Questions tagged [numerics]
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879
questions
0
votes
1answer
72 views
Numerical integrator for $a'(t)=e^{-a(t)}f(t)$
Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ ...
0
votes
0answers
24 views
How to efficiently perform this 2D integral in Quadpy?
I need to integrate a function defined in 2Dims (z and radius r), for which I don't have an expression.
I can just query the ...
1
vote
1answer
93 views
Accuracy gap for apparently stable solution
I was reasoning about the behaviour of the methods I'm using for my simulation and I noticed that, considering $h_s$ as the timestep over which I have unstable solutions and $h_a$ as the timestep ...
0
votes
0answers
35 views
Blown-up iterates in Gauss-Newton method
I am working on a non-linear least squares problem with standard form, in which I need to calibrate a parameter vector $\Theta$ to a set of inputs $\mathbf{x}$ and outputs $\mathbf{y}$:
$$\begin{align}...
1
vote
3answers
146 views
Linear solver recommendation(s) for small problems
I am interested in solving many linear systems $Ax = b$, where $A$ is symmetric positive definite and small (i.e. less than 25,000 rows) --- $b$ will be changing.
We can assume that $A$ arises from ...
0
votes
0answers
32 views
Error using scipy.integrate.solve_ivp: index error: the index 0 is out of bounds for axis 0 with size 0
I am recently working on the code based on the stick-slip phenomenon in Python. It's the stick-slip oscillator (chapter 3.4) in https://www.sciencedirect.com/science/article/pii/S0888327020301205.
And ...
1
vote
0answers
35 views
discretization of advection diffusion with variable coefficients
I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes
$$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
2
votes
0answers
22 views
Errors in Integral Estimate of Gaussian using Trapezoidal Rule
I'm trying to estimate the percentage error in computing the integral of a Gaussian via composite trapezoidal rule versus via an exact formula. To do this I've generated a gaussian with mean 0, ...
0
votes
1answer
47 views
finding boundary conditions when transforming a higher order ode to system of first order ode
given the following ODE:
$$\frac{d^{4}w}{dx^{4}} + B\frac{d^{2}w}{dx^{2}} = 1$$
with boundary conditions $w(0) =0 , w(1) = 0,w'(0) = 0,w'(1) = 0$
its possible to solve analytically but I am attempting ...
0
votes
1answer
46 views
High Running Time and Suboptimal Accuracy of 2D Wave Equation Solver with Finite Differences
Im trying to solve the following 2D wave equation: $$u_{tt} = u_{xx} + u_{yy}, \hspace{3mm} u(x,y,0) = \cos(4 \pi x) \sin(4 \pi y), \hspace{3mm} u_t(x,y,0) = 0$$ with the periodic boundary condition ...
2
votes
1answer
57 views
Initial condition precision
Is there a way to have an estimate of the error propagated on an ode numerical solution by the error of the initial conditions? I suppose this depend on the numerical method used and on the problem ...
0
votes
1answer
51 views
2 point BVP solver: how to compute errors
Background
I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/
I have build my own solver in Python to solve the 2 point BVP:
$$
\epsilon u''+u(u'-1) =0 , \\
u(0)...
1
vote
0answers
40 views
Solving two field Schrodinger-Poisson system numerically
I want to solve the system of Schrodinger-Poisson equations numerically:
\begin{align}
\chi_1''(r) + \frac{2}{r}\chi_1'(r)&=2U(r)\chi_1(r) \\
\chi_2''(r) + \frac{2}{r}\chi_2'(r)&=2\...
0
votes
1answer
104 views
Trouble Implementing 1d Wave Equation Finite Difference Solver
Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
2
votes
1answer
124 views
Solving Cahn-Hilliard equation using semi-implicit Fourier spectral methods
So, I have written both a C and python code to solve the 2D Cahn-Hilliard equation:
\begin{equation}
\frac{\partial c}{\partial t} = \nabla^2\left(c^3 - c - \kappa\nabla^2c\right)
\end{equation}
...
