Questions tagged [numerics]
The numerics tag has no usage guidance.
574
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Questions on the theory of distributed numerical algebraic computation
I'm trying to build a pure python distributed numerical algebra computation kernel based on GPU. but after I've learnt most of the software engineering, I realise that I'm seriously lacking in ...
0
votes
1
answer
78
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Using Crank-Nicolson to solve Non-Linear Schrödinger equation in Python
I aim to solve the (non-linear) Schrodinger equation using the Crank-Nicolson method in Python. Here are my two functions.
...
0
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0
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35
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recommendation on some papers/books about frontal solver used in FEM
I'm reading a program about computational plasticity, this program use frontal solver to solve the program, but I'm not familiar with frontal solver even after reading some papaers, so could you ...
- 21
0
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1
answer
43
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How do you build a polyharmonic discrete system?
Polyharmonic equations, to my understanding, are defined as:
$$\Delta ^k u = 0$$
i.e. one repeatedly applies the laplace operator to the function a certain number of times and the result must be 0.
...
- 263
0
votes
1
answer
53
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Discretization of generalized kinetic term in 2D Poisson partial differential equation
A typical 2D Poisson PDE is given as
$$\nabla^2\varphi(x, y)=f(x, y)$$
where the Laplacian term, $\nabla^2\varphi$, can to some degree be interpreted as the kinetic energy (given proper scaling) (...
- 53
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35
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Perfect Conducting (PC) and Perfect Insulating (PI) boundary conditions for magnetic induction in MHD problems in cylindrical coordinates
I understand the computational implementation of the PC and PI bcs in cartesian case. In terms of the magnetic field components for PI, we have $B_x = B_y = B_z = 0$, and for PC, we have $\partial_x ...
- 1
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0
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32
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Constructing generalized Laplacian matrix?
I am staring intently at this paper by Botsch and Kobbelt.
In particular, I want to make the matrix specified in equation 5. I am trying to understand the specific computations I must instruct a ...
- 263
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votes
1
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75
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Geometrical Nonlinearity in Abaqus
In Abaqus, there is an option NLGEOM to turn on the geometrical nonlinearity. But I'm not clear what it does specifically. Because it also works with UMAT written for small strain formulation, i.e. ...
- 229
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1
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61
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Solve bivariate polynomial system
Given a bivariate polynomial system with variables $(x, y, z)$ like
(1) $ f_1 = x * a_1 + y * a_2 + z * a_3 = 0 $
(2) $ f_2 = x * a_4 + y * a_5 + z * a_6 = 0$
(3) $ f_3 = x^2 + y^2 - 1 = 0$
how do I ...
- 11
5
votes
1
answer
727
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Taking derivative using FFT
I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code
...
1
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2
answers
90
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Can I find sines or cosines of rational parts of π without using radians? If not, how do I reduce the errors caused solely by the transcendence of π?
this may be irrelevant for people who need fast code. But for me it's just the opposite -- i.e. in the specific situations when I know that the extra time allows me to make my calculations more ...
- 13
3
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2
answers
130
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Solving systems of advection-diffusion-reaction equations with finite element methods
I have been doing a lot of self-study on numerically solving PDEs so that I can solve system of linear and nonlinear Advection-Diffusion-Reaction (ADR) systems on complex meshes.
I have been watching ...
- 287
1
vote
0
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84
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Stability for the 2d diffusion equation
I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
- 623
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2
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446
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PhD in scientific computing to be a scientific programmer
Intro and disclaimer: this question concerns developing a career in Scientific Computing in industry, starting from an (applied) mathematics background, say an MSc. It definitely arises from my ...
- 243
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1
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83
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Deformation matrix, Math hack for stability on large simulation steps?
So there is a numeric technique for updating a deformation gradient in MPM that goes as:
$$F_{n+1} = (I + \nabla \vec v \Delta t)F_n$$
This works for small time steps but for large time steps ...
- 263
1
vote
1
answer
88
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Improved euler on hybrid methods where both time and space are discretized?
I am trying to understand how to use the improved euler method on MPM simulations.
In the kind of MPM simulation I am doing with forward euler the order of operations is as follows:
Write particle ...
- 263
3
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0
answers
78
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Verlet integration on grids or how to get better stability in hyperelastic simulation
I am using MLS-MPM to simulate both solid and fluids. It works, but the amount of time steps I must do for hyperelastic solids is absurd.
To give you some perspective, I am able to simulate just the ...
- 263
2
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2
answers
241
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How to plan convoluted measurements
I have a physical function $f(x)$ which I intend to measure. Problem is that I cannot read it directly, but through a response function $g(x)$ which is known to me with great accuracy and any one ...
