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# Numerical Methods - Science method

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Questions related to Numerical Methods

I wonder if we can list direct methods to solve the optimal control problem. I would start with:

1. Steepest descent method

Can anyone add more? With appreciations.

Many Boundary Value problems can be solved by numerical methods ,l,am looking for the possibilty of combining some numerical methods with some integral transforms in order to speed the convergence

Fuzzy boundary broblems have many applications ,so it's very important to compined semi-numericale methods as HPM,VIM,and ADM,......with some artificial algorithms.

Hello all,

Source terms have been known to cause reliability issues in numerical methods affecting therein convergence and accuracy. I am currently facing a similar challenge when trying to solve a Poisson equation with a non-zero divergence velocity field. The source term that I am working which is a cavitation source term dependent on local value of pressure.

For the most part, linearizing that source term seems to solve the issue in the literature however even with linearization my Poisson equation does not converge, and even when it does, the solution is inaccurate and often oscillatory.

Any input from the experts would be helpful

It is believed that:

1- Fully implicit block numerical method performs extremely better than explicit methods and/or partially implicit methods in solving Stiff IVPs of ODEs.

2- Higher ordered methods tend to have better accuracy of the scaled error compared to lesser ordered methods, thurs the later got the advantage in less computational time in most cases.

3- In solving Stiff IVPs of ODE with any of the above methods, the smaller the choices of the step length (say h), the higher the accuracy (lesser error) and the faster the convergence of the method.

In line of the 1-3 above. Iam simulating and comparing a 4-point fully implicit method with 4-point explicit and 4-point diagonally implicit methods. But , iam not getting any advantage in terms of accuracy or less computatinal time. Despite the fact that all the method are zero and A-stables. Please, Sirs is the mistake from the simulation or any other idea ?

In general, there are numerous methods for estimating battery state in the literature. But, which method is more commonly used in real-world EV applications?

Can someone please explain how the time-lag and decrement factor of a building envelope obtained by the admittance method is different from the time lag and decrement factor obtained by any numerical method?

Some numerical methods (such as FDM and FEM) permit the use of high-order schemes for discretization of computational domain yielding higher-order accurate results. The accuracy attainable by a particular numerical method is a function of many variables and the proposed usage of such numerical solutions. From point of view of a Code Developer or Computational Fluid Dynamist,

**can it be said that high-order discretization method necessarily produce higher-order numerical solution?**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

Some commonly used numerical methods for resolution of unstructured complex geometry include Finite Volume Method (FVM), Finite Element Method (FEM) and Spectral Element Method (SEM). Complex turbulent flow in a convergent-divergent rocket nozzle involving fluid-structure interactions may present additional challenge with grid resolution near the nozzle wall, hence the relevance of this question.

I have to study the cyclic behaviour of RC Beam-column joints by numerical method. For this study, which software will be the best.

Can anyone please help me write a MATLAB program to find the temperature distribution (By numerical method) across a composite building wall subjected to periodic boundary conditions?

Hello everyone, I have a request about solving a system of equations (preferred with "Maple"). please tell me an effective numerically method for solving two nonlinear (higher order) equations with two unknowns. it's about a magneto static instability in dusty plasmas. THANKS

When the value of our Trading Account is PI and our cash is equal PI-qS.

q is number of shares of stock (S).Where q is between one and minus one.(why?)

we know:

- dPI=r(PI-qs)ds+qds
- ds=s(mu)dt+s(sigma)dX

My field of expertize is in CFD and not in climatology. But I would start a discussion about the relevance of the numerical methods adopted to solve physical models describing the climate change.

I am interested in details in physical as well as mathematical models and the subsequent numerical solution.

I'm undergoing a Masters degree dissertation on the topic Assessment of concentration and modelling of radionuclide fate-and-transport in groundwater water of Wurno LGA, Sokoto state. I'm facing challenge with the modelling aspect.

I study about numerical solution of fractional partial differential equation using radial basis functions (Multiquadric). According to properties of Multiquadic, if for determined shape parameter c of Multiquadric, numerical method is stability, can I say the numerical method would be stability for smaller values than it?

There are several numerical methods for solving IVPs (of ODEs). There are single step or multi-step methods. The methods may be classified as explicit and implicit methods. When applying numerical methods convergence, consistency and stability are among important points of considerations.

How to find the shape of natural pressure arch formation around underground opening?

kindly suggest the right approach using analytical and numerical methods.

