Some Diverse Directions in Higher Order Compact Methodology
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This dissertation is concerned with the application of some existing compact finite difference schemes to hitherto unexplored areas and further development of some new higher-order compact (HOC) schemes. A global stability analysis of the staggered lid-driven cavity flow for both parallel and anti parallel motion of the lids is carried out with a recently developed HOC scheme. To the best of our knowledge, this is the very first attempt to carry out the analysis of the flow stability inside a staggered cavity. Some more complicated flows are then analyzed, where the biharmonic formulation of the Navier-Stokes (N-S) equations is utilized for the first time. In doing so, a most recently developed pure streamfunction based compact scheme has been employed, again for the first time. This approach is seen to drastically reduce the computational effort vis-a-vis the and the primitive variable formulation of the N-S equations, specifically in computing the critical parameters from the generalized eigenvalue problem. Next, a family of implicit HOC finite difference schemes is developed for the transient three-dimensional (3D) convection-diffusion-reaction equations. They efficiently capture solutions of fluid flow problems governed by any of these fundamental processes. The results obtained for flows of varying complexities governed by the 3D incompressible N-S equations are in excellent agreement with established numerical results. The 2D formulation of one of these schemes is then applied to problems in Mathematical Biology to study pattern formation occurring in nature. Various patterns have been recognized with different model problems and in particular, spikes, spots, stripes and labyrinths patterns are obtained. Finally, a new family of HOC schemes has been developed for the 1D Euler equations of Gas Dynamics (compressible flows). These schemes produce highly accurate results and are seen to capture shocks and discontinuities very well on much coarser grids.
Supervisor: Jiten Chandra Kalita