13

There are many well-written books/notes on this topic including:
J. C. Strikwerda, Finite difference schemes and partial differential equations, SIAM, 2004.
R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, vol. 98, SIAM, 2007.
B. Gustafsson, H.-O. Kreiss, and J. Oliger, ...

10

This goes back to the beginning of the subject of unitary representations of locally compact noncompact groups. Wigner was looking for all possible generalizations of the Dirac equation to higher spin, and developing the representation theory of the Poincaré group is how he obtained his results (Bargmann did this independently, so they published together). ...

10

There is e.g. a book
Differential Galois Theory by M. van der Put and M. F. Singer, where in Appendix D one can find things on the PDE case (the book is mostly about ODEs).
In mathematical physics there are topics such as KZ equations
which are related to linear representations of braid groups.

9

Let $A$ be this matrix. Because of the formula
$$\int_D\sigma\nabla u\cdot\nabla v\, dx=\sum_{i,j}a_{ij}U_iI_j,$$
($U$ for voltages of $u$, $I$ for currents of $v$), we see three necessary conditions:
the matrix must be symmetric,
it must be positive semi-definite,
and $A{\bf1}=0$.
Actually, if $U\ne \mu{\bf1}$, then $u$ is not constant and the choice $v=u$...

answered Jun 21 '18 at 14:53

Denis Serre

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8

Peter Olver has an interesting book on Symmetry and PDEs. Another area to consider (that is particularly important for geometric PDEs) are exterior differential systems. Here are some notes on the subject by Robert Bryant (who sometimes posts here).

7

In dimensions 3 and higher, and without any constraints on $\sigma$, one can apparently obtain any symmetric matrix $A = (a_{ij})$ such that $a_{ij} < 0$ when $i \ne j$ and $a_{ii} = -\sum_{j \ne i} a_{ij}$, as suggested in Denis Serre's answer. That answer already explains why these conditions are necessary. To see that they are also sufficient, one can ...

answered Jun 21 '18 at 18:03

Mateusz Kwaśnicki

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6

This question (for two dimensional domains) was answered by Curtis, Ingerman and Morrow, "Circular Graphs and planar Resistor Networks" (1998). Let $a$ be the $n \times n$ response matrix. As already noted, we must have $a_{ij} = a_{ji}$ and $a(1\ 1\ \cdots\ 1)^T=0$. The additional condition is that, if $i_1$, $i_2$, ..., $i_k$ and $j_1$, $j_2$, ..., $j_k$ ...

answered Jun 21 '18 at 18:57

David E Speyer

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5

The book "D-Modules, Perverse Sheaves, and Representation Theory " by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki is the perfect source for this topic. The introduction gives a very nice (and elementary) explanation how representation theory of D-modules and symstems of partial differential equations are related. I just give a very nice excerpt ...

5

You have assumed an equidistant grid in one dimension, but the answer below can be formulated (and is true) for general grids in any number of dimensions. You also haven't specified the boundary conditions, but as long as the boundary conditions lead to a well-posed problem, the result below applies (assuming a stable discretization of the boundary ...

5

I will let the SciComp people address the issue of the actual numerical method. Instead I will point out that your PDE, as written, is not necessarily hyperbolic. In particular, you see that $\sin(xt)$ can change signs, and $u^3$ can change signs. This means that depending on the parameters your PDE is actually of mixed type: at some regions it is hyperbolic,...

4

For simplicity, let's assume initially that the support of $\mu$ is the whole real line, so that $F$ is a homeomorphism $ \mathbb{R} \to (0,1) $. Let's denote $b:=F(0)=\mu(-\infty,0]=1-\mu[0,+\infty)\in(0,1)$.
The relevant function is the strictly convex function $\Phi :(0,1)\to\mathbb{R}$ defined as
$$\Phi(t):=\int_{b}^t F^{-1}(s)\, ds $$
which is the ...

3

As already said by Kosh, you are trying to solve an inverse problem in the calculus of variation: in its classical formulation, given a system of PDE, the problem consists in finding a functional whose Euler-Lagrange equations are exactly the given PDE(s).
To my knowledge, a necessary and sufficient condition for a system of PDE (jointly with its boundary/...

3

One of the earliest papers is Application of machine learning algorithms to flow modeling and optimization (1999).
A model reduction can be accomplished by projecting the Navier-Stokes
equations on a properly selected lower dimensional phase subspace. A
reasonable choice for a “proper” selection criterion for the base of
this manifold is the ...

answered Feb 28 '18 at 7:03

Carlo Beenakker

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3

In the first case, when you are given a finite set of points, your problem is not well defined. There are in general arbitrarily many solutions if you are just given a finite set of function values.
In the second case you have to perform two steps. First you discretize your problem, then you solve the resulting linear system using an iterative method.
If ...

3

This question could be asked on MSE.
Using the rewording of the problem by Gerry Myerson, the analysis below shows it is not possible for some $n$. For $n=2$ it is possible, but for $n\geq 3$ it is not. Take the case $n=3$, and let the entries of the first vector be denoted $a,b,c$, with those in the second vector $d,e,f$, so that outer-product matrix is ...