Nonlinear algebraic or transcendental scalar equations are usually solved by using iteration methods. Because some of them are so intricate to solve analytically or an approximate numerical solution is required. Nonlinear algebraic equations can occur during several modeling processes or when solving nonlinear BVPs using FDMs......

Dear researchers,

In solving the problem, by using the numerical method, we already understand that the selection of the grid size strongly influences the calculation result, so we must perform a grid independence test. Is the transient case also affected by the time-step size (time delta)? If so, it means that we must also perform a 'time-independent test on problem-solving using numerical methods. What is your opinion?

Please give your reply.

Thank you very much.

Kind regards.

MCMC sampling is often used to produce samples from Bayesian posterior distributions. However, the MCMC method in general associates with computational difficulty and lack of transparency. Specialized computer programs are needed to implement MCMC sampling and the convergence of MCMC calculations needs to be assessed.

A numerical method known as “probability domain simulation (PDS)” (Huang and Fergen 1997) might be an effective alternative to MCMC sampling. A two-dimensional PDS can be easily implemented with Excel spreadsheets (Huang 2020). It outputs the joint posterior distribution of the two unknown parameters in the form of an

*m*×*n*matrix, from which the marginal posterior distribution of each parameter can be readily obtained. PDS guarantees that the calculation is convergent. Further study of comparing PDS with MCMC is warranted to evaluate the potential of PDS as a general numerical procedure for Bayesian methods.Huang H 2020 A new Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference, preprint,

Huang H and Fergen R E 1995 Probability-domain simulation - A new probabilistic method for water quality modeling.

*WEF Specialty Conference "Toxic Substances in Water Environments: Assessment and Control"*(Cincinnati, Ohio, May 14-17, 1995),Conference Paper Probability-domain simulation - A new probabilistic method f...

For explicit numerical numerical methods Courant number should be close to 1. How is the stability criterion defined for transient simulations using implicit numerical methods?

My quest is with respect to glass. How the numerical method application is useful in case of glass?

- How can I solve the following differential equation system (equations 30 and 35 highlighted in the image) with known initial conditions xp0 and xw0 using the Runge-Kutta method or any other numerical method? HP, HW, GP, GW, gammaP, gammaW, and gammaF are fixed coefficients.

The Scheme may refer to a FDM or another numerical method.

LBM is described as a powerful tool in treating fluid flows.

We always hear about its advantages as a numerical method.

The question is what are the limitations of LBM when it comes to complex flows?

How can optical memories based photonic crystal create a hysteresis loop in a structure?

(In the analysis of these structures using the numerical method, how can we change the power from minimum to maximum and vice versa)

Please introduce reference in the field

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?

I want to integrate two carbon fiber materials together and want to model it mathematically considering the processes of joining the materials and the possible stress around the joints due to external loads.

I will appreciate if someone can recommend and the numerical methods which one is better suited for this purpose.

Thank you all in advance

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 am reading this article to understand Richardson's extrapolation for NSFD. With MATLAB, I did the simulations and got the same result with the step size they used. But when I increase the step size, I will get negative results in the Richardson Extrapolation (I have used the Aitken-Neville algorithm). What is the problem? Is it the formula used? Or the code that I have used? that result in negative values.

Thank you!

Please, how can I output global displacement (U) in Comsol? I want to calculate the compliance of my model. (Please, see the attached Comsol code).

compliance = U transpose x Global stiffness matrix x U

I have used these lines of code to obtain the global stiffness matrix (Kf).

MA = mphmatrix(model, 'sol1', 'out', {'K','E'})

K = MA.K; % stiffness vector

Kf = full(K); % global stiffness matrix

Thank you.

Lagrange multipliers might become tricky for large number of variables and most of the examples on the internet are for small systems where it is possible to try every state of multipliers (either relative inequality constraints being active or not). Is there any suggestion for systematic numerical algorithm (iterative one) to solve optimization problems with inequality constraints by Lagrange Multipliers?

Considering the rate of contaminant removal as pseudo 2nd order from the batch study, column study is being performed. I would like to predict the removal pattern of the contaminant in a non-equilibrium condition. A partial differential equation is coming as advection-dispersion-reaction equation(mentioned in the attached doc file). How can I get the concentration distribution (c vs t) using this equation? I found that it can be solved in Matlab using a numerical method. But somehow I couldn't figure it out using Matlab till now. Any help will be highly appreciable. Thanks in advance.

I am developing a solver to simulate compression of oil in shock absorber filled with viscoelastic compressible oil. The model will consider the loss due to viscosity which will decide the response of damper to the shock it experiences. Please include any link or document you think curucial for this.

