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I'm a bit confused on how to Runge-Kutta time stepping methods for finite volume problems. With finite volume PDEs, we have essentially have discretized equations of the form $$\frac{\partial\bar{u}}{\partial t} = F(u(t))$$ where $\bar{u}$ is the average over a cell, and $F$ is usually some numerical flux function. However, for problems that implement RK4 they are usually stated as an initial value problem of the form: $$\frac{\partial y}{\partial t} = F(y(t))$$

I am trying to solve the viscid Burger's equation: $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x} + u^2)$$

via finite volume methods and am wondering if I can advance in time using Runge-Kutta methods.

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Yes, you can! The operation which consists in approximating the field values $u(t)$ by knowing the cell averages $\bar{u}(t)$ is called reconstruction, and is classically based on interpolation. Thus, a method of lines \begin{equation} \frac{\partial \bar{u}}{\partial t} = F(u(t)) \end{equation} rewrites as \begin{equation} \frac{\partial \bar{u}}{\partial t} = L(\bar{u}(t)) \, , \end{equation} where the operator $L$ combines $F$ and a reconstruction procedure.

For example, ENO and WENO finite-volume methods combine a Runge-Kutta method and a reconstruction procedure, in order to achieve high-order integration (see e.g. Shu, 2009).

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  • $\begingroup$ Thanks for the response! In fact, I've read this paper and have employed ENO reconstruction methods to reconstruct $u$ from $\bar{u}$, but got confused because her "spatial discretization operator" $L$ seems to be a function of $\bar{u}$ (pg. 95 of the pdf in the link) while mine is a function of $u$, which I thought would invalidate the use of Runge-Kutta. $\endgroup$ – hijasonno Feb 22 '17 at 18:38

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