Finite Volume Method (FVM)

The Finite Volume Method’s strength lies in its approach to discretization, which is particularly well-suited for fluid dynamics problems governed by conservation laws. The process begins by dividing the geometric domain into a set of non-overlapping control volumes, or cells, which collectively form a mesh. The governing partial differential equations are then integrated over each of these control volumes.

A key step is the application of the Gauss-Divergence theorem, which transforms volume integrals of divergence terms into surface integrals of fluxes across the cell boundaries. For a generic conserved scalar \(\phi\), the conservation equation in integral form is \(\frac{\partial}{\partial t} \int_V \phi dV + \oint_S \mathbf{F} \cdot d\mathbf{S} = \int_V Q dV\), where \(\mathbf{F}\) is the flux vector and \(Q\) is a source term. The FVM discretizes this exact equation, approximating the surface and volume integrals. The flux across each face is calculated, often using interpolation schemes to find the value of \(\phi\) at the cell face from the values stored at the cell centers.

This flux-based approach ensures that the quantity \(\phi\) is conserved perfectly at the discrete level, both locally for each cell and globally for the entire domain. This property of exact conservation is a major advantage over methods like the Finite Difference Method and makes FVM robust and physically realistic, especially when dealing with shocks or sharp gradients in the flow. It is also flexible in handling unstructured meshes, which are necessary for modeling complex geometries.