Finite volume method

Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain and $F \colon \R \to \R^d$ a $C^1$ flux. Consider the conservation law $u_t + \nabla \cdot F(u) = 0$, with $\nabla \cdot$ the divergence. Partition $\Omega$ into polygonal control volumes $K_i$, and write $e_{ij}$ for the common face of $K_i$ and $K_j$. Let $n_{ij}$ be the unit normal of $e_{ij}$ pointing from $K_i$ to $K_j$. A numerical flux is a function $H(a, b, n)$ with two properties:

1. consistency: $H(a, a, n) = F(a) \cdot n$ for all $a$ and $n$;
2. conservation: $H(a, b, n) = -H(b, a, -n)$.

Let $u_i(t)$ approximate the average of $u$ over $K_i$. The finite volume method evolves these numbers by $$\abs{K_i} \frac{d u_i}{dt} = -\sum_{j} \abs{e_{ij}} \, H(u_i, u_j, n_{ij}),$$ where the sum runs over the neighbours $K_j$ of $K_i$, and $\abs{\cdot}$ denotes volume or area. Boundary faces use fluxes built from the boundary conditions.