I have made models for incompressible flows but I am having a hard time to find resources for this case.

Thank You!

It's a very tedious job to write the charatcteristic equation and then find the eigen values, especially in large dense matrices.

Is there any general (so called) method by which one can find the eigen values of matrix?

- Due to the influence of university education and papers published in leading international scientific journals, young scientists, in particular, believe that this is the sole way to advance knowledge. Experienced and practically oriented hydrologists do not always share their enthusiasm.

Hello All,

I am a novice to numerical methods. I am trying to numerically solve the following two coupled equations using Matlab; any suggestions are greatly appreciated.

Flux balance in spherical coordinates:

d/dr(r^2*N(r)) = r^2*[Ko*Exp(-E/R/T(r))*(Vmax - V(r))].....................(1)

The term in [ ] is a source term.

N is zero at r = 0 (symmetric boundary condition);

Flux, N, is here is convective in nature and is due to pressure gradient as given below.

N = (K/u)*(P(r)/r/T(r))*dP(r)/dr..................................................(2)

P is Ps at r = R (surface).

I have already worked out the T(r) and V(r) profiles separately.

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

In compressible flow , five equation should be solved simultaneously .I need to solve these equations by numerical methods. Is there any references that explains this algorithms??

I am working in Fractional Optimal Control problem of Epidemic models, I am trying to solve the problem numerically but I can not do it, as there is no toolbox and any sample MATLAB code is available to solve fractional optimal control problem. Please share the MATLAB code to solve Fractional Optimal Control Problem. Please help me.

Reach me at ramashisbanerjee@gmail.com

Thank You in advance.

I am studying mass-spring-damper systems with coulombs friction. There are multiple discussions on simulating such systems using numerical methods and the problems that arise due the discontinuous excitation but I wanted to know if an analytical solution exists. To be mathematically clear about the problem, I am trying to analytically solve the following.

m*(d2x/dt2) + c*(dx/dt) + k*x = F*sign(dx/dt)

where the sign function is defined as:

sign(var) = 0 if var = 0

sign(var) = 1 if var > 0

sign(var) = -1 if var < 0

Note: I am aware of treating such systems as piece-wise linear nonlinear systems but I want to know whether a general solution exists that is capable of solving the problem without breaking it to a number of mini-problems.

I am interested in collaborating with any researcher working on modelling corona virus using fractional derivatives. If you are a researcher or you have a related project, please feel free to let me know if you need someone to collaborate with you on this research study. If you know someone else working on this research project, please share my collaboration interest with him.her. I would be very happy to collaborate on this research project with other researchers worldwide.

If an Eigenfunction expansion method is used to transform a nonlinear PDE (in space and time) into a system of truncated nonlinear ODEs in time which are then solved numerically (using Runge-Kutta method or any other numerical method), can we then say Eigenfunction expansion method is a total analytic method, a semi-analytic method or just a method of transformation of PDE to ODE ?

I'm reviewing a lot of papers where the authors take a 3-D autonomous chaotic system (think Lorenz) and add a fourth variable bidirectionally coupled to the other three and then report its unusual properties which typically include lines of equilibria, initial conditions behaving like bifurcation parameters, and sometimes hyperchaos. Usually these systems have two identical Lyapunov exponents (often two zeros) and a Kaplan-Yorke dimension ~1.0 greater than the dimension determined by other methods. Thus it seems clear that the system has a constant of the motion such that it is actually 3-dimensional with an extraneous variable nonlinearly dependent on the other three. Are there algebraic or numerical methods for demonstrating this by finding a constant of the motion?

Dear all,

I have a differential equation which encompasses a multiplication two unknowns (u &w) at the above manner (both are functions of time and space). I do not understand how it should be discretized? I follow the procedures as space discretization, time discretization, Newton-Raphson method and variatoinal derivative, but does not look like to be correct.

Thank you.

There are numerous methods I have been looking in the building energy benchmark field. However, there are different lookouts for every method applied. It is very difficult to decide what's the best method for benchmark residential buildings energy.

My approach is towards the in-house appliances energy for benchamrking.

Please suggest!!

I have already gone thought some of the related papers also I attached, please suggest any best method/ methods or any means by which I got insight.

Thank you in advance

How can we simulate the mechanical response of high entropy alloys?

I've used a code to calculate flow rate in DFN. Now I want to calculate permeability using the Darcy law. The flow rate is calculated in m3/m.s.

Should I multiply the flow across the mesh or divide the model width and then use Darcy law ?

