0
votes
0answers
35 views

Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and ...
4
votes
1answer
70 views

Explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one. But when I tried to add spin into account in this scheme, it ...
1
vote
3answers
113 views

What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states: $$ i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...
0
votes
0answers
34 views

How can one use stability analaysis of finite differences methods in linear Schrodinger to the NLS?

Specifically, I've seen a lot of analysis of grid stability for solving Linear Schrodinger with Forward Euler, Backward Euler and Crank-Nicolson. However, most of the usages I've seen for the same ...
4
votes
1answer
870 views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
32
votes
2answers
1k views

Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
3
votes
1answer
137 views

Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...
4
votes
1answer
479 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
6
votes
4answers
619 views

Applications of Hodge-de Rham Laplacian on p-forms ($p\neq 0,n$) in physics or engineering

The Hodge-de Rham Laplacian $L=(d+d^*)^2$, where $d$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use ...
1
vote
1answer
248 views

Stieltjes convolution with white noise

I'm looking for a reference that would discuss a Stieltjes convolution between a wiener process and a function of bounded variation. Additionally, I had a question about this sort of convolution. Is ...
2
votes
0answers
313 views

Gaussian type integral with inverse square root

Hi, I have encountered an integral of the following type in an engineering application: $\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$, where $a$ and $b$ are real ($a$ could ...
7
votes
2answers
469 views

Efficiently computing a few localized eigenvectors

Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$. The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
5
votes
3answers
420 views

Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel \[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
2
votes
0answers
697 views

Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...
8
votes
2answers
255 views

Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems

Hello, I am a PhD student who does not have extensive computational experience seeking advice from those experienced with computational modelling as to which method would be most appropriate for ...
10
votes
1answer
583 views

Regge calculus: Questions of consistency resolved?

Hello, Regge calculus is an approximation scheme for General Relativity, which has been introduced in early-sixties and has been adopted both in numerical relativity and numerical quantum relativity. ...
4
votes
1answer
278 views

Schrodinger's equation over a randomized grid

I am interested in solutions to $$ \frac{d}{dt} \Psi = -iH \Psi $$ for $H$ hermitian and time independent. This boils down to evaluating $$ \Psi(t) = e^{-iHt}\Psi_0 $$ at points of interest $t_n$. I ...
0
votes
1answer
479 views

Delta notation used for describing numerical stencil

While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order ...
0
votes
0answers
381 views

Splitting wave equation for application of CPML

A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) ...
10
votes
2answers
846 views

Are there any known quantum algorithms that clearly fall outside a few narrow classes?

I'm trying to refresh myself on quantum algorithms and have been skimming Childs and van Dam's 2008 RMP paper among other things. From my preliminary surfing it looks like the known quantum algorithms ...
2
votes
2answers
667 views

Application of coordinate-stretching transformation for Perfectly Matched Layer

A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems. PML note The PML ...
1
vote
1answer
1k views

Boundary conditions of wave equation near infinity

For the following wave equation $ \frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}} $ ...
0
votes
0answers
225 views

What's a good approach to model this system?

Edited 15 Jul 2010 Willie's points are well-taken. I apologize for the wordy description. It turns out have a relative who is quite knowledgeable about numerical problems like this and has offered ...
3
votes
2answers
518 views

Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution

Hi, I'm studying an ODE with a small parameter $\epsilon$ and I'm trying to find the solution in terms of a zeroth-order term and a boundary layer. The zeroth-order term has a logarithmic behavior ...