-2
votes
1answer
65 views
Forward Euler Adaptive Step Size Stability
Given
with a generalization using adaptive times-stepping as
then is it still reasonable to assume that to ensure stability of the Euler’s forward method we need the growth factor for all n to be ...
1
vote
0answers
35 views
Energy cannot decent during optimization despite non-zero gradient
Assume we have an (at least) 2nd-order differentiable energy $f(x), x\in R^n.$ And $n$ is very big. Mathematically, I think it is impossible to find a point $\bar{x}$ where the energy cannot be ...
-1
votes
1answer
55 views
illegal use of ODEINT
given the following system:
$$\frac{dP}{dt} = \alpha P(1-\frac{P}{K}) - \beta P I$$
$$\frac{dI}{dt} = \beta P I - \rho I$$
how do I solve the system numerically. as when I attempt to solve this is the ...
3
votes
0answers
45 views
Difference between wave vector and density matrix in numerical calculation of Schrödinger equation
I solved Schrödinger equation for a following tow-level time-dependent Hamiltonian numerically in two ways:
...
1
vote
1answer
117 views
Drawing saddle node bifurcation diagram for a non-linear ODE in Python
I'm trying to draw the bifurcation diagram of the following ODE,
This ODE leads to a saddle-node bifurcation (see wiki)
However what I get is not exactly right. There's a lot of "noise" as ...
0
votes
0answers
86 views
Numerov method for solving Schrödinger equation
I have just begun learning computer science to apply it to Physics and I am trying to write a code for solving Schrödinger's equation of the harmonic oscillator (setting $V=\frac{x^2}{2}$) in one ...
0
votes
1answer
143 views
Perturbation problem using Runge-Kutta 4
I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
1
vote
1answer
102 views
Reference request: C++ and numerical analysis book
I'm a master student with a good Numerical analysis background. I'm going to do a master thesis in the same subject, but I need to use C++ since my advisor loves it, and I also believe it's the best ...
1
vote
1answer
72 views
Calculate the arc length of a Steinmetz curve numerically
I'd like to know the length made by the intersection curve of two orthogonal cylinders of different radii a and b where a > b >0.
I came across this post that provides a solution with an ...
-2
votes
1answer
39 views
Solving differential equation by setting vectorization `on` in MATLAB
This is a follow up to my previous question posted here.
I've set up an ode system in MATLAB and I'm trying to vectorize the code to increase the speed of computation.
The follow is the code for my ...
2
votes
1answer
201 views
Comparing numerical solutions with very different time grids
I've read an article (Long-term integrations and stability of planetary orbits in our Solar system) in which the authors solved the problem of the absence of an analytical solution for the solar ...
0
votes
1answer
51 views
Solving differential equation by specifying jacobian pattern
This is a follow up to my previous question posted here
I'm trying to construct the sparsity pattern of the jacobian matrix to speed up the computation of a large system of odes. The following is the ...
2
votes
0answers
147 views
Best way to compute given functional with accuracy:
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1},
$$
Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
0
votes
0answers
63 views
Sparse linear solver in fortran working with REAL16
I need some (direct) sparse linear solver for fortran, which works with REAL16 data type. Any suggestions? Both Pardiso and MUMPS support only REAL8. (identical question: https://math.stackexchange....
1
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0answers
36 views
Are FDS scheme and ROE-FDS scheme the same?
In SU2 documentation about the numerical schemes (https://su2code.github.io/docs_v7/Convective-Schemes/) FDS is mentioned as a "standalone" method.
Yet, when looking online for more ...
3
votes
1answer
94 views
Anomaly in 2-body simulation error
I've generated a solution for a planet motion around the Sun (placed at the reference frame center). I wanted to test what would I get for the error of the method I'm using (Runge Kutta with 4 stages, ...
0
votes
0answers
61 views
Shape recognition of tangientially connected polygons using openCV
Lately, I have encountered the following problem, namely how to recognize the shapes of particular polygons. I have realized that quite straightforward to do so when the polygons are separated by ...