I am getting higher natural frequency value of a sandwich plate when computed on the basis of HSDT as compared to that of FSDT. But in general it is expected that the higher order kinematics (cubic in-plane displacement) should give lower frequency values than that computed using first order kinematics (linear displacement). I know that my result is correct, but I am unable to justify this point. kindly help

*Dear Colleagues*

*Can anyone suggest me a good numerical method for solving a system of fractional partial differential equations?*

I just started using gmsh for 2D triangulations and I'm observing that both the nodes and elements are being outputted in ascending order (consecutively). This is good as it makes reading the data easier but I want to know if this is always the case.

Secondly, I also observed that the physical-line boundaries are outputted as elements with two nodes (line-elements in a 2D triangular mesh). I have also figured out that the actual elements (triangles) are listed after these line-elements. This then means that the true number of triangles (elements) in the mesh is the number of elements oputputted in the mesh file less the number of line-elements. Kindly let me know if this is the correct idea.

Thanks in anticipation.

All,

I have two non-linear coupled differential equations. Is not a problem to solve them with Euler’s numerical method. The system is very stable and converge to a better solution every time you make h smaller.

But how do you prove the stability for this Euler’s numerical method for the two non-linear coupled differential equations?

Thanks

A Research on roundness of a circle: An investigation of pie π. Since π is a long digits irrational number can π actually be constant? Or is it gotten from iteration of closed figure using numerical method? We will be using a 2d circles from 3 d circular objects, we will check the ratio of it diameter to it perimeter after cut neglecting the reduction in length due to cutting chips.

π is use in the determination of some parameters and quantities in mechanical engineering design and fluid mechanism e.t.c. and most importantly in force distribution of rotation of earth F=2πf Alare's Formula. So exactness is important to know if the forces are evenly distributed or turbulent .

Do you have any reference on π ? Or have you done any research on investigation of π? Help me with solution. We need a team to join I and Kehinde Alare.

Hi everyone,

I have been working on a laminar compressible microtube flow analysis (supercritical fluid). Mass flow inlet, pressure outlet and constant wall heat flux boundary conditions are applied to the circular microtube (0.5 mm diameter). I get fluid properties by using CFD Post and there is a mistake at CFD Post total pressure results (it was used areaAverage at related cross-section plane) as can be seen on below figure. In addition, theoretical total pressures are calculated by using momentum equation between two control surfaces (the related values (density, velocity etc.) were taken from numerical solution). We are also suspicious about getting this mistaken result due to operating pressure condition which is 101.325 kPa. I would like to grateful if you could share your recommendations.

What are the possible methods to solve non-linear system of ordinary differential equations of n equations containing n variables?

List possible analytical method through which we could solve the nonlinear system. Also list possible numerical methods as well.

Share any material or article or book or even videos you know worth related about solving nonlinear equations.

Thanks.

I need to simulate the transition of Flow through Coarse Porous Media such as Rock-fill dams, to investigation of Water level profile in each distance from up-stream and determination of discharge of fluid seepage from the body of these media. Notice, I want to simulate a Single-Phase Flow.

How can I simulate this project? Please suggest and introduce a useful software for this issue to me...

What is your idea about Flow 3D, Fluent, ABAQUS,...

Conference Paper A numerical study of the flow through coarse and homogeneous...

Conference Paper Transition of Flow through Coarse Porous Media with Network ...

Please help me to solve numerically of the following 4th order nonlinear ODE

f''' + ff'' +1 -f'*f' +k(ff''''-2f'f'''+f''*f'')=0

boundary conditions

f(0)=0, f'(0)=0, f'(infinity)=1, f'''(0)= -[1+kf''(0)]

where k is a parameter and the solution exists for k<0.325786 and one may choose infinity= a finite number, say 10. I am unable to solve it through 4th order Runge-Kutta with shooting technique and Matlab solver 'bvp4c'. Here I have also the attached the article in which I found this equation.

Sometimes we hear somethings on stiffness problem of solution for chemical reaction modeling.

There is an option in Fluent software to figure it out.

Have you guys used it so far? tell us when and why did you use it?

Guys,
Sometimes after initializing the problem in Fluent and triggering the calculate button, a residual/residuals start from a big non-zero magnitude. For instance, to me, it in many cases happens for some convection-diffusion equation when I'm simulating combustion. Interesting point is that the results looks ok and have good agreement with experimental results. In this page:

It's said that:

"For most problems, the default convergence criterion in

**ANSYS FLUENT**is sufficient. This criterion requires that the scaled residuals defined by Equation 26.13-4 or 26.13-9 decrease to 10^{-3}for all equations except the energy and P-1 equations, for which the criterion is 10^{-6}."As it is said, the residuals need DECREASE TO 10

^{-3}. Then it means that it doesn't matter from where the residual starts, it should finally decrease to 10^{-3}.- Am I right?
- What's your opinion?

