Using Runge-Kutta to time-step finite volume PDEs 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.

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).