New answers tagged numerics
6
votes
Accepted
How to design a sin and an arcsin function such that arcsin(sin(x))=x, where x is a finite precision floating point number
That seems impossible to do with a small numerical error, because of cardinality reasons. Consider for instance the much simpler case of the function $f(x) = x^2$ over $[0,1]$. This function maps $[0,...
- 10k
3
votes
Storing Raw Simulation Data or Truncated Data?
I would recommend keeping the computation in your original precision and reducing the accuracy just when you write, unless the computation is too slow. If you reduce the precision of the computation, ...
- 565
2
votes
Finite difference approximation error
Take the Taylor series and re-arrange it to be
$$
f'(x) - \frac{f(x+h)-f(x)}{h} = - f''(x) \frac{h}{2} - f'''(x) \frac{h^2}{3!} - f^{(4)}(x) \frac{h^3}{4!} + \ldots
$$
Assuming that $f^{(n)}(x)$ is ...
- 1,741
3
votes
Chebyshev/Lagrange polynomials in spectral methods
To complete david's answer:
References: Canuto et al., Spectral Methods
Fundamentals in Single Domains
We consider the Burgers equation
\begin{align}
\text{advective form}: \qquad \frac{\partial u}{\...
- 875
4
votes
Accepted
Chebyshev/Lagrange polynomials in spectral methods
Your understanding is perfectly fine, except for the last statement that Lagrange polynomials turn out to be a more suitable choice. In fact, both methods, the modal and the nodal expansion, have ...
- 2,937
3
votes
Accepted
Faster than forward substitution?
Let's assume that the size $m$ of the individual blocks is fixed, but that the number of blocks $n$ grows. Then the one-step-at-a-time algorithm takes $O(m^3n)$ operations if you chose to invert the ...
- 52.4k
3
votes
Faster than forward substitution?
I am not an expert in the field, but since there are no other answers at least I can suggest you a keyword to start a literature search: "parallel-in-time integration methods". This is a ...
- 10k
4
votes
Accepted
C^1 continuous element for a triangle?
tl;dr: defelement link
There is an equivalence between all polynomial basis which span the same space so in theory you can use the standard monomial basis to fit the various parameters using a ...
- 1,741
3
votes
C^1 continuous element for a triangle?
The $C^1$ elements are all very challenging to implement, which is why you don't see them get used very often.
If you just want to see what the basis functions are on the reference (or other) ...
- 9,013
2
votes
Accepted
FreeFEM++ converting equation into code
In the piece of code that you mentioned from the FreeFEM documentation, the Galerkin Finite Element Method (FEM) is used for the spatial discretization and the Finite Difference Method (FDM) is used ...
- 194
1
vote
Accepted
Solving a boundary value problem with variable number of coupled equations
This is a situation where some variables only live on parts of the domain. This is not so different from multiphysics problems -- say, a fluid-structure interaction problem where velocity and pressure ...
- 52.4k
1
vote
Solve 1st order ODE in using `scipy`
One preliminary comment is that supplying at the initial point both $y$ and $y'$ is redundant. If $y'(t_0)$ is given then obviously the ODE fully defines $y(t_0)$. On the other hand, if $y(t_0)$ is ...
- 2,230
2
votes
What are the benefits of cutting by half the number of multiplications needed to calculate n?
What are the benefits of cutting by half the number of multiplications needed to calculate n!
The rest of your question doesn't really refine what you're asking and it's too long for me to want to ...
- 3,586
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