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Numerical analysis is used to calculate approximations to things: the value of a function at a certain point, where a root of an equation is, or the solutions to a set of differential equations. It is a huge and important topic since in practice most real problems in mathematics, science and technology will not have an explicit closed-form solution (and even ...


11

Such a simple question, but it opens up so many cans of worms. Here's my crack at a comprehensive answer simpler than what you'd find in a textbook. I'm sure a numerical relativist would have a much more inside scoop... this is coming from more of a mathematical perspective. First I'll try to clear up some general difficulties which I think were ...


8

The name DMRG is somewhat of a historical accident, and its modern day incarnations are not directly linked to the renormalization group or phase transitions. Instead, it is better understood as a variational technique based on matrix-product states (MPS) ansatzes. Still, there is a historical link between the two tools, and it can be useful to know about it....


6

The "solution" of the three-body problem can be written as the pair of differential equations, \begin{align} \vec{v}&=\frac{\mathrm d\vec{x}}{\mathrm dt}\\ \vec{a}&=\frac{\mathrm d\vec{v}}{\mathrm dt} \end{align} where the latter is usually written in terms of the force, $m\vec{a}=\vec{F}$. Then using the definition of the derivative, $$ \frac{\...


5

The infinities in QFT occur at infinitely large momenta (infinitely short distances, known as UV divergences), and infinitely small momenta (infinitely long distances, known as IR or soft divergences). When you put the dynamics of the QFT in a box of finite size, and discretize the lattice inside the box for computations, you automatically do away with both ...


5

The equation can be integrated once, then we have second order equation $$x''+x'+\frac {x^2}{2}=C_1$$ Make a substitution $x'=y(x)$, then $x''=y\frac {dy}{dx}$ as a result, we obtain the Abel equation , for which an analytical solution is known, see https://www.hindawi.com/journals/ijmms/2011/387429/#sec2 $$y\frac {dy}{dx}+y=C_1-x^2/2$$ The numerical ...


5

Implementing such a method is not a trivial task. Just FYI, there are multiple programs purpose-built to perform Hartree-Fock calculations. In terms of a good introduction to the theory of Hartree-Fock calculations, I found this pdf extremely helpful. First of all, you have given the expression for the psuedo-exact (psuedo, because we have made the Born-...


4

First, let me break your question into two pieces: What automated methods are used by physicists to (i) cull the vast amount of data collected and (ii) find results that are of interest? Regarding (i), the technique used is a so-called 'trigger system'. A trigger system decides which events to record to disk and which to discard. It is designed to ...


4

There are two generic sources of errors. The first is the choice of guess energy for your solution. Quantization of energy is the result of applying boundary conditions, and it is by forcing the boundary conditions that one recovers discrete energy levels. If you do this in the reverse direction, i.e. you start with some energy and solve the Schrodinger ...


4

This seems to be a common misunderstanding about Grover's algorithm. It is not about querying a magically encoded database. Rather, you have an efficiently computable function $f(x)\in\{0,1\}$ and you want to find some $x_0$ for which $f(x_0)=1$. Since you know how to realize $f(x)$ (i.e., you have a circuit), you can run $f$ on a quantum computer and use ...


4

The reason Verlet's algorithm is less suitable than midpoint method is due to the form of the force. Verlet's algorithm requires the acceleration at time step $t+\Delta t$ in order to update velocity at time $t+\Delta t$ (I refer to the so called "velocity" form of the algorithm, although the same considerations hold for the "position" form too): $$ \begin{...


4

In earlier DFT studies of ferroelectric materials, GGAs such as PBE were avoided as they tended to exaggerate the ferroelectric distortion. Instead, LDA calculations were performed and an artificial (offset) pressure was applied to compensate for LDA otherwise overestimating lattice constants Philippe Ghosez, Javier Junquera: cond-mat/0605299 "First-...


3

For completeness I'll summarize the answer here. After a fun conversation in the comments, we saw that it will be more illuminating to write $$H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h \sum_ {i=1} ^N \sigma_i^z $$ as $$ H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h\left( \sum_ {i=1} ^{N-1} 1_1 \otimes \cdots 1_{i-1} \otimes \sigma_i^...


3

This does not work. The reason is that physics is about understanding, not just making equations. For example, I can easily combine the law of gravity $F=GM_1m_2/r^2$ with Hooke's law $F=kx$ and get $$kr^3=Gm_1m_2$$ (here I assumed the software was smart enough to realize that the distance in Hooke's law should be equated to the radius in the gravitational ...


3

I'll go ahead and say that one can roughly classify systems into Systems with a unique ground state. Systems that have multiple (possibly infinitely many) degenerate ground states, but that will tend to select a unique one by some mechanism. Systems with topological ground state degeneracy. (I won't rule out there existing additional classes, and I'd ...


