Finite Difference - Science topic

Here’s a basic for your MATLAB code:(m-file)

% Parameters

L = 1; % Length of the domain

N = 200; % Number of spatial points (increase for better resolution)

dx = L / (N-1); % Spatial step size

x = linspace(0, L, N); % Spatial grid

nu = 1e-4; % Viscosity

dt = 0.0001; % Time step size (reduce for stability)

T = 1; % Total time

Re = 10000; % Reynolds number

% Initial condition

u = sin(pi*x);

% Time integration using RK4

for t = 0:dt:T

k1 = dt * burgers_rhs(u, nu, dx);

k2 = dt * burgers_rhs(u + 0.5*k1, nu, dx);

k3 = dt * burgers_rhs(u + 0.5*k2, nu, dx);

k4 = dt * burgers_rhs(u + k3, nu, dx);

u = u + (k1 + 2*k2 + 2*k3 + k4) / 6;

end

% Plot the result

figure;

plot(x, u);

xlabel('x');

ylabel('u');

title('Solution of Burgers'' Equation');

% Compute the Fourier transform

U_hat = fft(u);

% Compute the energy spectrum

E = abs(U_hat).^2 / N;

% Plot the energy spectrum

k = (0:N-1) * (2*pi/L);

figure;

loglog(k, E);

xlabel('Wavenumber k');

ylabel('Energy E(k)');

title('Energy Spectrum');

% Function to compute the RHS of Burgers' equation

function rhs = burgers_rhs(u, nu, dx)

N = length(u);

rhs = zeros(size(u));

for i = 2:N-1

rhs(i) = -u(i) * (u(i+1) - u(i-1)) / (2*dx) + nu * (u(i+1) - 2*u(i) + u(i-1)) / dx^2;

end

% Boundary conditions

rhs(1) = 0;

rhs(N) = 0;

end

Thank you Geraldo J B Dos Santos , for your contribution. I have myself generated a similar formula for non-uniform grids. However, by keeping track of the remainders I noticed my formula did not have a second order truncation error, which is why I posed the question. Interestingly, numerical methods showed the convergence was indeed second order and that confused me. When Filippo Maria Denaro gave me the reference above, I finally understood that even though the truncation error may be first order, the convergence is assymptotically second order. That clarified this for me, thank you all!

Hello, you may need to investigate a solution using the Laplace and/or Mellin transforms. I have not tried myself but I can see that you could try the Laplace and/or Mellin transform of derivatives in the left side and f(of integral) in the right side; I suspect you may need to know f(x) so you can transform the f(g(x)) using the relation between the Mellin transform and other integral types (Weyl, Stieltjes, Hankel, Laplace, etc).

explanation is quite good but you should point on that the value V in is taken from a new time layer

you can email me.: nnadikwe.johnson@cgrpng.org

info@globalscientificlibrary.org

Hi Mat! I hope that all is well.

I use NR to solve sets of discretised nonlinear equations all the time. The first type of problem that I used the method on were nonsimilar boundary layer equations where I used the Keller-box method. More recently I've solved a set of equations for what are, in essence, Fourier coefficients. This system is very highly nonlinear.

In both cases (and others) the Jacobian matrix is computed within the code rather than specified by the programmer.

It may be necessary for some applications to solve each NR iteration using a block tridiagonal solver.

I don't mess around with Matlab for this. Good old-fashioned Fortran works very well for me.

I will say that, once you've cracked it you'll be dancing. So yes, persevere!

HEC-RAS can solve the 2-D Shallow water equation using a finite volume approach.

implementation details are missing anyway, for analytical derivatives, use of flow thrust trajectory optimization problem via indirect optimal control is applicable Finite difference derivatives are not accurate enough to achieve convergence in some perspectives, this can be done through transition matrix using instance of chain rule to derive required analytic partial derivatives.

Chun Qin , you can use the discretization error in the extra states arising from finite difference approximation as an extra source of noise. This is reasonable and works well if the error is 'well behaved' (e.g. is not always positive for most or all operating points etc).

Thank you, Syed Haseeb Shah and .

Alireza Akbari

please write the ODE, not only the discretization You used.

however, for multi-step methods you need to create the starting values using a single-step method for all required nodes. Use a discretization of the same accuracy.

