Questions tagged [numerics]
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1,096
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Scheme Solving CCZ4 Formalism Numerically
For my thesis, I have to implement the CCZ4 formalism into an existing code (I tried to type the evolution equations here but does not seem to work).
However, I can't seem to come up with a scheme to ...
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1
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136
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Computing numerical derivatives
I am trying to create a sweeping surface, for which I need the frenet frame of a curve. I am trying to compute this for arbitrary curves but for testing I am just using the parametric unit half circle....
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Solving basic barystochrone problem in python
I am trying to solve $\frac{u''}{1+u'^2} - \frac{1}{2(1-u)} = 0$ subject to $u(0)=1, u(1)=0$.
If I understand how to do this properly, I first do the variable substitutions:
$u = y$, $y_1 = y; y_2 = y'...
- 263
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Integration of a singular kernel function over a triangle
Problem:
I am currently trying to integrate a singular kernel function of the type
$$G(x,y)=\frac{\exp(ik||x-y||_2)}{4\pi ||x-y||_2}$$
which lies at the centre of a triangle, over this triangle. $i$ ...
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discrete definition of curl $ \nabla \times \vec{A}$ on a 3D grid?
I have the data for 3D vector field $\vec{A}$ (with components $\vec{A_1}$, $\vec{A_2}$ and $\vec{A_3}$) sampled on a 3D grid with integer indices i, j and k.
Assuming that only the third component $\...
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79
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Is it possible to globally solve a general coordinate-wise monotone nonlinear system 3x3?
Consider a system
$$
F_i(x, y, z) = 0, \quad i = 1, 2, 3
$$
with $F_i$ monotonic w.r.t. $x, y$ and $z$.
The system 2x2 can be easily solved with alternating direction method that will find all its ...
- 281
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Numerical integration of a rapidly varying complex exponential
I have a function $f : \mathbb{R}^2 \mapsto \mathbb{R}^+$ and I wish to numerically evaluate the integral below over a finite domain $\Omega \subset \mathbb{R}^2$
$$
I = \int_\Omega e^{i k \cdot f(\...
- 131
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98
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Compact Finite Differences for the Heat Equation with Robin Boundary Conditions
I am trying to solve the Heat equation with Robin Boundary condition:
$$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$
for $ 0\leq x\leq1$ ...
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195
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Nonlinear Robin boundary condition involving square root
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
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computing higher order derivatives with linear elements
Consider the following equation on $(0,1)$, with Dirichlet boundary conditions on both ends.
$$
\frac{d}{dx}\left(k(x)\frac{du}{dx}\right) = 0
$$
Let us solve this using simple linear finite elements. ...
- 758
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463
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Solving a 2nd order complex-valued matrix differential equation in Python
I am trying to solve the following complex-valued matrix differential equation backwards (i.e. not starting at $r=0$, but rather at $r > 0$):
$F'' = 2ikF' + VF$.
Here $F=F(r)$ and $V=V(r)$ are 2x2 ...
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Numerical Method for Multivariate Inversion Formula
For my research, I need to evaluate the density of a random vector $\boldsymbol{X} \in \mathbb{R}^p$ using the multivariate inversion formula. Let the density function of $\boldsymbol{X}$ be $f (\...
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Finite-difference produces a derivative off by one order
I have a nonlinear 2nd order boundary value differential equation where I used finite-difference method (central finite-difference) to solve it.
$$z''(x)-\frac{\frac{1}{100} z(x)^4 \left(2 z'(x)^2+12\...
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How do the current FEM opensource libraries compare?
Almost all FEM libraries are good enough, but I want to start with a FEM package and stick to it for some time. Instead of trying all of them, or going with what everyone else is using, I want to ...
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Wrong Boundary Conditions Result Using Wavelet Collocation
I have a functional $S$,
$$S = \int_{x_0}^{x_b} dx \frac{1}{z(x)^d} \sqrt{1 + \frac{z'(x)^2}{f(z)}}, \qquad f(z) = 1-\left(\frac{z(x)}{z_h}\right)^{d+1}
$$
where $d=3$ is the dimension and $z_h$ is ...
