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As a model of a more complex problem, I am studying a linear advection equation with a rapidly oscillating velocity $$ \partial_t u + a \cos(\omega t) \partial_x u = 0 $$ with initial condition $u(x,0) = \phi(x)$. The exact solution is $$ u(x,t) = \phi(x-a\omega^{-1}\sin(\omega t)). $$ I am trying to develop a finite difference method that

  1. Remains accurate when the time step is of order the oscillation period $(\omega\Delta t \sim 1$)
  2. Remains stable when using spatial meshes such that $\Delta x < a\Delta t$

Criterion 2 obviously points me toward implicit methods, but it is less clear what to do about Criterion 1. I have been trying to explore time integration methods based on quadrature of highly oscillatory functions, e.g., Filon's rule. Based on this idea, I write $$ u(x,t+\Delta t) = u(x,t) - \int_{t}^{t+\Delta t}a\cos(\omega\tau)\partial_xu(x,\tau) d\tau $$ A trapezoid-like rule that integrates $\exp(i\omega t)$ and $t \exp(i\omega t)$ exactly is $$ \frac{1}{\Delta t}\int_{t}^{t+\Delta t} f(\tau)e^{i\omega\tau}d\tau\approx c f(t)e^{i\omega t}+c^*f(t+\Delta t)e^{i\omega(t+\Delta t)} $$ $$ c = \frac{1 + i\omega\Delta t - e^{i\omega\Delta t}}{\omega^2\Delta t^2} $$ Using this quadrature in the same manner as one would use trapezoid integration, I write a semi-discrete approximation to the advection equation $$ u^{n+1} = u^n-a\Delta t~\Re[ce^{i\omega t_n}] \partial_xu^n-a \Delta t~\Re[c^*e^{i\omega t_{n+1}}]\partial_xu^{n+1} $$ Now I have to decide what to do about my spatial differencing. I have already attempted central differencing to form a Crank-Nicolson-like method, but I found it produced unphysical trailing oscillations, just like the usual CN method. Rather than continue other methods by trial and error, I hope to do some analysis. Since the coefficients are time-dependent, I can't really use standard Fourier mode amplification analysis. So my questions are

  1. Is there an existing literature on methods for stepping over rapid oscillations in PDEs?
  2. What analysis tools are at my disposal to try to derive a suitable spatial difference scheme in the above case?
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Central spatial differencing with crank-nicholson like time integration schemes are already known to produce unphysical oscillations; see link. If you want to solve advection PDEs with time-dependent coefficients for larger time steps, the method of characteristics (see link for example) may be worth exploring.

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  • $\begingroup$ However, if shocks form the method of characteristics won't easily track the shock front. $\endgroup$ Dec 21, 2021 at 23:19

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