Questions tagged [crank-nicolson]
The crank-nicolson tag has no usage guidance.
39
questions
2
votes
1answer
68 views
Derivation of a parabolic PDE using Alternating Direction Implicit method
I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme.
If I have an equation of the form:
\begin{equation*}
\frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t)
\end{...
4
votes
0answers
89 views
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-...
2
votes
0answers
70 views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
3
votes
1answer
77 views
Maintain unitary time evolution for a nonlinear ODE
I want to solve a nonlinear ODE of matrix $A(t)$
$$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
0
votes
0answers
52 views
Crank-Nicolson solution of parabolic PDE with Newumann boundary conditions
I am trying to solve the non-linear parabolic PDE in $c(t,r)$
$$c_t=\frac{1}{r}(rDc_r-\alpha r^2 c)_r$$
with initial condition $c(0,r)=f(r)$
and boundary conditions $c_r(t,r_1)=\alpha r_1c_1/D$ and $...
0
votes
1answer
110 views
FDM on nonlinear PDEs
I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$.
In order to perform time discretization with FDM (finite ...
0
votes
0answers
246 views
Time-dependent Schrodinger equation implementation in FEniCS
For our Bachelors thesis we're trying to solve the Schrodinger equation $i\partial_tu = -\nabla^2u+Vu$ in FEniCS. Given the domain $[-5, 5]^2$ with an initial value of $u_0(x, y)=e^{(-2(x^2+y^2))}$ ...
1
vote
1answer
146 views
Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
5
votes
2answers
207 views
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
I am considering the following diffusion equation:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$
over a grid ...
1
vote
1answer
97 views
Finite difference methods
I am currently applying the finite difference method to the solution of the diffusion equation.
I think that a problem has occurred, and is as follows, my explicit method is the most accurate when ...
2
votes
2answers
107 views
Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$
I want to use the Crank-Nicolson scheme to solve the equation
$$u_t = iu_{xx}+2iu$$
Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
1
vote
0answers
39 views
Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme
I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$).
$\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$
Where $...
3
votes
0answers
59 views
Use of non-typical values of $\theta$ in theta-methods
The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step:
$$
\frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
1
vote
1answer
193 views
Finite difference - Explicit / Implicit / Crank Nicolson - Does the implicit method require the least memory?
Examine a dynamic 2D heat equation $\dot{u} = \Delta u$ with zero boundary temperature. A standard finite difference approach is used on a rectangle using a $n\times n$ grid. For the resulting linear ...
5
votes
1answer
456 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,...
3
votes
1answer
2k views
Applying Neumann boundaries to Crank-Nicolson solution in python
Consider the heat equation
$$u_t = \kappa u_{xx}$$
with boundary conditions of
$$u(x,0)=0\\
u(0,t)=100\\
u(l,t)=0$$
Numerical analysis by pyton can be done with
...
0
votes
1answer
956 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 ...
0
votes
1answer
197 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
0answers
296 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, ...
2
votes
1answer
668 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-...
1
vote
0answers
263 views
Why can I not solve the negative advection equation (backwards in time)?
Suppose we have the negative, inhomogeneous advection equation:
$$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
1
vote
1answer
676 views
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 $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
0
votes
1answer
492 views
Solving an equation in space and time using the Crank-Nicolson approach
Assume I have the following equation (light propagating in $z$-direction through the matter):
$$id_zu+d^2_ru=0$$
with $u(z, r)$ being a complex wave. The time scale in this equation is
$$t\equiv t_\...
6
votes
1answer
2k views
What is the origin of the spurious oscillations in the Crank-Nicolson scheme?
I was reading about the Crank-Nicolson method, and it is often said that it can produce "spurious oscillations" or that this method is prone to "ringing", especially for large time step and stiff ...
1
vote
1answer
285 views
How can I numericaly solve a convection-diffusion equation with a large diffusion term?
I want to numerically solve the advection-diffusion equation:
\begin{equation}
u_t(x,t) + cu_x(x,t) = v u_{xx}(x,t)
\end{equation}
for $x \in [0,1]$ and $t \geq 0$ subject to the boundary conditions ...
1
vote
1answer
262 views
Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicolson Method
I am trying to solve numerically the following 1D EBM:
$C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](...
4
votes
2answers
2k views
How to handle boundary conditions in Crank-Nicolson solution of IVP-BVP?
I'm trying to solve the PDE for $c(r,t)$
$$c_t=(1/r)(rJ)_r$$
using Crank-Nicolson, and I'm having difficulty with the boundary conditions. $J$ is the flux, the initial condition is $c(0,r)=c_{init}$, ...
0
votes
0answers
38 views
Useful Quantity for Heat Equation? [duplicate]
I'm interested in testing some algorithms on the heat equation, and I'd like to assess their accuracy. When evolving a Hamiltonian system, one has the energy to check the validity/correctness of the ...
1
vote
2answers
1k views
Numerical solution of burgers equation with finite volume method and crank-nicolson
I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is ...
6
votes
2answers
231 views
Is this system of diffusion equations well-posed?
I’m using a standard Crank-Nicholson algorithm to solve this system of two coupled diffusion equations:
$$\dot{u} - \dot{v} = \frac{\partial}{\partial x} \left( \alpha(x) \frac{\partial u}{\partial x}...
5
votes
2answers
1k views
Why is Crank-Nicolson considered implicit in time?
From Wikipedia:
Explicit methods calculate the state of a system at a later time from
the state of the system at the current time, while implicit methods
find a solution by solving an equation ...
0
votes
1answer
319 views
Numerical Solution of non-linear diffusion equation using Finite Differencing
I'm trying to solve the following non-linear diffusion equation:
$$
\frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0
$$ with the boundary ...
3
votes
1answer
2k views
How to solve the advection equation in 2 dimension using the Crank-Nicolson method?
I've an equation like this to solve with the crank-nicolson method
$$U_t -\frac{y}{2} U_x + \frac{x}{2}U_y = 0,$$
where $x$ and $y$ are: [-2,5:2,5] and the time $T$ ...
4
votes
1answer
587 views
My algorithm for the heat equation is unstable
I have implemented the 2D heat equation with what I thought was the Crank-Nicolson algorithm in the following way:
...
1
vote
0answers
1k views
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-...
5
votes
0answers
79 views
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 ...
8
votes
2answers
13k views
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?
10
votes
2answers
1k views
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 ...
28
votes
1answer
6k views
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)...
