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Questions tagged [numerics]

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1
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1answer
124 views

FFT Poisson Solver for non-uniform grid

I have a 3D solver for the incompressible Navier-Stokes equations which uses a FFT library for the Poisson equation with a uniform grid on all directions. In 2D the Poisson equation is given by: $$ ...
2
votes
3answers
223 views

Matrix vector multiplication performance

I have been learning about the impact of cache size on code performance. I wrote a small code to see how using a column major loop in MATLAB would be better than using a row major loop, since MATLAB ...
1
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0answers
77 views

(Approximate) Incremental Projection Method for Navier-Stokes equations

I am trying to implement an incremental projection method for the 2D incompressible Navier-Stokes. The type of projection method I am trying is $$ \frac{u^{*} - u^{n}}{dt} = - \nabla p^{n} - u \cdot ...
0
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1answer
351 views

Proper boundary conditions for potential flow around cylinder

I am computing the stationary, incompressible, inviscid and irrotational flow around a circular cylinder using a discretization in general coordinates. I derived a PDE and proper boundary conditions ...
3
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3answers
121 views

How to calculate $\arg(z_1z_2\cdots z_n)$ to minimize results error?

As in title, which method is the most optimal for numerical calculating value of: $\arg(z_1z_2\cdots z_n)$? Method 1: one can first calculate $Z=z_1z_2\cdots z_n$ and then calculate $\arg(Z)$. ...
4
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0answers
118 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
3
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0answers
85 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
0
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1answer
77 views

How to treat non-linear term in finite difference solution of $T''_x+T''_y+aT^2=0$?

Can we linearize $T^2$ When solving $T''_x+T''_y+aT^2=0$ by finite difference? I solved $T''_x+T''_y=0$ in Matlab using a finite difference explicit scheme. But when there is a source term, I come ...
1
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1answer
125 views

Integrating over $\mathbb{R}^{3}$ without a convex subset

I am working on a problem (solid state physics, I am stripping all the details for brevity but if more details can help I'll elaborate) where I need to numerically calculate an integral of the form: $$...
2
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1answer
162 views

How to show that Gauss-Seidel iterative method is equivalent to successively setting each component of residual vector to zero?

As stated in the title, it's said in the book that Gauss-Seidel iterative method is equivalent to successively setting each component of residual vector to zero. After rearranging G-S scheme, I got ...
0
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3answers
228 views

Why there are people that still prefer fortran 77 over new versions?

I am reading some notes from a course in numerical analysis for physical sciences and it is my impression that there are still people that prefer Fortran 77 over new version due to the implicit ...
0
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1answer
61 views

Normalization of MATLAB HermiteH

I was wandering - what kind of normalization does Matlab use in hermiteH, its implementation of the Hermite polynomials? It is certainly not the case that they use ...
1
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0answers
174 views

Vectorised root finding in Python

I have an array of size (254, 80) which I am trying to use Scipy's fsolve on. I have found that the speed of using fsolve on a vector is quicker than it is in a for loop but only for vectors upto ...
1
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1answer
170 views

What is a relative condition number of a sum of positive values?

We want to compute the relative condition number of: $$x_1+x_2+x_3+\cdots$$ We assume all values are positive, and we will do a limit of a large $x_1=10^{8}$, and smaller values for all the other ...
1
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0answers
153 views

Newton - Raphson method : maxima of function in 2 variables

I am computing the maximum of a function (with two-variables) using Newton-Raphson method. The function is : $e^{-(x \ - x_0)^2 - (y \ - y_0)^2}$, whose maxima exists at $(x_0,y_0)$. The Jacobian ...
0
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1answer
270 views

Crank-Nicolson method and mixed derivatives

I am curious if anyone had literature references or knowledge on how to apply the Crank-Nicolson (with approximate factorization) to the $$ \nabla \cdot (\nu (\nabla \mathbf{u} + \left(\nabla \mathbf{...
9
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2answers
251 views

Lagrange multipliers space is too rich in a mathematical view

Background: Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces. It is ...
1
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0answers
137 views

the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM

The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form: I am considering ...
1
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1answer
71 views

finite difference for a second order ode

I saw in a code for discretization of something like $\frac{d^2T(x)}{d^2x}$ , ( $x = sin(\theta)$ ) tries ...
3
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2answers
126 views

How does Mathematica compute real and complex solutions to single, non-polynomial equations?

