Questions tagged [numerics]

Question 1

I have this recurrence formula for $\pi$ with convergence order $2m+1$: $$ x_{n+1} = x_n + \sum_{k=1}^{m} \left[ (-1)^{m+k} \cdot \frac{1}{k} \prod_{\substack{j=1 \\ j \ne k}}^{m} \frac{j^{2}}{k^{2} - ...

Question 2

I am debugging some code I wrote and I am starting to question if I have a theoretical limitation. Assume I construct a tensor-product-real-fourier basis as $$ \phi_{i,j,k}(x, y, z) = \phi_i(x) \phi_j(...

Question 3

I implemented a paper that creates a sparse voxel oct tree hierarchy of cubes. Each cube is treated as a compact domain with a functional basis. The paper uses Legendre polynomials as the basis for ...

Question 4

Math preamble I am trying to create a functional basis for $[-1, 1]^3$ in $\mathbb{R}^3$. To this effect I take the real expression of the fourier basis and index it with an integer such that $$\phi_i(...

Question 5

I'm trying to code the path integral simulation described in this article. I followed all the instructions, but no matter how hard I try, my program won't work. Could someone please explain what I am ...

Question 6

I’m running computational fluid dynamics simulations where small perturbations lead to divergent trajectories (chaotic systems). Even with fixed seeds, compiler differences and rounding lead to ...

Question 7

Chaotic systems like weather models are extremely sensitive to initial conditions, and computers can only store finite precision. What algorithms or statistical techniques are used to keep results ...

Question 8

I have the occasion to be reviewing some of the "basics" of the finite element method. In particular, I am interested in several technical details related to the combination of the FEM with ...

Question 9

Taubin's matrix is a very useful mathematical fact used to extract curvature information out of data. Originally it was developed for meshes, but I know for a fact that it's commonly used in point ...

Question 10

What is the significance of having an explicit symbolic error formula for polynomials, instead of relying on traditional numerical error bounds? Take a simple example: integrating the function x² over ...

Question 11

I am implementing a 2D elasto-plastic phase-field fracture model under plane strain conditions using a staggered approach. The solver iteratively updates displacement and phase-field damage until ...

Question 12

I wanted to try numerical analysis of a chaotic system. So I decided to write my own code for the Poincaré section of a double pendulum in Python. The code works and the Poincaré section should be ...

Question 13

Say, for example, that you have a Gauss-Legendre quadrature rule. This rule is only defined for the interval $[-1,1]$. Say you have a different interval $[a,b]$. The mapping between the intervals is ...

Question 14

I am attempting the following. I have a method to produce one dimensional Legendre polynomials, I have a function from $\Phi: \mathbb{R}^3 \rightarrow \mathbb{R}$. The idea is to construct a three ...

Question 15

I have the function $f(x)=\tanh(x)$. I also know its derivative $f'(x)=\left(\mathrm{sech}(x)\right)^2$. I create a new function $g(a,b)$ defined in this way: $$g(a,b)=\left\{\begin{aligned}\frac{f(b)-...

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Questions tagged [numerics]

Efficient computation of $\sin(k x_n)$ in iterative algorithms

Suitability of tensor product fourier basis for a function

Good functional basis for tiled grid?

Issues deriving a periodic basis in r^3

Numerical Simulation of Path Integral

What are the current best practices for reproducibility when simulations depend on chaotic systems?

How do large-scale simulations maintain numerical stability when chaotic systems amplify rounding errors?

Does the Aubin-Nitsche trick apply for time-dependent PDEs?

Estimating Taubin's matrix ina point cloud

What’s the advantage of knowing the exact error in polynomial integration over estimated bounds?

Why does my elasto-plastic phase-field fracture model fail at low H/E ratios (linear hardening) and in all nonlinear hardening cases?

Poincaré section for double pendulum code improvement

How do you handle weights in a quadrature rule after shifting and scaling?

Projecting function on an orthogonal basis always yielding spheres regardless of input

Catastrophic cancellation in difference quotient of tanh

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