Numerical methods that can be written in flux conservative form

Question

I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have read that an approach is to use a so-called "high-resolution scheme" that includes a flux limiter function. Essentially, the scheme is employed by using a low resolution (first order) method near discontinuous regions and a higher order method elsewhere.

Before this can be done, the two chosen methods must be written in conservation form, which is stated as:

$$u_{m}^{n+1} = u_{m}^{n} - \frac{\Delta t}{\Delta x}\left(F_{m+\frac{1}{2}}^{n+\frac{1}{2}} - F_{m-\frac{1}{2}}^{n+\frac{1}{2}}\right)$$ The book also mentions that the flux $F$ is a function that depends on $u_{m}$ and some neighbors in space, and that the form of $F$ does not change from one point to the next.

I am a little confused by this specification. How "strict" is this form, exactly? It is unclear to me which parts of this definition are necessary.

Take for example, the leapfrog method.

$$u_{m}^{n+1} = u_{m}^{n-1} - \frac{\Delta t}{\Delta x}\left(f(u_{m+1}^{n})-f(u_{m-1}^{n})\right)$$

At first, this seems like a perfect candidate for a 2nd order method, but it uses $u_{m}^{n-1}$ instead of $u_{m}^{n}$. Does this mean that it is not flux conservative?

Which methods can be put in conservation form? I am planning to use Lax-Friedrichs for the first order, but I am still looking for a suitable higher order method that is easy to extend to non-linear problems.

Answer 1

The key feature to a conservative method is simply that the changes due to the fluxes cancel out (i.e., the flux leaving one cell is entering another), so the total mass is constant. Using the form you wrote for a standard conservative method, if we sum $u$ on a grid with $M$ cells, we have:

$$\sum_{m=1}^M u^{n+1}_m = \sum_{m=1}^M u^n_m - \frac{\Delta t}{\Delta x} \left(F^{n+1/2}_{M+1/2} - F^{n+1/2}_{-1/2}\right)$$

If the fluxes at the boundaries are zero (or if the domain is periodic), then the total mass at step $n+1$ is the same as at $n$.

For your leapfrog flux-differencing method, the same analysis shows that (if the flux at the boundaries is zero) $\sum_m u^{n+1}_m = \sum_m u^{n-1}_m$. So all the even steps will have the same total mass and all the odd steps will have the same total mass, but the mass for even and odd steps might be different (depending on whether you use a conservative method for the first step).