This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer. $$ \frac{0.125567841}{d^{2.25}} = \...
3
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1answer
70 views

SIRS Model doesn't depend on initial conditions?

So i have been working (as an undergrad, by working i mean "Redoing a few things my professor does") in a SIRS model for epidemies. SIRS stands here for: Susceptible -> Infected -> Recovered -> ...
0
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1answer
85 views

Error measure for a simple finite difference scheme

I am responsible for assisting with certain error measurements for an FE program. The idea is to set up some benchmark comparisons which we can use for future development of the program. To simplify ...
1
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0answers
87 views

Definition of CFL number in Arbitrary Lagrangian-Eulerian framework

In an Eulerian frame of reference, the CFL number is defined as $$\sigma=\frac{u \Delta t}{\Delta x}$$ with $u$ the magnitude of the fluid velocity. A restriction such as $\sigma<1$ for time ...
3
votes
1answer
201 views

How to compute the Frobenius norm of matrices whose entries are either too large or too small?

While implementing in Matlab the Frobenius norm of a matrix $$\| A\|_{\text F} := \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 },$$ a problem arises when numbers are too big or too small: If a number ...
3
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0answers
85 views

How to optimize for decay constant in exponential-like function?

I've got a data set of points $M_O .. M_N$ for time points $t_0 .. t_N$, where $N$ is approximately 10-20, and the spacing of time is not uniform (i.e., $t_{i+1}-t_i$ is not constant for all i). It is ...
3
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1answer
246 views

Numerical calculation sum of exponential functions

I have to repeatedly calculate a function which contains a sum of a large number (~100) of exponential terms: $f(x) = \sum_{r=1}^{100} C_r e^{b_r \cdot x}$ There is no relation between the $C_r$ ...
4
votes
1answer
180 views

Clenshaw-type recurrence for derivative of Chebyshev series

The naive summation of a Chebyshev series \begin{align*} f(x) = \frac{c_0}{2} + \sum_{k=1}^{n-1} c_{k}T_{k}(x) \end{align*} which employs the three-term recurrence for evaluation of the Chebyshev ...
0
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1answer
92 views

finding the growth rate from numerical data

Suppose i have a bunch of 10 data points and i have to conclude whether the increase is $n^2,n^3,\cdots,2^n,3^n, e^n,\cdots$. For example i have the image:- Now the increase is either polynomial or ...
1
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0answers
17 views

Does excitation type matter in a time-domain simulation/computation of a transfer function of an LTI system?

Let's say I am running a FDTD simulation of a wave-equation to determine a transfer function of an LTI system: \begin{equation} H(f) = \dfrac{Y(f)}{X(f)}\ \end{equation} where $Y(f)$ and $X(f)$ are ...
4
votes
1answer
109 views

Using GSL for basic operations

I am learning C/C++ for Scientific Computing and I have a question regarding the usage of scientific libraries for basic operations. Suppose I have to write a small program in C for a bioinformatics ...
4
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2answers
154 views

Literatures on numerical stability of optimisation algorithms

I am curious of whether optimisation algorithms (whatever simplex, active-set quadratic programming, interior point sequential etc.) can fail due to numerical errors and how to avoid them. But I ...
4
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1answer
108 views

Reference Request: Raviart Thomas with hanging nodes

I am interested in reading about the analysis (existence, uniqueness, error estimates) of elliptic problems solved with a Mixed method that uses the Raviart Thomas elements (so far so good, easy to ...
5
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3answers
1k views

Galerkin method: Test functions vs. Basis functions

I'm a novice to finite element and I'm finding quite hard to find the actual difference between Test function(s) and Basis function(s). I would be glad if somone could explain me that and point out ...
1
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1answer
86 views

interpolation 2D irregular nodes

Given a 2D irregular spaced data like shown in the figure, I would like to know how to find derivatives at '*' by interpolating the values at 'o'. Does lagrange 2D interpolation work at irregular ...
6
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1answer
207 views