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3
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How can I numerically integrate the Kepler problem?
I tried to solve a simple Kepler problem numerically.
I have discrete time steps, a starting position $(x_0,y_0)$ and starting velocity $(u_0, v_0)$.
I used this iteration by calculating the forces ...
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Given a set of 1d-points, find the most probable periodicity that models the points (with possible omissions) as equidistant occurences
I try to detect interference fringes in a bunch of pictures. I projected on one axis, and I was able to detekt the peaks that indicate one of the fringes.
So now I'm having a list with points (e.g. $(...
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How to exploit QR factorization implicitly
I meet a problem when I try to develop an iterative method for discrete inverse problem
$$Ax+e=b$$
where $A\in\mathbb{R}^{m\times n}$ and $e$ is a noise. I want to approximate the true solution $x_{...
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1
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75
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blown up solution for linear advection in upwind method with finite difference
I am going to solve this advection equation regarding the flow simulation of an energy tower
\begin{equation}
Y_t+ v(t) Y_x =0
\end{equation}
with the following boundary conditions which depend on ...
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3
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1
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Scipy solve_ivp sensitivity to random phase shifts
I am trying to solve a coupled system of ODE's using the solve_ivp function from scipy. The general form of the equation is given via
$$\dot{y}(t) = M(t)y(t).$$
The time dependence of matrix is ...
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221
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Accurate computation of logbinomial(x,y)
I need to compute the logarithm of the binomial coefficient,
$$\log\binom{x}{y} = -\log\mathrm{B}(y + 1, x - y + 1) - \log(x + 1)$$
accurately, where $\mathrm{B}(x,y)$ is the Beta function.
I'm aware ...
- 1,729
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180
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Solve discontinuous ODE with lsode
I am trying to solve a discontinuous ODE using the lsode solver. I tried setting the t_crit parameter to specify the time where the discontinuity is present, but it ...
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1
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130
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Deterministic SIR metapopulation model and coupling behavior
Context:
I am trying to reproduce a figure from Keeling 2007 that illustrates time lags that can occur between the peaks (maximum) of the infected solutions for two subpopulations of a metapopulation ...
- 253
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1
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217
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solve_ivp not giving out any output and no error while souple 3 coupled 2nd order ODES
Please, someone tell me what is wrong in my code it does not give any outputs ( No plot nor print).
The code is as below:
...
0
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1
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98
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Averaging oscillatory data
I have an oscillatory data generated vs time as shown below. Essentially, I want this data to be averaged and free of any oscillations. I am not satisfied with the results from a simple moving average ...
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Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?
recently I need to solve a 2D steady state PDE equation.
It’s not time dependent, and the only two independent variables are z and r direction.
So far for this solution, I was thinking using Method ...
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Dispersion of Runge-Kutta methods when applied to systems of ODEs
I am interested in computing the dispersion / phase error(s ?) of an (explicit) Runge-Kutta method when applied to a linear system of ODEs
$$ u'(t) = A u(t). \tag{1} \label{1} $$
To begin, consider ...
- 1,083
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Python bifurcation diagram of seasonally forced epidemiological models
TL:DR
How can one implement a bifurcation diagram of a seasonally forced epidemiological model such as SEIR (susceptible, exposed, infected, recovered) in Python? I already know how to implement the ...
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Numerical precision on tricky coupled nonlinear boundary value problem on infinite interval
I am trying to solve with high precision the following coupled system $(f,h)$ on $[0,\infty]$:
$$-h''-\frac{1}{r}h'+\lambda_c h(f^2-1)+4\lambda_h h^3=0$$
$$-f''-\frac{1}{r}f'+\frac{1}{r^2}f+ f(-\...
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Boundary value problem solver fails on trivial case
I am trying to solve a boundary value problem on $[0, \infty]$, using scipy's scipy.integrate.solve_bvp and I am seeing that the solutions are not converging even ...
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Correctness of direct numerical solution of ill-conditioned linear system
To what extent can you put trust in a numerical solution obtained by direct solver for an ill-conditioned linear system? In other words, how can you test the solution? Dropping it into the system says ...
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How to find armijo step length for a neural network?