Note that the second order ODE could be written as system of two first order ODE.

Not sure what you are looking for but I think you can use Automatic Differentiation libraries like #Casadi #yulmip, I prefer Casadi, because it is easy to use, you can simply define your objective function and your constraints

Spherical polars, rather than Cartesians.

In general, the name diffusion equation (time dependent or not) identifies a linear PDE, whose finite difference discretization leads to a linear algebraic problem that does not require a nonlinear solver like Newton Raphson. This means that your problem is a nonlinear one, for example with nonlinear viscosity (porous media equation and such), and that an implicit method is being used for the time discretization. The correct way to approach these problems is to write the space and time discretization (there is no mention of what time discretization method is used) which leads to a (potentially large) nonlinear system to be solved

at each time step. Then at each time step a nonlinear solver is used to compute the solution of thie algebraic problem. Any of these steps might have been coded incorrectly leading to the problem you mention. One simple way to check what is going on is to compare with the results of a simple explicit method (say forward Euler) employed with a very small time step, such a method would not require NR iterations and therefore any incorrectness in the NR solver would not affect it.

In addition to answer of Mesfin, the following link explain the answer in a simple clear way

For arbitrary unit cells, your best bet is an eigenmode simulation. Sweep the Bloch phase shifts across the various axes from 0 to pi and solve for the eigenfrequencies. The regions in which there are no found eigenfrequencies are your band gaps.

SOR will converge faster than GS or Jacobi methods if the relaxation factor is chosen well. If, by diffusion equation, you mean Fourier's equation, a linear equation, then SOR will be perfectly fine. Easy to encode and not complicated. BiCGSTAB etc are powerful methods but are unnecessarily complicated for something like this. Quasi-Newton methods are for nonlinear equations. But if you have a nonlinear diffusion equation then these more powerful methods may well be the way to do it.

I hope it helps you

See page 24, Chapter 3 on Convolutions. Another path for understanding your filter is to see them as wavelets. They are a version of a Haar wavelet. Normalization is the same.

A good choice here would be use of staggered grid. You can associate different quantities with different grid elements, e.g. diagonal stress components go at cell centers, off-diagonal stress components go at vertices, and velocity components go to cell edges (Vy sits at horizontal edges and so on). This way you'll have, for example, dVx/dx approximated at a cell center, exactly where sigma_xx is attached.

This is perhaps too general a question and it certainly doesn't have enough information to give a comprehensive answer. You haven't mentioned if you are solving an ODE or a PDE, if it steady or unsteady in time, what the boundary conditions, Cartesian or polar coordinates, and so on. So I'll make some observations in teh light of some of the thigs that I have been looking at recently.

I am not a all sure that one can formally compare rates of convergence because one isn't comparing like with like. A second order accurate finite difference method has an error which is proportional to h^2, i.e. quadratic convergence. The speed of convergence of a Fourier series depends on the degree of continuity of the function being approximated, and therefore one looks at the power of n in the denominator. Perhaps it would be better to consider how quickly one can evaluate an approximation to one's target function to, say, a relative accuracy of 10^{-6}.

If one is trying to compare the speeds of computation of the steady-state solution of a nonlinear PDE for a given accuracy, then there are many issues to consider. It is much quicker to write a code to employ finite differences to a nonlinear system than it is to write what is in effect a spectral solver. Part of this is due to the multiple interaction of the sinusoids leading, occasionally to triple or even quadruple summations. Such summations require extensive checking and careful coding. Perhaps all of this work may be justified if the resulting code is going to be used very many times. On the other hand, if one requires only a few solutions, then I would use a simple finite difference scheme. It's a bit like the vexed old question of, which is better, a mountain bike or a road bike. The answer depends on what you want to achieve.

As a generalisation, I would imagine that Fourier series solutions will be very quick compared with finite difference solutions, whereas they are likely to be much slower for nonlinear equations.

I guess that the best comparison between the methods themselves but with no account being taken of code creation time will be by means of an operation count for the various methods. Then, the number of iterations required to obtain solutions with a maximum error of 10^{-3], 10^{-4}, 10^{-5} and so on may be used show the variation of the time taken against the error.

