This article presents a new methodology for the development of Transient Interpolation for Capturing of Surfaces schemes suitable for the simulation of free-surface flows, which is given the acronym TICS. The newly developed approach is based on a switching strategy that combines a bounded high-order transient scheme with a bounded compressive transient scheme. Bounded high-order and compressive transient schemes are constructed by discretizing the transient term in the volume-of-fluid (r) equation over a temporal control-volume in a way similar to the discretization of the convection term over a spatial control-volume, allowing advances in building convective schemes to be exploited in the development of bounded high-order and compressive transient schemes. Following that approach, a bounded version of the second-order upwind Euler scheme is constructed (B-SOUE). The B-SOUE is used to develop a family of temporal compressive schemes that is denoted by the B-CE ~m family, where "m" refers to the slope of the scheme on a temporal normalized variable diagram. The TICS methodology is then applied to the B-SOUE scheme and the B-CE ~m family of schemes to create a new family of transient interface-capturing schemes that is designated by TICS ~m. The virtues of the TICS ~m family, in producing a steep interface for the volume-of-fluid (r) field that defines the volume fraction occupied by the different fluids in a computational domain, are demonstrated through results generated using two schemes of the family (TICS ~(1.75) and TICS ~(2.5)). The accuracy of the new transient TICS schemes is compared to the first-order Euler scheme, the Crank-Nicolson scheme, and the B-SOUE scheme by solving four pure advection test problems (advection of hollow shapes in an oblique flow field and advection of a solid body in a rotational flow field) and one flow problem (the break of a dam) using both the SMART and the STACS convective schemes. Results, displayed in the form of interface contours, demonstrate that predictions obtained with TICS ~(1.75) and TICS ~(2.5) are far more accurate and less diffusive, preserving interface sharpness and boundedness at all Courant number values considered.
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