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One example that appears in many areas of physics, and in particular classical mechanics and quantum physics, is the two-body problem. The two-body problem here means the task of calculating the dynamics of two interacting particles which, for example, interact by gravitational or Coulomb forces. The solution to this problem can often be found in closed form ...
32
First, if your undergraduates are like ours and had no prior introduction to computers, expect to spend some time teaching them how to use basic stuff like using a proper editor (i.e., not MS Word), the command line, etc.
I think the answer somewhat depends on where you set the focus of your course (or what you are required to teach). For example: How ...
27
A famous example is the boolean satisfiability problem (SAT). 2-SAT is not complicated to solve in polynomial time, but 3-SAT is NP-complete.
26
In one and two dimensions, all roads lead to Rome, but not in three dimensions.
Specifically, given a random walk (equally likely to move in any direction) on the integers in one or two dimensions, then no matter the starting point, with probability one (a.k.a. almost surely), the random walk will eventually get to a specific designated point ("Rome").
...
26
There are a few ways to conserve energy during ODE integration.
Method 1: Symplectic Integration
The cheapest way that is to use a symplectic integrator. A symplectic integrator solves the ODE on a symplectic manifold if it comes from one, and so if the system comes from a Hamlitonian system, then it will solve on some perturbed Hamiltonian trajectory. ...
24
The Schroedinger equation is effectively a reaction-diffusion equation
$$
i\frac{\partial\psi}{\partial t}=-\nabla^2\psi+V\psi\tag{1}
$$
(all constants are 1). When it comes to any partial differential equation, there's two ways to solve it:
Implicit method (adv: large time steps & unconditionally stable, disadv: requires matrix solver that can give bad ...
22
In 2014, I would've said Python. In 2017, I wholeheartedly believe that the language to teach undergraduates is Julia.
Teaching is always about a tradeoff. On one hand, you want to choose something that is simple enough that it is easy to grasp. But secondly, you want to teach something that has staying power, i.e. something that can grow with you. The ...
21
Here's one close to the hearts of the contributors at SciComp.SE:
The Navier–Stokes existence and smoothness problem
The three-dimensional version is of course a famous open problem and the subject of a million-dollar Clay Millenium Prize. But the two-dimensional version has already been resolved a long time ago, with an affirmative answer. Terry Tao notes ...
20
In social choice theory, designing an election scheme with two candidates is easy (majority rules), but designing an election scheme with three or more candidates necessarily involves making trade-offs between various reasonable-sounding conditions. (Arrow's impossibility theorem).
15
This question is very closely related to (and possibly a duplicate of) Is variable scaling essential when solving some PDE problems numerically?.
There are still good practical reasons to nondimensionalize equations, if possible:
It reduces the number of independent parameters for parametric studies (which was one of the original reasons for ...
13
This is not really 4D data. As Geoff said, it's 3D scalar data, i.e. you're visualizing a scalar function of three variables: $f(x,y,z)$.
There are several ways to visualize this kind of data, and many tools that will help you. I'll show you a few styles of plots you can make.
Contour plot showing one or more $f(x,y,z) = \text{(const.)}$ surfaces, ...
13
Atmosphere and ocean have highly-stratified flows in which the Coriolis force is a major source of dynamics. Maintaining geostrophic balance is extremely important and many numerical schemes are intended to be exactly compatible (at least in the absence of topography) to avoid radiating energy in gravity waves. Due to the stratification, limiting vertical ...
12
TL;DR: It depends on what kind of accuracy you need.
Energy conservation does not automatically equal accuracy. Suppose, you want to simulate the solar system, and you are using a solver that – to use an extreme example – just rotates the entire system by some angle every second. These solutions obviously conserve energy, but they are blatantly incorrect.
...
12
Further to Chris' answer:
Yes, weather (or the equations describing it) is extremely sensitive to the initial conditions. The fact that the weather system contains phenomena at pretty much all time and space scales (have a look at figures 1.1 and 1.2 of these notes for some examples) does not make predictions any easier. Also, the "weather" (at various ...
11
Simultaneous diagonalization of two matrices $A_1$ and $A_2$:
$$
U_1^T A_1 V = \Sigma_1,\quad U_2^TA_2V=\Sigma_2
$$
is covered by existing generalized singular value decomposition.