0
votes
1answer
98 views
Numerical solution of ill-conditioned differential equation
I want to solve the following Cauchy problem
\begin{equation}
y' = y^2 + \frac{t^4 - 6t^3 + 12t^2 - 14t + 9}{(1+t)^2}
\end{equation}
with initial condition: $y(0) = 2$ for $t \in [0,1.6]$ using a 3 ...
3
votes
1answer
63 views
How does the diffusion of a finite volume method with a WENO scheme compare with that of spectral methods?
I know that, in general, finite volume (FV) methods are more (numerically) diffusive than spectral methods. However, I can't find any information on how the advection scheme changes that.
For example, ...
-1
votes
1answer
67 views
Numerical solution of the advection equation with Crank–Nicolson finite difference method
I need to implement a numerical scheme for the solution of the one-dimensional advection equation
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
3
votes
2answers
156 views
Using the BDF and RK4 methods to solve this coupled system of ODEs in C++
I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ...
1
vote
0answers
53 views
Book recommendation on numerical methods for solving Integro-Differential equations
I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
1
vote
1answer
152 views
Trouble with backwards time integration in Python
I am struggling with a rather basic numerical integration task: Using Python's scipy.integrate.solve_ivp module to integrate an ODE sytem backwards in time. As a test, I am using the following ODE ...
6
votes
0answers
98 views
A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms
Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit.
I am studying the paper by Putnam ...
1
vote
2answers
99 views
Software to build a mesh of a surface from points on the surface
I have a set of points $(x_i,y_i,u(x_i,y_i))\in\mathbb{R}^3$, $i=1,\dots N$, over a surface $S$ (from experimental data). I need to calculate the integral of a function $F$ over that surface.
If the ...
1
vote
1answer
29 views
Numerical stability of taking the `mean` of outputs from the simulation of a discrete stochastic dynamical system
I am writing a simulation for a discrete stochastic dynamical system. Since the simulation is stochastic, I need to run the simulation multiple times and then average the values of each timestep. I ...
2
votes
1answer
71 views
How to include negative number in the log-sum-exp?
I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...}
If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
1
vote
1answer
85 views
preconditioner for $u''(x)=\sin(x)$
I am interested in finding preconditioner to solve the problem for one dimensional problem $u''(x)=\sin(x), u(0)=u(1)=0$ using Dirichlet-Neumann method.
The preconditioner $M$ coming from Dirichlet-...
1
vote
0answers
63 views
Upper bound on condition number in linear preconditioning
I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia:
Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
2
votes
1answer
104 views
Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular
I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
1
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0answers
19 views
Problem with recursive implementation of Subspace Iteration method in Numpy
I am having trouble with implementing the method of subspace iteration to find the eigenvalues and vectors of a random, symmetric matrix, A that is mxm with m = 10. ...
1
vote
0answers
273 views
Minimax optimization with an oracle
I have an optimization problem of the following form: $$\min_y\left[\max_x f(x,y)\right].$$ It is fairly straightforward to minimize $f(x,y)$ over $y$ with $x$ fixed, and similarly to maximize $f(x,...
0
votes
1answer
137 views
Conserve energy by message passing?
There are $N$ particles with positions $x_i(t)$ and velocities $v_i(t)$ and mass 1. There is a potential function $U_{i,j}(x_i, x_j)$ between each pair of particles, which is $0$ unless the particles ...
3
votes
1answer
279 views
Time Reversibility of Velocity Verlet Algorithm
I'm very new to computational Physics and am finding conflicting statements on whether the velocity Verlet algorithm, defined as:
$\begin{align}
x_{n+1} &= x_n + v_n \Delta t + \frac{1}{2} a_n \...
1
vote
0answers
38 views
How can I improve the accuracy of the calculation of the magnetic field in Gmsh/GetDP?
I need to calculate the magnetic field along a straight line in proximity of an array of 6 magnets. I used the tutorial files "magnets" included in Gmsh and I slightly modified the file in ...
