Singhal, Raghav2023-07-072023-10-202023-07-072023-10-202023ROLL NO.186123012https://gyan.iitg.ac.in/handle/123456789/2425Supervisor: Kalita, Jiten Chandra"This study is primarily focused on the development of explicit jump high-order compact finite difference immersed interface approaches for the purpose of solving incompressible viscous flows that are governed by the Navier-Stokes (N-S) equation on uniform and non-uniform grids on a Cartesian mesh. In all, three basic schemes have been developed in the process: one for elliptic problems and the steady state of N-S equations with discontinuities in the solutions, source terms, and coefficients across the interface; the next one is the transient counterpart of the previously developed one uniform grids; and lastly, a discrete level-set approach on non-uniform grids with complex interfaces. The overall accuracy of the scheme is four in space and two in time. Throughout the whole physical domain, a nine-point compact stencil is maintained by incorporating the jump conditions into the right-hand side of the matrix equation Ax = b resulting from discretization of the concerned equations. We use the streamfunction- vorticity ( - ) formulation of the N-S equation, and the jump conditions for and at the irregular point across the interface are taken into account by using a new method based on Lagrangian interpolation.enMATHEMATICSThesis