Finite Volume Method

Abstract

The Finite Volume Method (FVM) was introduced into the field of computational fluid dynamics in the beginning of the seventies (McDonald 1971, Mac-Cormack and Paullay 1972). From the physical point of view the FVM is based on balancing fluxes through control volumes, i. e. the Eulerian concept is used (see section 1.1.4). The integral formulation of conservative laws are discretized directly in space. From the numerical point of view the FVM is a generalization of the FDM in a geometric and topological sense, i. e. simple finite volume schemes can be reduced to finite difference schemes. The FDM is based on nodal relations for differential equations, whereas the FVM is a discretization of the governing equations in integral form. The Finite Volume Method can be considered as specific subdomain method as well. FVM has two major advantages: First, it enforces conservation of quantities at discretized level, i. e. mass, momentum, energy remain conserved also at a local scale. Fluxes between adjacent control volumes are directly balanced. Second, finite volume schemes takes full advantage of arbitrary meshes to approximate complex geometries. Experience shows that non-conservative schemes are generally less accurate than conservative ones, particularly in the presence of strong gradients.