1
vote
1
answer
96
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Isolating decaying solutions to nonlinear second-order ode
I need to solve a nonlinear ODE of the form
$$
\frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx}-\frac{1}{x^2}\sin(y)\cos(y)+\frac{2}{\alpha}\frac{\sin^2(y)}{x}-\sin(y)=0
$$
numerically, subject to the ...
- 11
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2
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147
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Approximating the solution of a non-linear ODE using Python
This is my first time asking a question here, so please tell me if I have made a mistake or if anything is unclear.
I am working on my high school research project on the motion of a ball falling ...
6
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1
answer
115
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Numerical artefacts in solution of spherical heat equation using FDM
I was attempting to solve the diffusion equation for a solid sphere using a naive FDM scheme. The governing PDE for the scalar concentration field $u(r,t)$ is
$$
u_t = r^{-2}(r^2 \alpha u_r)_r, \quad ...
- 454
4
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1
answer
132
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Is there a way to generate a matrix-free decomposition for a matrix-free operator?
Hypothetical question for some code that I'm writing. Suppose I have an matrix-free linear operator $A$, i.e. the only thing I know about it is the forward action $v \mapsto Av$. For simplicity, let's ...
- 141
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0
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61
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When is Lanczos tridiagonalization accurate?
Suppose that we are given a random, symmetric matrix $A$, and a random vector $q$. For concreteness, assume the dimensions of $q$ and $A$ are both $1,000$. I would like to use the Lanczos algorithm to ...
- 41
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1
answer
111
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How to numerically solve differential equations involving sines, cosines and inverses of the unknown function?
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find that the main idea is to ...
- 35
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0
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19
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MPM course, implicit updat clarifification
In the 2016 SIGGRAPH notes on MPM the authors explain how to do an implicit update in section 11.4 throguh by this formula:
It is not clear to me what the $j$ is over in this case (for eq 202), is it ...
- 263
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0
answers
77
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Why do we use modified pressure in incompressible multiphase solvers with gravity?
The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
0
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0
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30
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MPM implicit integration
Setup
I am having a hard time translating the formulas in the 2016 MPM course from siggraph into algorithmic computations.
In particular we have equations 200 and 201
$$h(v^{n+1}) = Mv^{n+1} - \Delta ...
- 263
1
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0
answers
37
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How can i study stability for a new method that solves second degree non lineara differential equations?
I developed a new method to solve this equations $\frac{d{y}^{2}}{dx^{2}}=g(x,y,\frac{dy}{dx})$ for a general g(x,y,z), wich their solution is a function f(x)=y. Obviously you need $y(x_0)=y_0$ and $y'...
- 111
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63
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Encountering blow-up when solving the one-way heat equation using Lax-Wendroff
This is my first time attempting to implement a finite difference method for a PDE in Python, and I am having a bit of trouble. The PDE I am trying to solve is as follows:
$$
\begin{cases}
...
- 153
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0
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36
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What is meant exactly by the domains in this context and how can I simulate their size?
I was working on a problem from "Computational Physics using python: Chapter 17." The problem is concerned with simulating the thermodynamics behind the spin of electrons and the resulting ...
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1
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118
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How do the navier stoke equations model materials who "forget" their original form?
Sorry for the screenshot but I don't want to try to format this on latex:
We have this annotation of the Navier-Stokes equations:
I am particularly puzzled by the viscosity/stress term. For an ...
- 263
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0
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54
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Numerical method for space fractional derivative in 1 dimension
I am very new to the subject of fractional derivatives which arise while characterizing the anomalous transport of passive scalar in turbulence.
I have found an equation of the following form, to ...
- 87
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0
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32
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Matrix for Marker and Cell grid?
I have an assignment question that reads:
Show that the combined matrix for the Marker and Cell (MAC) grid
for velocity and pressure for the steady Stokes equations is symmetric.
You can consider the ...
- 263
3
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1
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156
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Numerical scheme for the level set equation that solves inverse mean curvature flow problems
I am considering the problem of simulating the evolution of an interface given as a curve in 2D (or surface in 3D) that evolves according to a velocity specified at the interface of the form:
$$\vec{v}...
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1
vote
1
answer
100
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How to design a sin and an arcsin function such that arcsin(sin(x))=x, where x is a finite precision floating point number
As commonly known for programming on computer, if x is a finite precision floating-point number such as double/float in C language, arcsin(sin(x)) is usually not equal to x due to the numerical issue. ...
