Finite volume method

The finite volume method is a numerical method for the solution of conservation laws, so special, often hyperbolic, partial differential equations, which is a conservation law is based.

Most prominently, the use of the finite volume method in computational fluid dynamics, where it is so used as a standard method for solving compressible flow problems of the Euler and Navier -Stokes equations.

The procedure used in its derivation an integral form of the conservation laws, and thus also permits discontinuous solutions, which are typical for such equations. Furthermore, only slight requirements are placed on the grid cell, which allows unstructured and flexible geometry. In addition, the conservative variables of the equation are actually received.

Derivation

A conservation law is expressed by an equation of the type

Given to an area where the nabla operator called, resulting in the divergence here. The ordinary case we consider here is. The derivation of equations with additional terms is the same. First, the area is in a finite (finite ) number factored in grid cells ( the volume). In each of these cells the law of conservation. Complies with the terms of the Gaussian integral theorem, about Lipschitz continuity of the edge of each of the cells, it is apparent integration over a cell and converting the integral of the divergence into a surface integral:

The change of a received quantity (eg energy ) in a cell can therefore happen only by the absence or addition flow (in this case of energy) over the edge of the cell. In each cell is then calculated the average of the conserved in this cell is obtained in the case that the cell does not change with time, an equation that describes the change in the mean values ​​in the cells with the time:

In numerical methods are usually selects polygonal bounded cells, so that the integral ( in the two-dimensional case, straight edges ) can be represented, over the edge as the sum of surface integrals over simple structure.

Solution of the equation

To calculate the surface integrals of the second order Gaussian quadrature is taken as a rule. By averaging the values ​​in the individual cells, the problem arises that the numerical solutions along the grating edge is discontinuous. However, the situation can be regarded as at the edge of the Riemann problem. The use of an approximate Riemann solver then allows the calculation of the rivers. This consistency of the Riemann solver is required, which means on the continuity or even Lipschitz continuity in this case, as well as the condition that for identical data from two cells, it provides the physical flow.

This then provides the developed system of ordinary differential equations only if another condition is entropy added. For the purely mathematical consideration of the discontinuity at the cell edge allows, besides correct for the Riemann problem solution by virtue of a shock wave and the unphysical rarefaction shock. The entropy condition but excludes the dilution shock. The Riemann problem is then solved using numerical methods for ordinary differential equations (using the entropy condition ) may be approximated (eg Osher ) or iterative- exactly ( Godunov ).

Higher -order process

The process thus far described is by averaging the values ​​in each cell, only the first order. Higher order is achieved by higher-order polynomials are recognized in the cells means that It is believed ( constant, linear, parabolic, etc.), a distribution which receives the integral value.

The central difficulty here is that compression shocks or shocks can lead to oscillations in the solution. To avoid the total variation Diminishing Method ( TVD ) method are used, which do not increase the total variation and thus allow no new extrema ( Since polynomials discontinuities ia interpolate overshoot ). The main classes of methods here are the flux- limiter method and the ENO method (or WENO ).

Convergence theory

Finite volume method can be understood as special finite element methods in which one chooses piecewise constant or piecewise linear shape functions that live on the cells and not on the grid points for elliptic equations. This allows using the local comprehensive theory a convergence analysis.

For parabolic or hyperbolic equations such as the Euler or Navier -Stokes equations, the mathematical theory of convergence, however, is less advanced.

Hyperbolic Conservation Laws

For hyperbolic problems, in particular shocks occur, which greatly complicate the analysis. A further problem here is that the solution of the equations is not generally unambiguous. The Lax - Wendroff provides that a finite-volume method with convergence actually converges to a weak solution of the equation. Entropiebedingungen or numerical viscosity are then used to show that this is indeed the physically meaningful.

Another statement that applies to all finite-volume method, is the necessary CFL condition that the numerical range of dependence must include the actual range of dependence. Otherwise, the process is unstable.

In particular, for multi-dimensional equations of the theory of convergence is difficult. In the one-dimensional, there are also higher order methods results that are related to the fact that the space of functions with bounded variation provides for compact sets in L1.

Software

The most widely used commercial software package for numerical flow simulation using FVM is Fluent from Ansys Inc.. 's Aviation and aerospace are different codes in use, including developed by NASA codes flo codes of Antony Jameson, as well as at EADS codes ELSA and the TAU code of the DLR. OpenFOAM is a published under the GPL software package.

History

The basic theoretical and practical ideas were developed in the 1950s for space, especially from the Russians Godunov and the Lax Hungary. The first finite volume method are those of Richard S. Varga (1962) and Price Mann ( 1961). The term finite volume method were then introduced independently in 1971 by McDonald and 1972 by MacCormack and Paullay for solving the time-dependent two-dimensional Euler equations.

The idea of the approximate Riemann solver first appeared in the 1980s, when Roe, Osher, van Leer and others also independently presented such procedures.