Hi!

To be honest, I have no idea why you have these issues. Does your method have a formal order of accuracy? If it doesn't, then Richardson's Extrapolation won't do anything. Is your set of equations a stiff set? If so, then explicit methods can and will diverge irrespective of their order of accuracy. Direct methods, where the ODEs are converted into a matrix/vector system will often yield good solutions. Even if they are nonlinear, then a Newton-Raphson approach can work well.

Dear Dr. ,

Thank you for suggesting me this article to me. I will read it.

You can check the paper published by https://scholar.google.com/citations?user=-VIX0oEAAAAJ&hl=en

Dear Nischal Sehrawat, The PML mathematical implementation consists of the introduction of a modified coordinate system, where an expansion coefficient is a complex number with an evanescent imaginary part. This generalization is achieved through the substitution provided in the screenshot:

I am surprised that a known scholar with a long experience in the transportation domain maintains such a hard stance on heuristic search. Obviously, we live in a world where extreme opinions are those which are the most echoed. Truth is, assuming that all practical optimization problems can be solved to optimality (or with approximation guarantees) is essentially wishful thinking. Given this state of art, better integration of exact and heuristic algorithms can largely benefit the research community. At the risk of repeating myself, here are some important remarks to consider:

• CPLEX and Gurobi (the current state of the art solvers for mixed integer programming optimization) rely on an army of internal heuristics for cut selection, branching, diving, polishing, etc... Without these heuristic components, optimal solutions could not be found for many problems of interest. CPLEX has even recently made a new release permitting a stronger heuristic emphasis (https://community.ibm.com/community/user/datascience/blogs/xavier-nodet1/2020/11/23/better-solutions-earlier-with-cplex-201). MIP solvers also heavily depend on the availability of good (heuristic) initial solutions to perform well. For many problems, cut separation is also done with heuristics. In the vehicle routing domain, we have a saying: heuristics are the methods that find the solutions, exact methods are those that finally permit to confirm that the heuristics were right (sometimes many decades later, and only for relatively small problems with a few hundred nodes despite over 60 years of research on mathematical models)...

• The machine learning domain is quickly taking over many applications that were previously done with optimization. Among the most popular methods, deep learning applies a form of stochastic gradient descent and does not guarantee convergence to optimal parameters. Neural networks currently face the same scrutiny and issues as the heuristic community, but progress in this area has still brought many notable breakthroughs. Decision-tree construction and random forests are also largely based on greedy algorithms, same for K-means (local improvement method) and many other popular learning algorithms.

• Even parameter tuning by the way is heuristic... I'm sorry to say that, but most design choices, even in the scientific domain, are heuristic and only qualify as good options through experimentation.

thanks for your help.

I understood the reason of this trouble. there was a tiny mistake in my Implementation.

I don't know this specific physical problem but you have to be aware of the mathematical character of this system of partial differential equations. Try to see from the matrix form if the eigenvalues problem admits real solutions and the system is hyperbolic. That will define the type of BCs and the numerical discretization.

Hint: The second compatibility equation can be substituted directly in the first two equations so that you have only time derivatives of u and v.

This has also been studied extensively and you can find articles on the Web discussing it. The fact that the momentum equation (i.e., Navier-Stokes) is second order and the continuity equation is first order but must both be solved simultaneously gives rise to some interesting effects and computational challenges. The implementation you're suggesting has been tested and found to be problematic. If you search for "checkerboard" and "pressure" you will find many discussions on this subject. I have covered this and similar problems in several publications including, open channel flow using implicit, explicit, and hybrid differences compared to the MacCormack method, which came out of aerospace work. If your truncation errors are of different magnitude, you will get unwanted artifacts in the solution.

The reason is very intuitive: SPH captures free surface in a natural way due to its fundamentals (Lagrangian method).

Two main problem in Shallow water wave solvers are a) wet-dry algorithm in domain boundary and b) free surface fluctuations. Those are handled perfectly with SPH method. For example problem of DAM break which is highly nonlinear can tackle very nice by SPH

BTW: There are two down side to SPH: Computationally Expensive and measures of turbulence and bed shear are not readily come out of your model like classic CFD solvers.

