Another important task in the qualitative theory is to obtain a pattern of the behaviour of the family of solutions throughout the domain of definition of the equation. In the case of the autonomous system (9) the problem is the construction of a phase picture, . a qualitative overall description of the totality of phase trajectories in the phase space. Such a geometric picture gives a complete representation of the nature of all motions which may take place in the system under study. It is therefore important, first of all, to clarify the behaviour of the trajectories in a neighbourhood of equilibrium positions, and to find separatrices (cf. Separatrix ) and limit cycles (cf. Limit cycle ). An especially urgent task is to find stable limit cycles, since these correspond to auto-oscillations in real systems (cf. Auto-oscillation ).
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem . These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.
While there are many general techniques for analytically solving classes of ODEs, the only practical solution technique for complicated equations is to use numerical methods (Milne 1970, Jeffreys and Jeffreys 1988). The most popular of these is the Runge-Kutta method , but many others have been developed, including the collocation method and Galerkin method . A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics.