20
votes
What programming language should I choose and why?
You should definitely check out Julia. Julia is a programming language which is similar to Python or MATLAB but utilizes a strong type-inference algorithm + JIT in order to optimize code. If types can ...
- 12.2k
19
votes
Accepted
Use of machine learning in computational fluid dynamics
It's a long-running joke that CFD stands for "colorful fluid dynamics". Nevertheless, it is used -- and useful -- in a wide range of applications.
I believe your discontent stems from not ...
- 12.1k
13
votes
Accepted
Why is the central difference method dispersing my solution?
I'll write the equation short as $$\ddot x(t)+c\dot x(t)=a(t,x(t))$$ to separate the "easy" linear parts from the non-linear and forcing terms.
On the first method
The claimed order of the ...
- 5,194
12
votes
Accepted
When and when not to use automatic differentiation
Given code that computes a function $f(x)$, automatic differentiation tools produce a code that can compute $f(x)$ and its derivatives at the same time. Solving a differential equation is an entirely ...
- 18.4k
12
votes
Finite-difference software for solving custom equations
I'm going to assume since you mention electrodynamics that you're interested in PDEs.
You've already mentioned FEniCS in your comment. FEniCS offers a domain-specific language (DSL) called UFL.
This ...
- 9,813
11
votes
What programming language should I choose and why?
What is it you want to achieve?
If you want to develop methods/algorithms you might prefer a language that is flexible, and that you are familiar with. As you stated in your question, the Fortran ...
- 549
11
votes
What programming language should I choose and why?
Start simple. Learn Python.
I have been paid to write programms for over forty years and I have used all the languages mentioned in other answers (except Julia - I had never heard of it before now). ...
- 211
11
votes
Is using iterative methods to solve a linear system always superior to inversing the matrix?
First off, there are basically no scenarios where one would ever actually compute and store $A^{-1}$ in memory, even for small problems. An LU factorization offers both superior efficiency and ...
- 1,832
10
votes
Numerical derivative and finite difference coefficients: any update of the Fornberg method?
Overview
Good question. There is a paper entitled "Improving the accuracy of the matrix differentiation method
for arbitrary collocation points" by R. Baltensperger. It's no big deal in my opinion, ...
- 3,042
10
votes
Non-hermitian discretizations in quantum mechanics
If your discrete equations represent your problem properly you should have the right eigenvalues, although numerically you can get a small imaginary part when solving the eigenvalue problem.
As an ...
- 8,312
10
votes
Why FVM can handle unstructured meshes while FDM cannot?
The difference between finite volumes and finite differences is really more about the form of the equations solved. In typical FV methods, the conservative form is ...
- 534
9
votes
Accepted
Correct eigenfunctions of Laplace operator by Finite Differences
You should specify the eigenvalues you want with which="SM", for example.
Check the following snippet. I also changed the solver, since your system is symmetric.
<...
- 8,312
8
votes
ODE $x''(t)+\eta x'(t)+x(t)=0$ with the $\eta$ extremely small
The problem you have is what is called "stiff": it has two time scales, namely one for the oscillation (which is ${\cal O}(1)$) and one for the damping (which is ${\cal O}(\eta^{-1})$). If $\eta$ is ...
- 53.6k
8
votes
Accepted
Solving Ax = b with sparse A and sparse b
When looking at the solution of your system, you will find that almost all entries of $x$ are nonzero although the right-hand side is "sparse". Hence, whatever algorithm you use, it'll have to visit ...
- 3,108
8
votes
Why is my simulation of a first-order wave equation not stable?
Numerical solution of the advection equation with centered differences in space and forward Euler in time is unconditionally unstable. So the behavior you are seeing is expected.
Here is a nice ...
- 5,934
8
votes
Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
I think you're probably seeing artifacts that are due to numerical dispersion. In brief, in the discrete case different (spatial) frequencies of a wave function will propagate at different phase/group ...
- 4,794
8
votes
Accepted
Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?
The preconditioner for the FDM method that corresponds to the one you outline for the FEM (i.e., the Sylvester-Wathen approach) will still contain the Schur complement of the FDM matrix. The Schur ...
- 53.6k
8
votes
Accepted
Computation of diffusion time
It's easy to derive that equation from Fick's law. You have this diffusion equation as:
$$\frac{\partial C}{\partial t} = D \nabla^{2} C$$
The mean square displacement weighted by the concentration ...
- 2,322
8
votes
Finite-difference software for solving custom equations
You might want to check out DifferentialEquations.jl. It supports ODEs, PDEs, stochastic equations, delay equations, and basically everything else. It also has really good automatic sparsity detection ...
- 244
8
votes
Accepted
3D laplacian operator
Yes, that finite difference is correct. You can obtain it using a finite difference in each direction for the Laplace operator in each coordinate.
\begin{align}
\nabla^2 u =& \frac{\partial^2 u}{\...
- 8,312
7
votes
Accepted
Error in result of finite-difference approximation when refining
High order methods are not necessarily always more accurate than low order methods; they simply have a more rapid convergence rate. For example, if you take the Taylor expansion of a function at a ...
- 3,102
7
votes
Use of machine learning in computational fluid dynamics
I think you are mixing a couple different ideas that are causing confusion. Yes, there are a wide variety of ways to discretize a given problem. Choosing an appropriate way may look like "voodoo" ...
- 4,607
7
votes
Derivatives Approximation on non uniform grid
So not quite sure what you're missing, but here's how you go about doing this sort of thing.
So first, I am assuming this is 1D due to your description. Second, I'm assuming you know the relationship ...
- 3,878
7
votes
ODE $x''(t)+\eta x'(t)+x(t)=0$ with the $\eta$ extremely small
Let us write the equation in state-space form
\begin{equation}
\frac{d}{dt}\begin{bmatrix}x_{1}(t)\\
x_{2}(t)
\end{bmatrix}+\begin{bmatrix}\eta/2 & -\omega\\
\omega & \eta/2
\end{bmatrix}\...
- 2,455
7
votes
Advection equation with finite difference: importance of forward, backward or centered difference formula for the first derivative
Okay. Let's begin with the first situation.
Your equation is:
$$ \frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}=0\tag{1}$$
$\textbf{Previous comments}$
There are plenty of webs that ...
- 1,628
7
votes
Accepted
What is the maximum attainable accuracy with a given set of $\alpha,\beta$?
If you are looking for general accuracy of your stencil, it could be extracted by using Taylor expansion. Basically, if I want to write down your stencil explicitly, it contains terms of $u(x_{i}-2h)$,...
- 2,322
7
votes
Time complexity of numerical finite differences
As pointed out in the comments, the cost of evaluating $f$ is critical, and in most practical cases will be the dominant cost. Lets suppose it takes $C$ operations to evaluate $f$. For nontrivial ...
- 999
6
votes
Accepted
Accuracy of finite differences
I think you probably already know all this, and maybe it's just the wording that is confusing. I'm going to rephrase what they're doing to make it more explicit. It's exactly the same calculation of ...
- 11.4k
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