Dear;

For produce singular system you use eigenvalue and with Dirichlet’s or Neumann’s boundary conditions will .

Regards

I think the method depends on how you are using the waveguide and what you want to learn. Do you want to calculate the modes that this waveguide would support? If so, the link pointed out by Ahmed is perfect. However, that is a rigorous analysis. If the PPW is to be used as a microwave transmission line, it may be perfectly valid to analyze it using an electrostatic approximation. Under Topic 10 at the following link I provide a MATLAB function tlcalc() to analyze many different types of transmission lines.

https://empossible.net/academics/emp3302/

You might also want to simulate waveguide circuits and look at different waveguide discontinuities. This is actually a large 3D problem. Consider using finite-element or finite-difference time-domain. Here are some links to start learning FDTD:

https://empossible.thinkific.com/courses/1D-FDTD

https://empossible.net/academics/emp5304/

Hope this helps!!

Hi,

In fact the matrix H is not explicit in equation (19), it is present in the equation giving the estimate and the covariance which are then updated using equations (5).

Best regards

Shahriar Shahrokhabadi y Zoubair Boulahia , Vahid Moosavi necesito una ayuda en mi simulación por "Simulink" la ecuación de richards ,

As your system matrix most probably is sparse, you might be better off with numerical linear algebra routines especially designed for large, sparse matrices. I did stuff on that in the late 80s, through the Cray-2 machine, but there must be more streamlined ways to do it nowadays. (In those days one had to "cut" the lengths of the vectors in order to match the smartest size of sub-vectors for the linear algebra routines, but nowadays that will most surely be automatic, to a degree, nowadays.) Talk with numerical linear algebra experts, and they will help you.

The accuracy of the numerical results are mainly restricted to accuracy of the models, although sometimes the numerical stablilization might due to the detalied numerical methods. GPU might be treated as one possible way to decrease the CPU times in numerical simulations. Further, for a detaield numerical simulation problem, BC might be very important.

Hello Hamid Reza Attar ,

Basically, there are at least three methods to perform such task. The problem is that most of them works only for Explicit anylysis. Below is a brief description of them all.

1) Using *FILTER "to stop the analysis when the value of any variable in the output request reaches a specified upper bound or lower bound".

For more information read this:

2) Using *EXTREME VALUE with *EXTREME ELEMENT VALUE "to stop the analysis at the first occurrence of a variable exceeding its user-specifed bound"

For more information read this:

3) Using utility routine XIT "within any Abaqus/Standard or Abaqus/Explicit user subroutine, respectively, to terminate an analysis."

You need to include this command in some user subroutine which would check the PEEQ value, and if it reach some extream value, run the command XIT. I would recommend to use USDFLD subroutine with utility routine GETVRM for Abaqus/Standard.

For more information read this:

Hope this information will help you.

Sincerely,

Pavlo

Dario Fasino ,

Thank you for the clarification, however I'm a bit not very sure of whether to choose F(f + 2h𝜂) or F(f + h𝜂1 + h𝜂2) in analogy to f(i+2) (for normal functions).

Dear Onur,

I am also trying to solve the waveguide propagation problem using x,y, and z co-ordinates using FDBPM. Could you please share your matlab code which might serve as a starting point for me.

my email is vishwatoshmishra34@gmail.com

Thanks in advance.

Recommended

Oliver Mangold

personally I never used the CIP method and I don't like the semi-Lagrangian approaches at all. But the historical MAC method by Harlow & Welch (I attached) used a particle time-advancing method that is then re-located on an Eulerian staggered grid. In general, the projection step does not depend on the discretization of the convective term but has a source term containing the divergence of the intermediate velocity you redistribute on the staggered grid.

The compact stencil for the pressure (central second order discretiziation of Div Grad operator) ensures you have no spurious modes that, conversely, appear on a large stencil. If you consider my papers (but there is also a section in the textbook of Ferziger and Peric) , you will find that the compact stencil for the pressure is the same you can use in the Exact projection method on staggered grid and in the Approximate projection method on colocated grid. The difference is in the resulting Div v residual.

