Volume Preserving Numerical Methods

The next question is what happens to the volume of a set of points in phase space, which would be conserved by the dynamical system, when we use a numerical method to approximate its evolution. 

Yet it turns out that there are some numerical methods that always conserve the volume when this is conserved by the flow of the differential equations, or for some particular classes of differential equations. For example, consider the planar system 

The components of the Jacobian matrix can be obtained using implicit differentiation, they are  

In the event that the vector field is divergence free, we have fu + gv = which implies that the numerator and denominator are identical, and it follows that 

It would be valuable to have a general method for deriving volume conserving methods. It turns out that volume conservation is, itself, most readily obtained as a consequence of a more fundamental property of Hamiltonian flows, the conservation of the symplectic form. 

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Hamiltonian systems. Thus, one has to treat this non-volume-preserving piece of the integrator a bit more carefully. To ensure numerical stability, higher order reversible integration schemes in conjunction with multiple time step methods are preferred. The details of implementing this scheme are provided in Ref. 28. 

The basic idea behind the VOF method is to discretize the equations for conservation of volume in either conservative flux or equivalent form resulting in near-perfect volume conservation except for small overshoot and undershoot. The main disadvantage of the VOF method, however, is that it suffers from the numerical errors typical of Eulerian schemes such as the level set method. The imposition of a volume preservation constraint does not eliminate these errors, but instead changes their symptoms replacing mass loss with inaccurate mass motion leading to small pieces of fluid non-physically being ejected as flotsam or jetsam, artificial surface tension forces that cause parasitic currents, and an inability to calculate accurately geometric information such as normal vector and curvature. Due to this deficiency, most VOF methods are not well suited for surface tension-driven flows unless some improvements are made [19]. 

The principal objective of numerical methods is to solve even more complex flows, to preserve the properties of the flow locally, and to save computational time. Among the most widely used methods to solve flows, there are the methods of flnite difference, finite volume, and finite element. Each of these methods has its own advantages and disadvantages, which are well discussed in the literature. It is understood that, for reactive flows of technical interest, an approximation of second order in space and in time is frequently sufficient. 

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