Mathematical Physics - Volume II - Numerical Methods

4. Finite volume method

4.1 Introduction The finite volume method was introduced as clearly recognizable procedure of numerical fluid mechanics at the beginning of the eighth decade of twentieth century, independently by McDonald [26] and MacCormack and Paullay [25], who applied the mentioned method in solving non-stationary two-dimensional flow of non-viscous fluid. Applying it to the calculation of a three-dimensional flow, the method was later generalized by Rizzi [39], when finally the path to more universal application was open. The method is based on the integral formulation of the basic laws of fluid mechanics and discrete solution in physical space. The method can be applied to all types of computational grids, with great flexibility in terms of the efficiency and accuracy of flux calculations through the appropriate control surfaces. Direct discretization of integral equation forms in fluid mechanics allows a very simple formulation of discrete changes of basic quantities such as mass, momentum and energy , which describe fluid flow. The integral form of continuity equation, momentum and energy equation for arbitrary control volume Ω , bounded by surface ∂ Ω , reads: ∂ ∂ t y Ω W d Ω + x ∂ Ω F · d S = y Ω Qd Ω . (I) By applying equation (I) to the control volume Ω J , in which required quantity is W J , and Q J is volume source, its discrete form is obtained: ∂ ∂ t ( W J Ω J )+ ∑ I ( F I · S I ) = Q J Ω J , (II)