All Questions
Tagged with pde crank-nicolson
27
questions
2
votes
1
answer
278
views
Asking advice for implementation of Conservative Finite Difference Scheme for numerically solving Gross-Pitaevskii equation
I am trying to numerically solve the Gross-Pitaevskii equation for an impurity coupled with a one-dimensional weakly-interacting bosonic bath, given by (in dimensionless units):
\begin{align}
i \frac{\...
- 21
1
vote
0
answers
42
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
1
vote
0
answers
207
views
Solving PDE on a non-uniform grid with Crank-Nicolson scheme
I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
- 21
1
vote
0
answers
125
views
Crank-Nicolson vs Spectral Methods for the TDSE
The time-dependent Schroedinger equation (TDSE) depends linearly on the system's initial state $\vert \psi(0) \rangle$, such that the solution can be generally written as
$$ \vert \psi(t) \rangle = \...
- 111
0
votes
0
answers
89
views
Transparent Boundary Conditions for Finite Difference ADI PR 2D TDSE solution
I want to put (non-dirichlet) boundary conditions inside the code I wrote to solve the 2dim TDSE using the alternating direction implicit Peaceman - Rachford method.
$$
(1 + iB\Delta t/2 ) \psi^{n+1/2}...
- 141
0
votes
0
answers
101
views
Crank-Nicolson solution of parabolic PDE with Newumann boundary conditions
I am trying to solve the non-linear parabolic PDE in $c(t,r)$
$$c_t=\frac{1}{r}(rDc_r-\alpha r^2 c)_r$$
with initial condition $c(0,r)=f(r)$
and boundary conditions $c_r(t,r_1)=\alpha r_1c_1/D$ and $...
- 171
1
vote
0
answers
609
views
Time-dependent Schrodinger equation implementation in FEniCS
For our Bachelors thesis we're trying to solve the Schrodinger equation $i\partial_tu = -\nabla^2u+Vu$ in FEniCS. Given the domain $[-5, 5]^2$ with an initial value of $u_0(x, y)=e^{(-2(x^2+y^2))}$ ...
- 119
2
votes
2
answers
122
views
Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$
I want to use the Crank-Nicolson scheme to solve the equation
$$u_t = iu_{xx}+2iu$$
Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
- 139
1
vote
0
answers
53
views
Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme
I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$).
$\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$
Where $...
- 11
3
votes
0
answers
88
views
Use of non-typical values of $\theta$ in theta-methods
The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step:
$$
\frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
- 153
5
votes
1
answer
779
views
Finite Differencing schemes for Convection-Diffusion equation
I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger.
The flow/convection is always 1D,...
- 265
0
votes
1
answer
1k
views
Crank–Nicolson method for nonlinear differential equation
I want to solve the following differential equation from a paper with the boundary condition:
The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
- 217
0
votes
1
answer
245
views
implicit method (crank-Nicolson) I not understand the procedure [closed]
I'm trying to understand the passage through this equation can be written for easily solved with the fortran alghorithm in particular i don't understood the meaning of L_x and L_xx ... what (-1,0,1) ...
1
vote
0
answers
339
views
Crank-Nicolson scheme in space for advection equation
Consider the equation
$$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$,
for $t,x\in\mathbb{R}$.
I'd like to solve this equation forward in space and backward in time, ...
- 133
2
votes
1
answer
785
views
Crank-Nicolson algorithm for coupled PDEs
Assumed I have the following two coupled equations
$$\begin{split}
\partial_tA&=a_0AB\\
\partial_tB&=b_0AB
\end{split}
$$
but I am not sure how to calculate them. One approach is a crank-...
- 563