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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 $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (with boundary conditions not yet specified).

Assuming that we had no $v$, i.e. the homogeneous part of the equation, the Crank-Nicolson method would yield

$$-c\frac{\mu}{4}u^{n+1}_{\ell-1}+u^{n+1}_{\ell}+c\frac{\mu}{4}u^{n+1}_{\ell+1}=c\frac{\mu}{4}u^n_{\ell-1}+u^n_{\ell}-c\frac{\mu}{4}u^n_{\ell+1},$$

where $\mu=\frac{\Delta t}{\Delta x}$ and $u^n_\ell=u(n\Delta t,\ell\Delta x)$.

I don't know how to deal with the inhomogeneity in these schemes though.

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Your equation can be written in the following fashion (any spatial derivative approximation is valid), once space is discretised: $$\frac{1}{c}\frac{du_i}{dt}=-\left(\frac{\partial u}{\partial x}\right)_i(t) + v_i(t) \tag{*}$$ Keep in mind that $v_i(t) = v(x_i,t)$.

Now the system of equations depends only on time $t$ you can apply Crack Nicholson method to solve the system of ODEs. Fo simplicity let the rght hand side of $(*)$ be named as: $$F_i(t) = v_i(t)-\left(\frac{\partial u}{\partial x}\right)_i(t) $$ and therefore $(*)$ cab ne rewritten as follows: $$\frac{1}{c}\frac{du_i}{dt}=F_i(t) \tag{**}$$

The Cranck-Nicholson method can be deduced from a $\theta$ method setting $\theta=1/2$, therefore, the time discretised equation $(**)$ is:

$$\frac{1}{c}\frac{u_i^{n+1}-u_i^{n}}{\delta t}=\theta\, F_i(t^{n+1}) + (1-\theta)F_i(t^n)=\frac{1}{2}\left[ F_i(t^{n+1})+F_i(t^n)\right]$$

Do not forget that we had the expression for $F_i(t)$, in which the function $v_i(t)$ is also present.

Hope you now know how to deal with that term.

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  • $\begingroup$ Thank you for your answer, but suppose we want to propagate the equation in $x$, then I guess discretising the $t$ axis would yield: $$u^{n+1}_i=u^n_i+\frac{\Delta x}{2}(F_i(x^{n+1})+F_i(x^n)),$$ where $$F_i(x)=v_i(x)-\frac{1}{c}\left(\frac{\partial u}{\partial t}\right)_i(x),$$ right? $\endgroup$
    – Jason Born
    Jun 19, 2017 at 9:32
  • $\begingroup$ I think you want to solve the system in a Lagrangian frame of reference?, where your equation is simply: $$\frac{du(t,x(t))}{dt}=v(t,x(t))$$ with $$\frac{dx(t)}{dt}=c$$. In this case you only have the path of a single particle through time. The time only goes forwards, you must provide an initial condition $u(0,x)=u^0(x)$ and then solve for the next time. You can not solve time before space. $\endgroup$
    – HBR
    Jun 19, 2017 at 11:38
  • $\begingroup$ For implementation and the $F_i(t)$ term, I guess I would have to discretise the differential operator using a finite difference approximation and assign an initial condition for $v$? $\endgroup$
    – Jason Born
    Jun 21, 2017 at 17:06
  • $\begingroup$ $v$ is a given function of time. Imagine that $u=\exp{(-t)}\sin{x}$, for example. You must only provide an initial condition for $u$ $\endgroup$
    – HBR
    Jun 21, 2017 at 17:31
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    $\begingroup$ Yes you are right! $\endgroup$
    – HBR
    Jul 2, 2017 at 18:56

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