3

I wrote some open-source Python code for that purpose: https://github.com/bcrowell/karl . I believe the code used in the actual movie is not open source. There is a very complete discussion of the techniques by Riazuelo: https://arxiv.org/abs/1511.06025 . I don't think Riazuelo's source code is available. Mine only currently handles Schwarzschild black holes,...


3

Well, the comments taken together are starting to look like an answer to your questions... so I'll collate them all here. Is the only equation needed the one shown? The answer is, unfortunately, no. You have written the steady, inviscid momentum equation for an incompressible flow. As noted in the comments, your velocity profiles and wall boundary ...


3

The precession is just one symptom of a larger problem - inaccuracy of the calculation. The inaccuracy is because of two things: 1) your algorithm and really all usual integration algorithms is only an approximation to the mathematical model that is the differential equation in time; this approximation can be made better by using smaller steps in time and ...


3

At least to address the computational aspects of the turbulent, compressible flow solver parts, I will quote Kyle Kanos: On a more programmatical aspect, Toro's Riemann Solvers and Numerical Methods and LeVeque's Finite Volume Methods for Hyperbolic Problems are pretty much the bible for how to write code that will accurately model fluid flows. In both ...


2

My impression is that the two methods are the same, in both cases the position is updated using $$x_{n+1}=2x_n-x_{n-1}+a_n\Delta t^2$$ with $a_n$ the acceleration. Usually, however, I hear "Verlet integration" to be about the velocity Verlet method, where the velocity at the half-step is used: Compute $x(t+\Delta t)=v(t)+\frac12a(t)\Delta t^2$ Compute $a(t+...


2

You need a lot. To answer your question directly, yes you need to calculate the "force" vector that is driving a change of state of the system. 1) write down the second order equations of motion in whatever coordinate you want, Cartesian, spherical, etc. 2) define a new set of variables px = dx/xt, etc. 3) the last step will reduce the order of the ODE ...


2

$\textbf{Quick hint :}$ Interpret spin operators (for chain of length-N) this way (spin operators act on tensor product space) : $$\sigma_{i}^{\alpha} \rightarrow\underset{N_{}^{\text{th}}\text{order direct/kronecker product of identity and pauli matrices}}{\mathbb{I}_{}^{}\otimes\cdots\otimes\underset{i_{}^{\text{th}} \text{position}}{\sigma_{}^{\alpha}}\...


2

The second picture is from a bubble chamber and life was then simpler because the recorded data of the event were there. The first picture is from a detector experiment and to get at the image already programming has been used so that an image of track can be given in the computer. See this Atlas talk of how they detect and identify a track, to understand ...


2

The lid-driven cavity does not need to have cavitation (ie. bubbles forming in liquid). It can be a single fluid, which can be liquid or gas, and you will see the characteristic recirculation regions form in the corners. Depending on the Reynolds number, you may see a separation region on the top of the side wall opposite the direction of wall motion (as in ...


2

In complement to Anyon's answer, let me add two things: You can re-run your DMRG code with different initial conditions; you would assume that this gives different ground states in every run. If you have a system with symmetry breaking, you will typically get the symmetry broken states in DMRG, but even if not, you can compute their overlaps and thus ...


2

The total energy of the system $E = \frac12mv^2 + \frac 12m\omega^2x^2$ So the graph of $v$ against $x$ will be an ellipse with $-A\le x\le +A$ and $-\omega A\le v \le +\omega A$ with $E=\frac 12 m\omega^2 A^2$ where $A$ is the amplitude of the motion. So the graphs in your diagram are a series of ellipses with different total energies of the ...


2

One partial solution would be to use adaptive time steps, i.e., when you detect multiple collisions with the same particle at time $t+\Delta t$, your time step is reduced so that only one collision occurs with that particle, i.e, you compute the positions at $t+\epsilon \Delta t$, with $0 < \epsilon < 1$. It does not solve all the problems, but it ...


2

Soon a later, computational methods bring computational physicists in unknown lands never visited before. So, at some point one has to learn how to trust numerical results without the guidance of known cases. Of course, before trusting a numerical code on new systems, people extensively test it on well known cases, by comparison with i) exact results (if ...


2

A seminal paper on Kerr Geodesics is Wilkins. The necessary equations are found at 2 and 3. Note that it is not trivial to implement these equations by plugging into an RK4 integrator because of the square roots in the R and Theta "potential" functions.


2

The equatorial circular local velocity in natural units of $\rm G=M=c=1$ is $$\rm v_{\phi}=\frac{a^2 \pm 2a \sqrt{r}+r^2}{\sqrt{a^2+(r-2)r} \ (a \mp r^{3/2})}$$ where the larger solution is retrograde, and the smaller one prograde. This can be obtained by setting $$\rm \ddot{r}=\dot{r}=\ddot{\theta}=\dot{\theta}=\ddot{\phi}=0 \ , \ \ \theta = \pi/2 \ , ...


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