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I want to solve the advection equation ($v_0 \in \mathbb R$) $$ \frac{\partial f}{\partial t} + v_0 \frac{\partial f}{\partial x} = 0 $$

using 2nd order Runge Kutta like the midpoint rule for the time integration and an upwind (backwards finite differences) for the spatial derivation.

Here is a general Runge-Kutta implementation in python

def rk2a( f, x0, t ):
    n = len( t )
    x = np.array( [ x0 ] * n )
    for i in range( n - 1 ):
        h = t[i+1] - t[i]
        k1 = h * f( x[i], t[i] ) / 2.0
        x[i+1] = x[i] + h * f( x[i] + k1, t[i] + h / 2.0 )

    return x

And that makes sense and if I'd have to solve something like $y'=f=x$ I'd know how to pass $f$ but I'm confused because now we have $f=-v_0\frac{\partial f}{\partial x}$.

Since I want to use an upwind scheme for the spatial derivative, we have

$$ \frac{\partial f}{\partial x}|_{i-1/2} = \frac{q^{n}_i - q^{n}_{i-1}}{x_i - x_{i-1}} $$

whereas $n$ denotes the $n$-th timestep and $i$ denotes the $i$-th grid point.

Now to implement that, I'd have to write down f( x[i], t[i] ) and f( x[i] + k1, t[i] + h / 2.0 ) resp. provide a general f.

Now while I think I can write down f( x[i], t[i] ), I have no idea how I'd write down f( x[i] + k1, t[i] + h / 2.0 ).

So the question is: How do I use a upwind scheme in the midpoint rule?

Edit: I assume equispaced grid.

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    $\begingroup$ I think you would find it very helpful to read a good introductory book on numerical methods. For instance, chapter 10 of LeVeque's finite difference book will answer your question and give you a lot of other important information about what you're doing. $\endgroup$ Feb 8, 2022 at 4:02
  • $\begingroup$ What you have written here is a bit confusing because you are writing $f$ to mean two different things (the solution of the PDE and the RHS of the ODE). $\endgroup$ Feb 8, 2022 at 4:03
  • $\begingroup$ Ah, that's an issue stemming from rewriting the topic. I know that there's a difference between the f's. $\endgroup$
    – xotix
    Feb 8, 2022 at 7:12

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