Questions tagged [crank-nicolson]
For questions about the Crank-Nicolson method, an approach for discretizing and solving partial differential equations.
17
questions with no upvoted or accepted answers
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Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered
Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
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Order of convergence of Scrodinger eq. with CN scheme
I'm trying to solve numerically the 1-dim time dependent Schrodinger equation using the Crank Nicolson scheme and the Thomas algorithm to solve the tridiagonal matrix.
The physical system consists of ...
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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:
$$
\frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
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Error in implementation of Crank-Nicolson method applied to 1D TDSE?
Some context, I've posted this question on physics SE and stack overflow. The former had nothing to offer, the latter had a great commenter that agreed with the phase looking off being one of the ...
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Advection diffusion equation using Crank-Nicolson with total flux and Diriclet BCs
I am trying to model the 1D advection-diffusion equation:
$${\partial c \over \partial t} = D_c{\partial^2 c \over \partial x^2} -u{\partial c \over \partial x}.$$
With Robin boundary conditions that ...
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86
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Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
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42
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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 ...
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Deriving order of accuracy and interpreting a given discretization scheme when underlying method ( finite difference/volume) not known
If a spatial grid is given with time levels like this:
to solve the following model problem
Now consider the following discretization schemes:
Scheme 1
Scheme 2
Usually, to determine order of ...
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1
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207
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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
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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
$$ \vert \psi(t) \rangle = \...
- 111
1
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Time-dependent Schrodinger equation implementation in FEniCS
For our Bachelors thesis we're trying to solve the Schrodinger equation $i\partial_tu = -\nabla^2u+Vu$ in FEniCS. Given the domain $[-5, 5]^2$ with an initial value of $u_0(x, y)=e^{(-2(x^2+y^2))}$ ...
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Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme
I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$).
$\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$
Where $...
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339
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Crank-Nicolson scheme in space for advection equation
Consider the equation
$$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$,
for $t,x\in\mathbb{R}$.
I'd like to solve this equation forward in space and backward in time, ...
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Why can I not solve the negative advection equation (backwards in time)?
Suppose we have the negative, inhomogeneous advection equation:
$$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
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Crank-Nicolson for 2nd- and 4th-order finite differences
I modeled the heat equation,
$$
u_t = au_{xx}
$$
using the common 2nd-order Crank-Nicolson scheme,
$$
\frac{u^{n+1}_i-u^{n}_i}{dt} = \frac{a}{2\,dx}\left(u_{i-1}^{n+1}+u_{i+1}^{n+1}-2u_i^{n+1} + u_{i-...
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