Questions tagged [crank-nicolson]
For questions about the Crank-Nicolson method, an approach for discretizing and solving partial differential equations.
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Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation
I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
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How to discretize the advection equation using the Crank-Nicolson method?
The advection equation needs to be discretized in order to be used for the Crank-Nicolson method. Can someone show me how to do that?
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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
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2
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Can the Crank-Nicolson Method Be used to Solve The Schrodinger Equation with a Time Varying Potential?
I have been following an excellent article about how to use the Crank-Nicolson method to solve the Schrodinger equation. In the article, it starts with a $V(x, y, t)$ but the potential seems to become ...
- 33
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Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?
I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
- 105
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1
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Crank-Nicolson method for inhomogeneous advection equation
Suppose we have the inhomogeneous advection equation
$$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$
for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
- 133
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vote
1
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Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
- 172
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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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