Another question:

3. Why does it happans? (I mean why does the residual start from a non-zero magnitude?) and How to figure it out?

I've attached two screenshots of the residual of a project for instance.

I need to use matlab to get the displacement and velocity time history of the 3 degree of freedom system (image attached). i can use any numerical method or ode45 but one of the element in third equation has

*sigma of udot which cannot be taken as constant.*Kindly help how to solve these type of equations?Any help will be greatly appreciated.. Thanks in advance.

Yanai (1973) fluxes computation methods for atmospheric data at grid points available at equal time interval.

As we are aware of the importance of eigen values of matrix and also the mathematical efforts needed to solve for the same. But can we reduce those efforts or can we at-least generalize the method to compute all the eigen values. As we know:

1. power method is mostly used numerical method to compute eigen value , but as the size of matrix increases it's not smart approach to use this method.

2. Using characteristic polynomial , computationally, is even cumbersome approach because one needs numerical method to find the roots of characteristic polynomial.

Can there exist any algorithm which can be used on general matrix to find the eigen value??

When the buoyancy effect is significant in supercritical fluids passing in the tube, some eddies can form in any location of tube, especially at the vicinity of supercritical temperature. So, what is the numerical method to solve like the compressible problem? As my experience and literature, there is no way to calculate this type of problem by using RANS models, so I have just started using LES models. Although it was used a very low time step size (about E-5) and tried all of the subgrid models, I couldn't solve this problem. I would like to be grateful if you could share your recommendations. Thanks

*Computational costs are important in numerical methods, as if they were obtained by counting the number of operations(flops). The order to count this number has been asked by MATLAB that I have recently encountered.*

We know that , In applied mathematics,

**discontinuous Galerkin methods (DG methods)**form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications . so i want to know ,What is the difference between the discontinuous Galerkin method(DGM) and discontinuous Galerkin finite elelment(DGFEMs) method?Dear Professor Baines,

We need the DOI number for you book edited after the Reading Conference in 1985 :

Conference on Numerical Methods for Fluid Dynamics,

Reading, U.K., april 1985.

J.M. VANEL, R. PEYRET, P. BONTOUX, “A pseudospectral solution of vorticity-streamfunction

equations using the influence matrix technique”, Num. Meth. Fluid Dyn., II, Clarendon, ed. Morton &

Baines, 463-475, 1986.

Would you be kind to give us Roger Peyret and myself the DOI reference number requested by ISI Web Knowledge and Publons.

Sincerely, with our best regards,

Patrick Bontoux and Roger Peyret

How can one find a vector field B that satisfies the following equation for a known solar function phi which operates on 3D real space?

curl(B) = grad(phi)

What kind of equation is this, are there known methods to solve this equation for B (either analytic or numerical methods)?

Thanks

Dear colleagues

Hi,

As you know, for determining the order of accuracy of a numerical solution on a uniform mesh one normally needs to successively double the number of grid points in each direction (for a rectangular solution domain) and calculate the error at a specific location, say x ,which is common to all the grids (i.e. coarse, fine, and finer, if two levels of mesh refinement is employed) then use Log(E2/E1)/Log2=m, where m is the point wise order of accuracy, E2=ABS(Fine solution(x)-Coarse solution(x)), and E1=ABS(Finer solution(x)-Fine solution(x))

However, when a structured boundary layer-type non-uniform mesh is employed the situation is much more complicated. For example, doubling the grid points in each direction (in a rectangular domain) may not guarantee that at a particular location, say x, the grid spacing will also be halved. Not to mention that doubling the grid points may also not guarantee that the reference point (x) coincide with one of the grids in the fine mesh.

My question is how can we calculate the point wise order of accuracy of the numerical method in this case?

To elaborate, I should say that I am using the LBM to simulate a natural convection problem.

Hi all,

I am looking for some scientific exchange with researcher that have developed or even used an implicit 2d RANS solver.

I have there following questions in this context ....

1. Which numerical methods is used?

2. How is positivity treated in the implicit world?

3. What kind of equations solver is used?

4. How did the parallelization worked out?

In the literature for this problem is very limited, I know two papers, where on of those is only 1st order in space and time.

Cheers

Aron