix form(e.g. ea, da, c, etc). However, I don't konw how to enter a matrix in the coefficient form PDE interface. The matrixs are undefined variables that can't be used in this interface.

So how can I enter a matrix variable as a coefficient in the coefficient form PDE interface? Thanks for your help!

I need the basics for including energy loss in Hamilton formulation for Finite element analysis for vibration of viscoelastic materials. The papers I read use complex modulus to represent viscoelastic losses or convolution integrals. Can someone give me a link where the formulation starts from Hamilton's principle?

A lot of time i found different papers in which numerical methods execute to find the solution of given model these numerical methods apply with initial condition but in analytical methods we used wave transformation to convert PDE into ODE. Here my question is raised in any case there is numerical method in applied and wave transformation is not given and we want to apply the analytical method, analytical method need wave transformation so, how can we make a wave transformation?

I'm looking for a way to solve the diffusion equation with a non-linear source term, Equation 1 in the attached document, using the Crank-Nicolson method.

One of the first things we learned about PDEs, is how the cylindrical harmonics (e.g. the Bessel functions J, I, K, and Y ) can be used to solve cylindrical boundary conditions. However I have never seen a paper discussing elliptical boundary conditions.

1) Is there another function that can be used for elliptical boundary conditions (NOT an ellipsoid but a prism with an elliptical cross-section) ?

2) What is the relation of these elliptical harmonics to the cylindrical harmonics?

3) If the answer to the above is yes, how might I compute these efficiently in MATLAB?

4) Is there a corresponding analog to Graf's addition theorem within the open literature?

Best Regards and Thanks!

Hi there. Im writing a program in C to solve the viscous burgers equation in 1D using the finite volume method. I am currently using a structured uniform cartesian grid as my computational grid. However i want to implement adaptive mesh refinement in my program. Could anyone tell me the steps i should take in order to do this. Would i have to make i have to make an additional function like say Adapt_Mesh() as well as my Init_Mesh() function and call the Adapt_Mesh() function in the main time loop every time the mesh needs refinement and how could i go about determining if my mesh needs refinement or how would i go about this. Im relatively new to the realm of CFD hence im struggling a lot by thinking how to go implementing this. Could anyone help me. Id be willing to share my current code if anyone could point me in the right direction in regards with how to get started.

*EDIT: Also i have a current gird with 100 cells. If was to adapt the mesh would it refine the mesh so that NX is always 100 ie. by making areas of less density and areas of higher density so that the total points add up to 100 or does the adaptive mesh increase the number of total cells everytime the mesh is refined. This is something im unsure of.

If anyone is curiosus i have attached my program below just incase anyone wanted to take a look

If anyone could help me. It is my dissertation work. Thank you. I am looking for the matlab code to solve PDE using RBF.

(Note: Problem Solved)

I am trying to define the phase field equation in the picture, using Coefficient form (or maybe convection-diffusion) PDE in COMSOL. However, I have a problem defining the second line, as there is no equivalent for these two terms in the COMSOL pre-defined equation. How would you do that?

#COMSOL #PDE #Phase-Field #PF #phasefield

I am dealing with quasi-dynamic seismic cycle simulation, which involves 4 PDEs (or ODE). According to the Buckingham Pi theorem, there are many non-dimensional numbers in this system. How can I extract meaningful ones from such a PDE system? All examples in textbooks only consider one equation.

I solved a nonlinear PDE by linearization and using finite difference and spectral methods. the errors for simple u_exact= exp(-t)(-8x^2-8y^2+7/2) don't change when the meshes become finer, in other word the error for all spatial meshes (20, 40, 60,...) is equal approximately, but for a more complex example u_exact=exp(-t)sin(x)cos(y) the results are very good and order of convergence is correct(i.e. order=2).

can anyone pleas tell me the reason of this trouble that happens in test whit a simple u_exacts?

Would prefer a book for learners.

I am looking for an analytical solution of the 2nd order PDE as in the title. The problem would seem simple: I am referring to the case of the desorption of a volatile component from a solution layer in a closed vessel of finite geometry (p, T = normal).

I just try to find the time to establish a gas-liquid equilibrium. I mean vertical 1D model with no horizontal gradients. Unfortunately the literature research was ineffective. The only suggestion I found in a certain monograph does not make any sense. (Concentration disappears when time goes to infinity).