The armijo step length formula states that
f(x+lr*descent_direction) <=f(x)+c*lr*f_gradient*descent_direction
In the above formula lris the learning rate and f ...
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How to calculate Landau levels for a system e.g., Graphene numerically?
I am currently trying to calculate Landau levels for any system numerically in python. But, I am clueless about where to start rather, how to construct the Hamiltonian. Can anyone help?
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Differential equation for radioactive cooling in fortran
Today I'm trying to evaluate this differential equation for internal energy in a gas in Fortran:
$$
\frac{du}{dt} = - \frac{n_H^2}{\rho}\frac{\Lambda}{n_H^2}
$$
Where nH is the density of hydrogen in ...
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What are the various methods in adding an additional constraint to the quadratic spline interpolation problem
I am taking a class on numerical analysis. While the professor was deriving the theory behind quadratic splines, the professor mentioned that a quadratic spline function has the form:
$$
p_{i}(x)=a_{i}...
- 153
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Rectangular problem to use for benchmarking a large dense linear solver?
Can anyone suggest a rectangular problem I can use for evaluating a large dense linear solver? ($m,n>1000$) IE, ideally something that's easy to download and has been evaluated for some solvers ...
- 2,655
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How to calculate a product of two real functions of large, but opposite magnitude?
I need to evaluate a product of two real functions, namely $F(x)\cdot G(x)$. The function $F(x)$ is a diverging and obscure member of scipy.special, while $G(x)$ is a gaussian. While the product is a ...
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Efficient and stable QR factorization of partially orthonormal matrix
Let $U \in \mathbb{C}^{m \times n_U}$ be an orthonormal matrix, let $A \in \mathbb{C}^{m \times n_A}$, and $m \geq n_U + n_A$. I want to compute a QR factorization $X = \left[U A\right] = QR$, with $Q ...
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Error in implementation of Crank-Nicolson method applied to 1D TDSE?
Some context, I've posted this question on physics SE and stack overflow. The former had nothing to offer, the latter had a great commenter that agreed with the phase looking off being one of the ...
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Inaccurate results of integration using scipy solve_ivp
I am trying to use solve_ivp to solve the following 1st order ODE:
$$ \frac{d \rho}{d z} = \frac{m \theta}{(1+\theta z)} \, \rho, $$
subject to $\rho(z=0)=1$, where ...
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What are the known or used numerical methods for integration over the sphere $S^2$ ? and what about over $S^3$?
What are the numerical methods available to compute integrals over $S^2$, for the particular integration :
$$\int_{S^2} f(\omega)\,d\sigma(\omega)\ , \quad \text{ where $d\sigma$ is the usual measure ...
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finding discretization error in Burger equation
I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
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Why aren't Krylov subspace methods popular in the Machine Learning community compared to Gradient Descent?
Historically, iterative methods for solving relatively simple-structured systems $Ax=b$ with $A$ being a $4\times 4$ matrix or to find the eigenvalues of that matrix assuming in both problems that $A$ ...
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Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?
I am looking for possible numerical methods to solve the PDE
$$u_t+c u_x= \frac{-c}{x}u$$
I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
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A priori FEM estimates without $H^2$ regularity
In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's ...
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Efficiency of developing PDE solvers using sparse matrices versus loops
I am new to solving PDEs, but have been looking at different implementations of finite difference and finite volume schemes. One thing I have noticed in different implementations is that some ...
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A question on the Poisson equation with Neumann and periodic boundary conditions on a rectangular region
I am trying to solve the following PDE by using finite difference
\begin{eqnarray*} \Delta u&=& f ~~on~~(0,1)\times(0,1)\\
\frac{\partial u}{\partial y}(x,0)&=&0=\frac{\partial u}{\...
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Reference request for finite elements theory
Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how:
without needing isoparametric ...
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0
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FEM applied to heat equation and incompatible conditions
Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$
with $g$ NOT vanishing on the boundary. If I ...
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2
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Secant Method for finding $\sup f^{-1}(0)$
Let $f \in C^0[0, 1]$, and suppose $f \ge 0$. How can I compute $\sup f^{-1}(0)$ efficiently?
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