Robust Treatment of Interfaces for Fluid Flows and Computer Graphics

Researchers have used numerical techniques to solve partial differential equations describing physical phenomena for many years. One challenging area, the numerical treatment of interfaces, motivated the creation of a topologically robust interface capturing algorithm, the level set method of Osher and Sethian [29]. The level set method has been used to track interfaces in a wide variety of applications. Utilizing geometrical information about the interface, which is naturally obtained from the level set function, an accurate treatment of material discontinuities across the interface can be obtained via the Ghost Fluid Method [15]. Discontinuities are implicitly enforced with the ghost fluid method, avoiding any numerical smoothing of discontinuous quantities across the interface. The ghost fluid method and related techniques have been used to model discontinuities in compressible and incompressible flows [15, 24, 22, 3], flames and detonations [27, 16], solid fluid coupling [14] and Stefan problems [19, 18, 4]. A newly proposed, fully conservative ghost fluid method has been used to track contact discontinuities, inert shocks and detonation waves [25]. Accurate modeling of the motion of a contact discontinuity itself for incompressible flows has been a challenge for level set methods. Recently a new method, the “particle level set method” [10], has been proposed to accurately track contact discontinuities for incompressible flows.