I suggest starting from the general idea of using the interpolation as described in the paper by Harlow.

Roughly, numerical stability is to do with how well the numerical solution matches the exact solution, i.e. whether the error grows or not. So the Backward Euler method is a stable method when solving a linear equation such as Fourier's equation. However, if the equation being solved is nonlinear, then iterations are required when applying Backward Euler, and these iterations may diverge if the timestep is too large.

Finite element method (FEM) is a powerful and popular numerical method on solving partial differential equations (PDEs), with flexibility in dealing with complex geometric domains and various boundary conditions. So it has a wide range of applications in Mechanical Engineering, Thermal and Fluid flows, Electromagnetic, Biomathematics, Geo Mechanics, etc.

Please refer the attachments for FEM

Reducing oscilltory vibration in any system by damping must result in a total loss of energy. Any effect that contributs to this effect is not damping.

I think that the error can be reduced when using finite difference approach to solve differential equations by increasing the number of cells in the domain ,for example calculation of potential distribution by solving Poisson's equation .

Dear Dr.

Rashid Zafar

Greating!

You can see the following links

with best wishes

Here it is

J. C. Tannehill, D. A. Anderson and R. H. Pletcher, “Computational Fluid Mechanics and Heat Transfer,” 2nd Edition, Taylor & Francis Ltd., Oxfordshire, 1997.

Primitive variables, streamfunction/vorticity or velocity potential? Laminar/turbulent? Steady/unsteady?

All of this could have been stated in the original question!

Doesn't ADI give you good performance? It is fairly easy. You could use multigrid, which can be very fast but it isn't easy. Why do you wish to go for an easy method?

You must also achieve stability. One way to both require stability and accuracy is by solving feasibility in a constrained linear program: for example, see section 2.2 in

Alternatively, use a framework that has some assurance that the discretisation has the same stability as the original PDE, especially when that framework and assurance also applies to nonlinear PDEs. See one approach in

especially Section 3 that addresses advection-diffusion, and another cognate approach in

The choice of method depends on the desired accuracy, as well as, concerns about the stability and robustness of the system while maintaining computational efficiency and universality. As rightly pointed out by Butusov, sometimes versatility, convenience and elaboration of method mean more, than precision. Finite analytic method possesses several additional restrictions that limit its wide applications and universality.

I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books

Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.

Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.

There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!

This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.

Hussein ... thanks ... I'll check it out.

In my opinion, overlapping elements can dramatically change the accuracy of the results. Mesh distribution has to be as much as regular to obtain accurate results.

Dear Megahed.

Let me introduce first. I work about 14 y with plastic zone analysis of plane framed structures. My software PPLANAV run based on NR, which mean solve some very simple problems, like columns, beams and portals.

After all this time, I didn't worked with bigger structures because of time consummed processing, which becomes higher when you have many numerical data involved and many NR iteration too.

Saying this, I may ask you how numerically the computer arrived to this negative number? I guess that you must check it precisely, to guarantee that there is no mistake in your matrices, in the answers produced first. Therefore, it recommends to slow down and analyze.

How the computer program produced this answer? There is a possibility of any kind of buckling is involved or coupled? It is there any numerical bug? It is some interdependency between the tensors that lead to an negative value?

Actually, in a case of a simple supported steel beam with a single vertical load at the mid-length, after the plasticity has formed a potential zone where will be the collapse, but there is some little elastic part yet, hundreds of iterations happen till reach the end. Making a plot of its deformed shape, between two consecutive iterations, you will see it like a string, downward and upward. Therefore, the iterations of NR, trying to null the residual force vector, make this change of direction because the structure is very sensible at this time.

So, you must certify yourself that the unstability phenomena or the resistance limit are occurring at that point, that there is a sound explanation for this numerical correction. If this is the case, everything is fine. Otherwise, look to your code and search if there is no other thing producing the unespected answer.

Good luck.

Hi Mahdi, I saw that professor Tapan and Denaro answered the question. Who am I to disagree with them? Nevertheless, I will my two cents. If I am mistaken, please correct me because I am willing to learn.

