Implementation of Level Set Methods in Cactus Framework

Level set methods are a general and powerful technique to represent an object's boundary by the means of an implicit function that has a specific value on the boundary. This property permits the easy modeling of interfaces between different fluids that interact under prescribed/evolving conditions or that act under specified constraints. The Cactus Code is a general framework for developing computational solutions for a varied class of partial differential equations (PDEs). Amongst many Cactus applications are: numerical relativity, astrophysical flows and computational fluid dynamics (CFD). The purpose of this research work is to test and implement level set methods for solving problems with evolving surfaces within the Cactus framework. Once this method is implemented in Cactus it can be further used and upgraded to implement free surface flow and multiphase CFD problems making it easier for other scientists to conduct research in this field. To benchmark and validate the implementation, a simple problem of a notched disc moving in a rotational velocity field (known as Zalesak's Problem) is solved using Cactus. A simple implementation of a specific level set method must maintain constant volume of the disc after a full rotation. The results show the importance of reinitializing the signed distance function which is used to represent the interface of the notched disc. The shape of the notched disc changes to decrease sharp corners and becomes smoother at the end of a full rotation. Although the disc preserves its volume throughout the rotation by using this simple level set method, its shape changes drastically. Implementing advanced versions of simple level set methods 8,15 that preserve the shape of the interface over arbitrary velocity fields is deferred for future research. The results of Zalesak's Problem on different grid resolutions and under different flow conditions will be presented to validate as well as verify the correct implementation of the algorithm.