I'd be greateful for any suggestions.

First of all I should say that I work in probability and my knowledge about PDEs is quite small, so this question could make little sense, please let me know if something is not well stated.

While dealing with approximations methods for SDEs a I've noticed a particular connection between an SDE and a deterministic PDE of the form:

u_t+σ(t)⋅ u_x+x⋅u=b(t,x,u)

where b is a Lipschitz continuous function in u.

I've tried searching online for application of this kind of equation but unfortunately I wasn't able to find anything concrete. In the book by Moussiaux, Zaitsev and Polyanin "Handbook of first order PDEs" they discuss methods for solving this kind of equations but they don't provide examples of applications.

I suspect this could be connected somehow to the transport equations, but I am not entirely sure. Do you know of some references for applications of this particular equation?

Hi,

I build a model which includes 2 parts, each has a specific constant material property. For example, part/domain 1 has the absorptions coefficient a=50 and the other part/domain of the model will be a=100. Now I use the general coefficient form PDE and want to assign those absorptions coefficients in the PDE dialog. How can I do this?

Thank You, Chironjeet

UPD (29/08/2018): Is the RANS/LES way of turbulence modeling a converging process, i.e. will it hypothetically converge to a hypothetical big but single PDE system? Or will it inevitably disassemble into several non-overlapping PDE systems/approaches each of which is valid only for a certain turbulent regime? 

After more than 5 decades of active search, it seems that the hierarchy of moment equations for the Reynolds stress and higher moments derived from the Navier-Stokes equations does not have a universal closure that could be applied to more or less wide range of flows. Numerous closures have been proposed since then. All of them contain heuristic arguments which are usually changed from case to case. So, does it mean that our understanding of turbulence will never be complete? Or do people still believe that a “magic closure” exists which can explain everything?

hello

I need to solve a set of the nondimensional equation in comsol (PDE module). for this reason, I need to change the gradient operator(in z-direction I need to add coefficient).how can I do this?

thanks.

I want to find the analytical solution of a radial diffusion system (1D cylindrical coordinates). Here what is considered is pressure diffusion. The conditions are simple and similar to the following.

Initial P = P0

P(r=infinite) = P0

P (r= 0) = P1 when t > 0

I need the solution of P at any r_i and t.

The flow rate term is unknown.

Could anyone give any suggestion? Thank you.

Dear colleagues,

I would like to know how to implement the elasticity equation for an isotropic solid material ( elasticity equation , attached) using the PDE weak form interface in COMSOL Multiphysics. The PDE weak form requires three dependent variables [u, v, w] to represent the displacement vector U. Therefore three weak expressions must be defined in comsol instead of the general weak expression for the elasticity equation (weak form attached, where u is the displacement vector and w is the test function).

How can I replace the general weak form expression(weak form attached) into three weak expressions that would be implemented on COMSOL.

Many thanks.

Hello all,

I am looking for an method / algorithm/ or logic which can help to figure out numerically whether the function is differentiable at a given point.

To give a more clear perspective, let's say while solving a fluid flow problem using CFD, I obtain some scalar field along some line with graph similar to y = |x|, ( assume x axis to be the line along which scalar field is drawn and origin is grid point, say P)

So I know that at grid point P, the function is not differentiable. But how can I check it using numeric. I thought of using directional derivative but couldn't get along which direction to compare ( the line given in example is just for explaining).

Ideally when surrounded by 8 grid points , i may be differentiable along certain direction and may not be along other. Any suggestions?

Thanks

For example: Lu+0.5u=f(x,y)

or Lu-au=o (elliptic PDE).

Actually I am looking some PDE for which I should get a singular system after applying any finite difference scheme.

I want to solve PDE equation using Python. I have used codes of finite difference method for solving.

Could you suggest any solver Partial differential other than FiPy.

The problem is let's say if I finite-difference one out of the many equations then through clank Nicholson, we separate the (t+1) time step left-hand side and (t) right side, then there are some other terms which are yet to be calculated.

In short - In coupled partial equation we are going to need the values of another dependent variable at the same time as we are calculating the value of the required variable.

If the above text won't clearly explain the problem then refer attached photo it will clearly speak of problem.