There is nothing preventing you from getting such interaction with low order schemes. At least theoretically, you can demonstrate that all schemes tend to the same solution as dt and dx decrease. At the end of the day, this is what Roache's work is based on! You can converge to the same solution faster with a high order scheme. For example, refer to the Taylor Expansion series. The thing is the fidelity of your solution. In addition, after a brief literature review, you will see that most of the current research is carried out using High order schemes. Here is the catch, a second order scheme can be more ACCURATE than a 6th or 7th order scheme. For example, refer to the DRP.

Having said that, you can obtain your interaction using first order in time and space, but the temporal and spatial discretization will be extremely small. For example, if your Re_cell = 1 ( I would say < 1) You round off and discretization error will be smaller than your cutoff (Nyquist). However, to obtain such resolution, provided that you are reproducing professor's Tapan's paper, the mesh will be huge. Here is the point I want to make: You can obtain your interaction, but your solution will not be accurate. I am not surprised at all, and Professor Tapan gave a pristine answer for that.

There are multiple versions of Arc Length Method. I think the Cylindrical version could help you. Please have a look at the following reference:

A Unified Library of Nonlinear Solution Schemes by Sofie E. Leon, Glaucio H. Paulino, Anderson Pereira, Ivan F. M. Menezes, and Eduardo N. Lages

in ansys workbench for static analysis, ERROR response show the released energy between proximity elements, so if your error response is minimum, your mesh, so your mesh quality is good and then your other response have accuracy

You can make any geometry rigid in Abaqus.

It will be easier to model the tool in Solidworks,

or for much more detail and control I suggest Z Brush.

(But importing z brush model into Abaqus may be a little hard.)

And also you have to use lots of small elements on both rigid and deformable parts to keep the details.

by digraph

Thanks Dr. Yoisell Rodrguez

The correct answer depends on some factors:

1) the mathematical character of the PDE. Elliptic equations have no specific direction of propagation and central FD should be used). Conversely, hyperbolic equations have specific characteristic directions and forward/backward derivatives are often used according to these directions.

2) Numerical stability properties. Sometimes forward/backward derivatives are used also to stabilize the numerical solution, preventing oscillations that can onset the numerical instability.

Often, a time-dependent PDE problem can be discretized using several stencils, that happens particularly close to the boundaries where non-symmetric stencils must be used.

Dear Filippo Maria Denaro

DN-differentiation does not have truncation errors. It is an exact differentiation method. It can be interpreted as some kind of automatic differentiation, because you "hide" the derivatives of elementary functions in the dual part of the number.

Please check out some work I made with my students to see more details about it.

As Richard Epenoy said, complex-step directly gives you an O(h^2) error with one single (complex) evaluation of the function.

I also attached Vincenzo's Master thesis. You can see some tests we did about several differentiation techniques there.

Hello,

I have written simple 1D drift-diffusion solver examples using finite difference and Scharfetter-Gummel discretization which you can find here:

I'm happy to answer any questions.

Following..

I am not a person in this field, but I guess this could be a very good criteria for the person identification together with other established parameters. There are several waves which individually or and in combination should be used as a candidate to evaluate and set up as a benchmark for your aim.

Ok, it wasn't clear from the original question. I shall take a closer look at Cahn-Hilliard equation. Regards

Dear Dr. Ali Namadchian

Greating!

The following links may be useful to answer

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With best wishes.

there are several methods ( using finite difference method) to deal with, for example, crank-Nicolson method , ADI approach

Dear Sjoerd van Acht and Suresh Kumar Govindarajan ,

Thank you for your answers. It didn't make sense to me that the simulation becomes slow only when the contaminants starts to breakthrough, since I have considerable heat effects along the column.

Somehow, some lines in my code that are not working properly:

This lines:

DO IPX = 1, NumPtosX

Press(1,2,IPX,1) = ......

END DO

should be the same as

DO IPX = 2, NumPtosX-1

Press(1,2,IPX,1) = ......

END DO

IPX=1

Press(1,2,IPX,1) = ......

IPX=NumPtosX

Press(1,2,IPX,1)= ......

However, using the second approach, the simulation does not stop!!! I have no idea why this is happening. I didn't transcribe all the lines of the code, but the values used for the calculations are exactly the same!

Well, it is working now!

Thank you!