# Questions tagged [crank-nicolson]

For questions about the Crank-Nicolson method, an approach for discretizing and solving partial differential equations.

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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 ...
88 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{...
200 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-...
75 views

1 vote
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### 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 ...
561 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,...
3k 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 ...
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 ...
206 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
323 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, ...
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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-...
1 vote
291 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
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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 $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
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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 $$t\equiv t_\... 6 votes 1 answer 2k views ### What is the origin of the spurious oscillations in the Crank-Nicolson scheme? I was reading about the Crank-Nicolson method, and it is often said that it can produce "spurious oscillations" or that this method is prone to "ringing", especially for large time step and stiff ... 1 vote 1 answer 316 views ### How can I numericaly solve a convection-diffusion equation with a large diffusion term? I want to numerically solve the advection-diffusion equation: \begin{equation} u_t(x,t) + cu_x(x,t) = v u_{xx}(x,t) \end{equation} for x \in [0,1] and t \geq 0 subject to the boundary conditions ... 1 vote 1 answer 298 views ### Numerical solution of non-linear heat-diffusion PDE using the Crank-Nicolson Method I am trying to solve numerically the following 1D EBM: C\frac{\partial T[x,t] }{\partial t} - \frac{\partial }{\partial x}\left ( D(1-x^2)\frac{\partial T[x,t] }{\partial x} \right ) + I[T] = S[x,t](... 4 votes 2 answers 2k views ### How to handle boundary conditions in Crank-Nicolson solution of IVP-BVP? I'm trying to solve the PDE for c(r,t)$$c_t=(1/r)(rJ)_r$$using Crank-Nicolson, and I'm having difficulty with the boundary conditions. J is the flux, the initial condition is c(0,r)=c_{init}, ... 0 votes 0 answers 38 views ### Useful Quantity for Heat Equation? [duplicate] I'm interested in testing some algorithms on the heat equation, and I'd like to assess their accuracy. When evolving a Hamiltonian system, one has the energy to check the validity/correctness of the ... 1 vote 2 answers 1k views ### Numerical solution of burgers equation with finite volume method and crank-nicolson I'm having difficulty with numerically solving the inviscid burgers equation.Godunov's scheme is used in most of what I've found in literature . Now my question is if using a crank nicolson shceme is ... 6 votes 2 answers 233 views ### Is this system of diffusion equations well-posed? I’m using a standard Crank-Nicholson algorithm to solve this system of two coupled diffusion equations:$$\dot{u} - \dot{v} = \frac{\partial}{\partial x} \left( \alpha(x) \frac{\partial u}{\partial x}... 2k 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 ...
343 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 ...
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 $T$ ...
613 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
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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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### 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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### 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?