All Questions
Tagged with finite-difference poisson
40
questions
0
votes
0
answers
46
views
Vector poisson equation in cylindrical coordinates. What's wrong?
I am trying to solve this equation:
$$
\frac{\partial^2 A}{\partial r^2} + \frac{1}{r}\frac{\partial A}{\partial r} - \frac{A}{r^2} + \frac{\partial^2 A}{\partial z^2} = -g
$$
This is basically the ...
- 101
2
votes
1
answer
116
views
Are there necessary conditions for SOR algorithm to converge when used for 2D discrete Poisson problem, with PBC, besides solvability condition?
I am trying to solve a 2D poisson problem that is supposed to represent diffusion of chemicals on a grid: $\nabla^2 R_{ij}=f_{ij}$. I am discretizing the problem with the standard central ...
3
votes
1
answer
769
views
Solving Poisson's Equation with Periodic Boundary Conditions
So, I've been attempting to design a simple solver for a problem of finding the gravitational potential of a system using Poisson's equation (let's call the potential phi, $\phi$). The goal is that I ...
- 33
2
votes
0
answers
176
views
Poisson equation solution in a semiconductor structure
I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied).
$\textbf{Background}$
\begin{equation}
\frac{d^2V}{dx^2} = -\...
- 33
3
votes
2
answers
391
views
Solving 2D Poisson equation with nonhomogeneous boundary conditions (Dirichlet) and a source
I am a physicist who is fairly new to numerical analysis, currently, I am trying to simulate a non-linear paraxial equation, and part of my calculation involves solving a 2D Poisson equation with ...
- 33
1
vote
1
answer
132
views
Discretization of Poisson's equation with 2d permittivity tensor
I have to discretize a generalized Poisson equation in 2D which is
$$\nabla\cdot(\varepsilon \nabla \phi )=-\rho$$
My problem is that here $\varepsilon$ is $2\times2$ permittivity tensor
where
$$\...
- 11
2
votes
2
answers
258
views
Poisson equation with discontinuous variable coefficient
Trying to solve numerically the 2D Poisson equation with variable diffusion coefficient $K$, that in general can be discontinuous.
$$ \frac{\partial}{\partial x} ( K(x,y)\frac{\partial C}{\partial x}) ...
- 445
4
votes
1
answer
252
views
ON the Kronecker product form of the laplacian matrix
It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I $$ where $I$ is the identity ...
- 405
1
vote
1
answer
301
views
Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)
I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form:
$$\nabla(\epsilon\nabla\varphi)=\nabla\...
- 83
5
votes
1
answer
322
views
Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
- 2,267
1
vote
1
answer
413
views
A question on the Poisson equation with Neumann and periodic boundary conditions on a rectangular region
I am trying to solve the following PDE by using finite difference
\begin{eqnarray*} \Delta u&=& f ~~on~~(0,1)\times(0,1)\\
\frac{\partial u}{\partial y}(x,0)&=&0=\frac{\partial u}{\...
- 143
0
votes
0
answers
170
views
Finite difference solver for the 2D Poisson's equation with an integral boundary condition
I wanted to attempt an implementation of a finite-difference-based solver for the 2D elctrostatic Poisson equation when metallic objects are present. Also, I hope to take as input, the location of ...
2
votes
2
answers
100
views
Possible to use Iterative FD methods to solve a transformed non square domain [matlab]?
For the 2-D Poisson equation $$-(u_{xx}+u_{yy}) = f \ \ \text{where} f = 1$$
For boundary conditions
$$\frac{\partial u}{\partial n} = 0 \ \text{on AB and AD}$$
$$ u = 0 \ \ \ \text{on BC and CD no-...
- 23
1
vote
0
answers
121
views
Finite Difference Approximation for the Laplacian in 2D that produces a nonsymmetric matrix
Consider the following PDE
\begin{align}
-\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\
u &= 0 \ \ \text{en} \ \partial ((0,1)\times (0,1)) \label{P2}
\end{align}
if we ...
- 83
2
votes
1
answer
832
views
Tensor product representation for the 9-point finite difference approximations for the Poisson equation
If we use 5-point finite difference approximations in a uniform rectangular grid to solve the Poisson PDE
\begin{align}
-\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\
u &= ...
- 83