  The hyperbolic tangent function is usually used as a reliable approximation
of the equilibrium density distributions of a system with phase transitions.
However, analyzing the accuracies of the numerical derivatives, we find that
its numerical derivatives computed by central difference method (CDM) may
deviate significantly from its analytical solutions, while those computed by
high-order difference method (HDM) can agree very well. Therefore, we introduce
HDM to evaluate the interparticle interactions instead of popular CDM, and
propose a pseudopotential multiphase lattice Boltzmann model based on
high-order difference method. The present model not only retains the advantages
of the pseudopotential model, such as easy implementation, high efficiency,
full parallelism and so on, but also achieves higher accuracies. To verify the
performances of this model, several multiphase flow simulations are conducted,
including both stationary and dynamic situations. Firstly, full thermodynamic
consistencies for the popular equations of state have been achieved in large
temperature range and at large density ratio, without any combining interaction
and any additional adjustable parameter of interaction. Secondly, with
high-order difference, either the interparticle interaction proposed by
Shan-Chen or by Zhang-Chen can equally depict the phase transitions of the
fluids with all selected equations of the state. These numerical agreements
based on HDM are consistent to the theoretical analysis that the two models are
mathematically identical. Thirdly, the present model is stable and accurate at
a wider temperature range. Lastly, our newly proposed model can be easily and
reliably applied to various practical simulations and expected to obtain some
more interesting results.
