finitevolume

FiniteVolume is a numerical method for solving conservation laws, typically used for computational fluid dynamics and related fields. The domain is partitioned into a finite set of non-overlapping control volumes. The governing equations, written in their integral conservation form, are integrated over each control volume. By applying the divergence theorem, volume integrals of spatial derivatives are converted to surface integrals of fluxes across the control-volume faces. The result is a discrete balance that relates the rate of change of the conserved quantity inside a cell to the net flux through its faces and any sources within the cell.

Unknowns are usually cell-centered averages; fluxes are computed at faces by reconstructing values from neighboring cells and solving or approximating a Riemann problem. Temporal discretization can be explicit or implicit, with stability often governed by a Courant–Friedrichs–Lewy (CFL) condition in hyperbolic problems.

FiniteVolume methods are compatible with structured or unstructured meshes, including polyhedral and highly nonuniform grids, making them well suited to complex geometries. They are typically first-order accurate in space, with higher-order reconstructions (such as MUSCL, WENO) used to improve accuracy while preserving conservation and reducing nonphysical oscillations.

Applications span incompressible and compressible fluid flow, heat and mass transfer, and multiphase problems. The method is valued for its strict conservation properties, robustness in the presence of shocks, and ease of handling complex geometries. It sits between finite difference and finite element methods, sharing the emphasis on local control volumes and flux-based fluxes, and is implemented in many open-source and commercial CFD codes.