2

Not an answer, but I'll just note that there are some special cases that reduce to ODE's.
With $u(x,t) = v(x+at)$, the differential equation becomes
$$ (1+a) v' - a^2 (v'')^2 = v \tag{1}$$
In particular, for $a=0$ we have solutions $u(x,t) = c e^x$ (by symmetry, $u(x,t) = ce^t$ is also a solution), and for
$a=-1$ we have $u(x,t) = -(x-t+c)^4/144$ as ...

2

The most natural way to derive Lax-Friedrichs's scheme is to consider the discretisation where the approximate state $U$ is constant ($\equiv U_i^n$) in cells $((i-1)\Delta x,(i+1)\Delta x)\times((n-1)\Delta t,n\Delta t)$ where $n+i\in2{\mathbb Z}$. To pass from the array $(U_i^n)_i$ to the next one $(U_i^{n+1})_i$, you start from the piecewise constant data ...

answered Jul 13 '17 at 11:35

Denis Serre

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2

Pretty much any textbook method should work on a monotonic and convex function. Bisection, for instance, if you want to keep it simple (once you manage to find upper and lower bounds for the solution, which shouldn't be hard).
I recommend Newton's method, because your derivative is easy to compute and one can prove that (on a decreasing convex function) if ...

answered May 18 '17 at 17:51

Federico Poloni

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2

If I understand correctly, what you're doing amounts to:
Starting with initial data $\psi(x,0)$ that is represented as a truncated Fourier series.
Computing the exact (up to roundoff errors) time evolution of this initial condition on a periodic domain, using discrete Fourier transforms.
This is often referred to as a Fourier spectral method. Note that no ...

na.numerical-analysis fourier-analysis fourier-transform schrodinger-operators numerical-analysis-of-pde

2

I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of ITEs. Said that, however, I think I can be of some help.
Could you recommend me any articles or book with a brief overview of some methods (maybe classical one), please?
Perhaps a ...

2

In general, the problem can be complicated because some nonlinearities arise as Lagrange multipliers. Therefore, an energy functional (whose Euler-Lagrange equations coincide with the nonlinear PDE) can exist provided that some constraints to the energy functional are imposed.
Anyhow, the general problem is called The Inverse Problem of the Calculus of ...

1

I understand the question as a request for pointers in the literature to research in the discretization of spacetime. There are two reasons why this is an active research topic, a fundamental and a practical reason: Fundamentally, spacetime might be discrete at the smallest levels (Planck scale); practically, to simulate relativistic quantum field theories (...

reference-request soft-question mp.mathematical-physics na.numerical-analysis numerical-analysis-of-pde

1

Anthony Carapetis has code on his website that runs curve shortening flow. You can find that at http://a.carapetis.com/code and it links to a github account which has the source code. His demonstration is for closed curves though.
The issue with open curves is that there is not going to be a unique solution if you don't impose some sort of boundary ...

1

Thanks to @Andrew I dug out the following solution:
$$
u(\vec{x},\vec{\mu},t) = \frac{1}{(4\pi~ t)^{d/2}~\sqrt{|\mathbf{D}|}} \exp \large(\frac{-(\vec{x}-\vec{\mu})^{T} \mathbf{D}^{-1}(\vec{x}-\vec{\mu})}{4~t}\large).
$$
It seems the root of the determinant is not dependend on the problem dim. Take it with a grain of salt.

1

I know this method under the name lagged diffusivity. I learned it from the paper
Vogel, Curtis R., and Mary E. Oman. "Iterative methods for total variation denoising." SIAM Journal on Scientific Computing 17.1 (1996): 227-238.
and the paper
Chan, Tony F., and Pep Mulet. "On the convergence of the lagged diffusivity fixed point method in total ...

1

A finite-element method should work; for an implementation in two spatial dimensions, see Numerical solution of the 1 + 2 dimensional Fisher's equation by finite elements and the Galerkin method.
A Galerkin Finite Element method in two space dimensions is presented,
which discretises a 1 + 2 dimensional version of this partial
differential equation, ...

answered Nov 29 '17 at 6:45

Carlo Beenakker

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1

As a benchmark for comparison, one can use an explicit variable step size finite difference scheme. This is a well established numerical technique for 1D PDE problems even with unbounded domains.
I will focus on the spatial discretization of the term $c(x) \partial_{xx} f(x)$, because the temporal discretization is standard. Even though $c(x)$ may have ...

1

I'll summarize the comments above that settled my confusion;
An affine function on a triangle cannot, in general, satisfy a Dirichlet condition on two sides and a Neumann condition on the third side.
Thanks to Christian and user35593!

1

Let $v$ be a smooth, non-negative function supported on the interval $[1,2]$.
Consider the function in polar coordinates
$$ u_{\rho,\sigma}(r,\theta) = v( r - \rho) \sin( \sigma \theta) $$
A direct computation gives
$$ \triangle u_{\rho,\sigma} = \sin(\sigma\theta) \cdot \left[ v''(r - \rho) + \frac{1}{r} v'(r - \rho) - \frac{\sigma^2}{r^2} v(r-\rho) \...

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