Is there any point to using hypot() for $\sqrt{1+c^2}$, $0 \le c \le 1$ for real numbers

It is conventional wisdom that programmers should use std::hypot whenever one implements an expression of the form $r = \sqrt{x^2 + y^2}$ In my example, my ...
0
votes
2answers
206 views

Boundary conditions for streamlines in enclosed flow

I am trying to solve Lid driven square cavity flow problem of Stokes equation using finite element method. I have boundary conditions for velocity as zeros on every boundary but u=1 on top boundary. ...
2
votes
1answer
119 views

How to deal with numerical errors in electrostatic field calculations

I want to trace electrostatic field lines emerging from 2D surfaces in 3D space. Eventually I want to find their intersection with an (uncharged) mesh. The charge distribution $\sigma(x), x \in \...
0
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3answers
135 views

How to determine a PDE which is structure-preserving (energy, mass conserved)?

How to determine if a PDE is structure-preserving (energy,mass conserved)? Are there some standards in judging the preserving-structure? Or rather, how to derive the formulation of energy-preserved ...
1
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3answers
501 views

Pressure boundary condition in lid driven cavity using finite element method

Thank you all 1.) I am trying to solve lid driven cavity problem for an incompressible Stokes and Navier Stokes equations using general "Mixed" finite element method. dirchlet boundary conditions are ...
1
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0answers
75 views

Computing the change of function at two close points without cancellation

I want to compute the difference $\Delta f(x_1,x_2) = f(x_1)-f(x_2)$ of a smooth function $f(x)$ at two points $x_1$ and $x_2$ which are close to each other. The magnitude of the expected result, $|\...
1
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0answers
43 views

Discrepancy in estimating boundary stencil for finite difference method

I am trying to estimate the FD stencil for boundary as mentioned in this paper (section 4.1) using MATLAB. The stencil order (6th) is higher than the one mentioned in paper (4th). $$ f_1' +\alpha f_2'...
2
votes
2answers
301 views

Runge Kutta (RK4) to solve coupled harmonic oscillators [duplicate]

I have the same problem as in this question. But can someone elaborate on the answer? The poster says that: Solving this system of 4 ODE's with rk4 will solve for all your state variables ...
0
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1answer
203 views

Trying to plot 1D wave equation for benchmarking

I am trying to plot a reference solution for the 1D wave equation using python. The above link states the following: For a rod fixed at the right end and free at the left end and subjected to a ...
4
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2answers
245 views

Difference in performance of preconditioned GMRES and MINRES

I have two matrices $A, B$ coming from a finite element discretization of a system of partial differential equations. $A$ represents the system matrix and is symmetric and indefinite. $B$ is symmetric ...
0
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1answer
391 views

How can this multidimensional integral be efficiently implemented in python using Gauss-Hermite quadrature

I'm playing around with dynamic programming and need to calculate a multidimensional integral $E[V(W)]$ where we assume $W$ has a log normal distribution. I was looking at the following example in ...
1
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2answers
139 views

How to deal with big numbers in intermediate calculations?

I have a rather long expression (https://pastebin.com/jUsxdCCs) that is an analytical solution of a set of differential equations generated symbolically from Maple. I need to solve a set of equations ...
-1
votes
1answer
730 views

Using scipy.odeint to solve coupled equations [closed]

I have a set of three coupled autonomous equations: ${y_{1}}\prime = y_{1}(\frac{\Omega_{m}}{y_{1}^3} + \frac{y_{3}^2}{6.0} + \frac{V(y_{2})}{2.H_{0}^2})$ $y_{2}\prime = y_{3}$ $y_{3}\prime = -3\frac{...
1
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0answers
121 views

Time-dependent Schrodinger equation(time-dependent Hamiltonian)

This question was already asked here and this is a suitable form of the equation for numerically solving. \begin{align*} i\frac{\partial}{\partial t}u_{\ell}(r,t) = \Bigg(-\frac{1}{2} \frac{\partial ^...
4
votes
3answers
602 views

How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$

I'm interested in finding the roots of the following equation: $\tan(x) = \frac{2x}{x^2-1}$. It is easily seen that 0 is a root and the roots are symmetric w.r.t. 0. I wonder if an analytical ...