A study of blow-ups in the Keller-Segel model of chemotaxis

We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to accurately resolve singular solutions. The numerical findings (in two dimensions) are then compared with analytical predictions regarding formation and interaction of singularities obtained via analysis of the stochastic differential equations associated with the Keller-Segel model

Unsteady incompressible Navier-Stokes simulations on deforming domains

Mesh generation has been an important topic of research for the past four decades, primarily because it is one of the critical elements in the numerical simulation of fluid flows. One of the main current issues in this regard is mesh generation and flow solution on domains with moving boundaries. In this research, a novel scheme has been proposed for mesh generation on domains with moving boundaries, with the location of boundary nodes known at any particular time. A new set of linearized equations is derived based on a full nonlinear elliptic grid generation system. The basic assumption in deriving these new equations is that each node experiences only a small amount of disturbance when the mesh moves from one time to the next. Comparison with grids generated by the full elliptic system shows that this new method can generate high quality grids with significantly less computational cost. Inherently, the flow on such a domain will be unsteady. The Navier-Stokes equations for unsteady 2D laminar incompressible flow are expressed in the primitive variables formulation. A SIMPLE-like scheme is applied to link the pressure and velocity fields and ensure conservation of mass is satisfied. The equations are discretized in a pure finite difference formulation and solved by implicitly marching in time. The flow solver is validated against results in the literature for flow through a channel with a moving indentation along one wall