MUSCL scheme

Definition

Let $f \colon \R \to \R$ be a $C^1$ function and consider the conservation law $u_t + f(u)_x = 0$. Take a uniform grid of cells $[x_{i-1/2}, x_{i+1/2}]$ of width $\Delta x$, and let $u_i(t)$ be the cell averages of a finite volume method. Let $H \colon \R^2 \to \R$ be a numerical flux, for example one given by a Riemann solver, with $H(v, v) = f(v)$ for all $v$. Let $\phi \colon \R \to \R$ be a flux limiter with $\phi(r) = 0$ for $r \le 0$. A MUSCL scheme is the semi-discrete scheme built in three steps.

1. Slope ratio: $r_i = \dfrac{u_i - u_{i-1}}{u_{i+1} - u_i}$, with $\phi(r_i)$ read as $0$ when $u_{i+1} = u_i$.
2. Limited reconstruction at each interface: $$u^L_{i+1/2} = u_i + \tfrac{1}{2} \phi(r_i) (u_{i+1} - u_i), \qquad u^R_{i+1/2} = u_{i+1} - \tfrac{1}{2} \phi(r_{i+1}) (u_{i+2} - u_{i+1}).$$
3. Flux update: $$\frac{d u_i}{dt} = -\frac{1}{\Delta x} \Big( H(u^L_{i+1/2}, u^R_{i+1/2}) - H(u^L_{i-1/2}, u^R_{i-1/2}) \Big).$$

The system of ordinary differential equations is then advanced in time, for example by a Runge–Kutta method.