However, when the simultaneous reduction of three matrices to a canonical form (weaker condition compared to the above) is required:
$$
Q^T A_1 Z = \tilde{A_1},\quad Q^T A_2 Z = ...
10
Kyle Kanos's answer looks to be very full, but I thought I'd add my own experience. The split-step Fourier method (SSFM) is extremely easy to get running and fiddle with; you can prototype it in a few lines of Mathematica and it is, extremely stable numerically. It involves imparting only unitary operators on your dataset, so it automatically conserves ...
10
Suppose that the solution $u$ of the PDE lives in some function space $X$. We'll write the PDE as a bilinear form
$A(u, v) = f(v)$
for all $v$ in $X$, where $f$ is some element of the dual space $X^*$.
To approximate the solution $u$ of this PDE, we can instead look for some field $u_N$ that lives in a finite-dimensional subspace $V_N$ of $X$. Typically, ...
9
The traditional approach for scalar field-based data (temperature, velocity magnitude, pressure, density, etc.) plotted over two or three space dimensions uses color. It's important to note that choice of color scheme can distort your impressions of the data. For this reason, do not use a rainbow color scheme. (For why, see here, here, here, and here.) ...
9
From skimming the table of contents to the book you listed, I'd say that computational books of that type for physics (or in my case, engineering, since that is my background) tend to sacrifice depth and quality of explanation for breadth. The best book of this ilk I can think of is probably Strang's Computational Science and Engineering, because he's a ...
9
Errors grow exponentially in a chaotic system, and most people believe weather is chaotic. So even if you get a fairly exact numerical approximation, the fact that your input data (temperatures, etc.) and your model are slightly off from reality will cause results which exponentially diverge from reality as time goes on. Thus long time prediction of weather ...
9
There are plenty of examples in quantum computing, although I've been out of this for a while and so don't remember many. One major one is that bipartite entanglement (entanglement between two systems) is relatively easy whereas entanglement among three or more systems is an unsolved mess with probably a hundred papers written on the topic.
The root of ...
8
Many of us in scientific computing simply have well-equipped laptops for regular software development tasks, some multicore workstations for smaller-scale testing, and access to clusters for larger runs.
To give you an idea:
My laptop is a Dell M3800 (4-core Intel i7, hyperthreading, 16GB of RAM). This is good enough to regularly compile my software and do ...
8
Just simply by being consistent in all of my code?
Yes this is the only way. Matlab or any other programming language does not know about units. They only know about numbers.
As an example consider incompressible flow. If you set your velocity in m/sec, length in meters (how you generate the grid), pressure in Newton/m^2, kinematic viscosity in m^2/sec, ...
8
I think it's generally true that there are no advantages of Fortran 77 over either newer versions of Fortran or in fact any number of other programming languages that are widely used in scientific computing.
The reason it's still used is because there are millions of lines of code around that are written in Fortran 77. Now, recall that it takes a good ...
8
You can use Morton keying to sort the coordinate locations by binning them into cubes of some specified size $d$. This is an $\mathcal{O}(N\log N)$ operation. Then, given any point P, you can use its Morton key to search only in a small number $(\mathcal{O}(1))$ of nearby boxes, bounded by your distance criterion. The search for each box is $\mathcal{O}(\log ...
7
There is no shortcut. Just like there is no shortcut to becoming an "engineering expert in a short time".
The thing is that to be an expert in civil engineering, you need to understand load analysis, use cases of buildings and bridges, materials, designs, and regulatory issues. For computational science, you need to understand the mathematical background, ...
7
I would say that you have, mainly, two methods:
Being consistent in all your code, as already suggested in another answer. For that purpose I always keep a table like this one with me, since it might prove really useful.
Use nondimensional equations. That way, all my parameters and variables are consistent already. For that purpose I suggest reference 1.
...
7
One issue that you should be aware of is that NVIDIA has a market segmentation strategy in which it sells relatively inexpensive GPU's to the gaming and graphics workstation markets (GeForce and Quadro) and different higher-priced models (Tesla) to the high-performance computing market.
The GPU's sold for use in gaming and graphics have limited double-...
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