Isobaric-isothermal flash calculations are at the heart of any process calculation. In this paper, we propose two new methods for flash calculations at a specified temperature and pressure. The governing equations derived can not only be solved rapidly but also converge correctly even very close to the critical point with the methods presented in this paper. The derived equations show that simplified equations have been used in the past by other workers. The use of full equations results in faster convergence. Two different approaches, the pressure and density-based, are presented in this work. The density-based approach is particularly useful in reducing the computational time substantially in the case of multiparameter equations of state such as the PC-SAFT and the Helmholtz energy equations of state. Many examples are provided to demonstrate the efficacy of the methods. The methods presented have immediate use in process simulation software and thermodynamic programs. © 2021 Elsevier B.V.