Development of efficient numerical methods to solve complex system of Partial Differential Equations (PDEs), frequently arise in the field of Science, Engineering and Technology, is my main topic of interest. We develop efficient high order accurate numerical schemes, validate them with standard problems and then apply them to study different real life problems frequently arise in Computational Fluid Dynamics, Immersed Interface, Image processing and in many other fields. In each case, the governing differential equations are very complex, highly coupled, having irregular boundaries, or sometimes discontinuous and/or highly non-linear. No analytical solution exists for this type of problems, or if it exists it is very difficult to find. So, one has to depend on the numerical solution. Development of an efficient numerical method to solve this type of problems is itself a big challenge. Therefore, many researchers are working in this field worldwide to develop robust, efficient, highly accurate numerical methods.

Till now, we have developed a class of efficient, higher order accurate numerical methods using Finite Difference (FD) and Finite Volume (FV) discretizations. During my PhD, we have developed a class of efficient Higher Order Accurate (HOC) finite difference schemes on non-uniform polar grids and applied them to solve different fluid flow problems to study their complex flow behavior. During my Post-Doc, we have developed unstructured finite volume method (UFVM) with exact C-property for shallow water over dry and irregular bottom. We also have updated an existing two-phase sediment transport model to study the sediment transport in Gironde estuary (France). At IIT Mandi, we have developed a class of higher order accurate finite difference schemes for solving immersed interface problems, in both Cartesian and Cylindrical polar coordinates. Then, we extend our scheme to study moving interface problems and applied it to study droplet problem.