I am interested in the broad areas of Applied Mathematics and Scientific Computing. The main aim of my research is to apply state of the art mathematical and computational tools to real world problems and to develop smart and fast numerical algorithms that can simulate real world situations. My specific research interests are in the following areas,
Computational Fluid and Plasma Dynamics
Some Ongoing Research Projects are
Two-Phase flows in Heterogeneous porous media
Oil Recovery by water flooding can be modelled by a single conservation laws (degenerate convection - diffusion equations in the presence of capillary pressures) for the saturations coupled with elliptic equations for the pressure. Most reservoirs are highly heterogeneous consisting of several rock types arranged in layers across which the permeabilities, capillary pressures and porosites vary rapidly. Sometimes, this variation is discontinuous leading to conservation laws and convection-diffusion equations with discontinuous coefficients.
Even in One space dimension where the model is a single conservation law with discontinuous fluxes, both the mathematical theory as well as design of numerical schemes is difficult. My PhD thesis was devoted to a study of this problem and we were able to obtain a proper framework of entropy solutions for conservation laws with discontinuous coefficients and obtained existence and stability of the entropy solutions. The entropy framework was also used to design simple and efficient finite volume schemes of the Godunov type (based on exact Riemann solvers). These schemes were shown to converge to the entropy solutions and were implemented on a host of test as well as practical problems. This work was able to justify many adhoc theories to treat two-phase flows for heterogeneous porous media. Many of the ideas developed for this problem were subsequently used in models for continuous sedimentation and traffic flow with changing road conditions.
Currently, i am interested in analyzing the effect of capillary pressures on the flow and two-phase flows in a multi-dimensional reservior. My collaborators on this project were Prof. Adimurthi , Jerome Jaffre and G.D.Veerappa Gowda
Multiphase flows in porous media
Enhanced Oil and Gas recovery is increasing based on three or more phase flows in a reservoir. Examples include water flooding in presence of gas, polymer flooding and gas floods. The mathematical model is based on a system of conservation laws (degenerate convection-diffusion equations in the presence of capillary pressure) for the saturations and elliptic equations for the pressure. It is hard to design appropriate numerical methods for three phase flow simulation as the equations fail to be strictly hyperbolic and can infact contain elliptic regions in many models. The effects of capillarity and heterogenous media are much harder to include in this case.
My main effort in this project is to design efficient finite volume schemes for flow simulation. We have devised a new Semi-Godunov scheme for a reduced three-phase flow model where the saturation of one phase doesnot depend on the others. It is a reasonable assumption in some situations. We have also shown that the semi-Godunov schemes converge to a weak solution of the model and can be easily extended to more general triangular systems of conservation laws. We are trying to design a split Godunov type solver for a general three phase flow model.
Our future plans are to design a robust three-phase flow solver based on a split Godunov method and to extend it to include capillarity, heterogeneous media and multi-dimensional flows. My current collaborators on this project are Kenneth H. Karlsen and Nils Henrik Risebro
Conservation laws with stiff and singular sources
Many physical and engineering models of interest involve conservation laws (or convection-diffusion equations) with stiff/singular sources. Scalar examples include traffic at a light. More practical systems of interest include Shallow water systems with non-trivial bottom topography and/or coriolis forces and Euler equations for the flow of gas with a duct of variable cross-sectional area for simulating flow inside an engine.
The main difficulty in designing numerical schemes is to preserve steady states of interest while still maintaining desired accuracy at transients. This is done by the so called Well-balanced schemes. We have designed novel Well-balanced schemes based on a locally discontinuous flux formulation. These schemes are efficient both at steady states as well at resolving transients and are very simple to implement. We have rigorous convergence results for singe equations. The task of implementing these schemes for interesting systems like shallow water equations, Euler equations etc in 2-3 space dimensions is ongoing.
Another part of this project is to try and simulate stiff sources like those occuring in combustion problems. This project is jointly with Kenneth H. Karlsen and Nils Henrik Risebro
Computational Plasma Dynamics
Many plasmas can be modelled by the equations of MHD - a system of hyperbolic conservation laws in 3-D. It is difficult to design numerical methods for this system due to its complexity (extremely complicated wave structure), non-convexity and non-strict hyperbolicity. Our aim is to design Riemann solvers which are both stable and efficient. Another aim is to couple physical effects like radiation, real gas equations of states, source terms and heat transfer to the MHD equations and our simulation algorithm.
The primary motivation is to simulate the outer atmosphere of the sun by physically realistic models and efficient computational schemes. This is a joint project between the Theoritical astrophysics group and the PDE group at CMA and includes among others
Kenneth H. Karlsen , Andrew McMurry and Nils Henrik Risebro.
Among other projects, i have participated in the design of new algorithms for image restoration and segmentation by using tools from topological optimization. I have also worked on existence of symmetric steady states for the singular Vlasov-Poisson equations modelling flat galaxies and on designing schemes to compute blow ups for singular semi-linear parabolic equations.
I am also interested in numerical flow control and optimization, developing large time step numerical methods for convection-dominated flows and in designing frameworks for computing coupled multi-physics in a High performance computing environment.