Taylor expansions simulation techniques

As already mentioned, two-dimensional simulations can be very CPU and memory intensive, so techniques which can improve the efficiency (i.e. accuracy for a given number of nodes) are important. Using higher-order Taylor expansions to represent derivatives is a possibility, but the only area where a significant improvement in accuracy may be achieved is in the evaluation of the flux when calculating the current (Britz, 1988 Gavaghan, 1997). 

Another technique to determine the vapor-liquid equilibrium of pure substances or mixtures, which has some similarities with what is described in [190, 204-206], is the grand equilibrium method [192]. It is a two-step procedure, where the coexisting phases are simulated independently and subsequently. In the first step, one NpT simulation of the liquid phase is performed to determine the chemical potentials p] and the partial molar volumes v of all components i. These entropic properties can be determined by Widom s test molecule method [207] or more advanced techniques, such as gradual inserticMi [208-210] (see below). On the basis of the chemical potentials and partial molar volumes at a specified pressure po, first order Taylor expansions can be made for the pressure dependence ... 

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