- 205
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0
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90
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Solving 2D Poisson equation with mixed boundary conditions in Python
I am trying to numerically solve the Poisson's equation
$$
u_{xx} + u_{yy} = - \cos(x) \quad \text{if} - \pi/2 \leq x \leq \pi/2 \quad \text{0 otherwise}
$$
The domain is the rectangle with vertices ...
- 119
2
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0
answers
110
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Solving 2D Poisson equation with Dirichlet boundary conditions in Python
I am trying to solve the following PDE:
$$
\begin{align*}
u_{xx} + u_{yy}
=
\begin{cases}
- \cos(x) \quad -\pi/2 \leq x \leq \pi/2, \\
0 \quad \text{otherwise}
\end{cases}
...
- 21
1
vote
1
answer
156
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Storing Raw Simulation Data or Truncated Data?
I have a simulation that can generate quite a bit of data when it runs, for example $650\cdot 400 \cdot 400$ floating point numbers. Without compression, that's a few gigabytes worth if I want to save ...
- 33
0
votes
1
answer
59
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Finite difference approximation error
I was reading Scientific Computin, An Introductory Survey, by Michael Heath. In the Example 1.11, he madr a Finite Difference Aproximation, with the usual approxination : $f’(x)\neq \frac{f(x+h)-f(x)}{...
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4
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2
answers
267
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Chebyshev/Lagrange polynomials in spectral methods
I am currently trying to familiarise myself with (Pseudo-)Spectral Methods for solving differential equations. Now, I am struggling to understand some obviously crucial concept of this approach. The ...
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Are my boundary conditions in my stiffness matrix correct?
I am trying to find a P1 Lagrange finite element solution to the following ODE:
$\begin{cases}-u''+u'+u=f~~~~~~~~\text{in}~~(0,1)\\ u(0)=1, u'(1)=0\end{cases}$
Where $f(x)=-2e^{x}+2\left(1-x\right)e^{...
4
votes
2
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226
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Faster than forward substitution?
I have a matrix of the form:
$M:=\begin{pmatrix}
S_1 & & & \\\ Q_1 & S_2 & & \\\ & ... & ... & \\\ & & Q_n & S_n\end{pmatrix}$
where the blocks ...
- 127
1
vote
2
answers
158
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C^1 continuous element for a triangle?
I am looking for an element for FEM that is piecewise $C^1$ continuous across triangles (i.e. $C^1$ continuous on the edge separating 2 triangles of the mesh).
I have heard about the Bell element:
...
- 263
3
votes
0
answers
96
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A way to solve nonsmooth stiff ODEs
Let us considered the following ODEs
\begin{align*}
\dfrac{dX}{dt} = F(X), \tag{1.1}
\end{align*}
where the unknown $X\in \mathcal{D} \subset \mathbb{R}^l$ and it can be stiff. These ODEs can be ...
- 131
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105
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FreeFEM++ converting equation into code
I am trying to solve the following problem nuemrically:
$$u_t = \Delta u + \sin t$$
To that effect I scanned the documentation of FreeFEM, the closest example I can find to my problem is the Thermal ...
- 263
1
vote
1
answer
65
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Solving a boundary value problem with variable number of coupled equations
Let's assume the equation
$$
\nabla^2u_n(\vec{r})+a_n(\vec{r})u_n(\vec{r})=\sum_{m=1}^{N}b_{nm}u_m(\vec{r}),\quad n=1,2,\dots,N,\quad \vec{r}\in\varOmega\tag{1}\label{eq1},
$$
is to be solved for $u_n$...
- 194
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Pyton get fem solving simple 2D differential equation?
I need to solve the differential equation:
$$u_t = \Delta u + \sin t$$
On a 2D domain with homogeneous Dirichlet boundary conditions. Tod o this I am trying to use the python package getfem.
I am ...
- 263
1
vote
0
answers
70
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Strategies to solve an equation with a polynomial and a numeric function
I have to solve numerically an equation of the following form:
$$
\sum_{n=0}^m c_n x^n = f(x) x^k
$$
Where the $c_n$ are real values, $k$ is an integer and $f$ can only be evaluated numerically.
The ...
- 153
0
votes
1
answer
134
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Solve 1st order ODE in using `scipy`
I've been trying to solve the following equation
$$
y(t)=-A\cdot\frac{\mathrm{d} y}{\mathrm{d} t}+B\cdot\left(\frac{\mathrm{d} y}{\mathrm{d} t}\right)^{2}+C
\\
y(t=0)=y_{0}\\
$$
where $A$, $B$, and $C$...
0
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0
answers
30
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How is the transfer function of a state-space representation numerically computed?
This question is a duplicate of this question that I asked on dsp stack exchange. However, nobody had the answer there but the question seems more appropriate on this forum (If not feel free to tell ...
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