BackgroundPhD Research InterestsHighOrder Numerical Methods for conservation lawsHere the focus is on highorder accurate numerical methods for partial differential equations (PDE), in particular those of the hyperbolic type. I am interested in suitable highorder accurate algorithms for general unstructured meshes.There is considerable demand for the numerical solution of differential equations in rather complex computational domains. For example, the Euler or NavierStokes equations are routinely solved for full aircraft configuration by lower order methods. This has motivated the use of unstructured mesh technologies, which makes meshing of the computational domain simple, but the formulation of highorder accurate numerical algorithms much more challenging. Hyperbolic PDE are known to admit discontinuous solutions in finite time, even for smooth initial data. A well known example are the compressible Euler equations, which allow compression shocks for sufficiently high Mach numbers. This leads to shock capturing schemes, which have matured over the years for lower order accuracy, and are widely used in industrial applications. Extensions to high order schemes, however, still leave a lot to be desired. I have been involved in the development of the Spectral Difference (SD) Method. The SD method uses a local pseudospectral representation of the solution on unstructured meshes to achieve arbitrary order of accuracy. Liu, Vinokur and Wang proposed the baseline scheme for hyperbolic PDE in 2004. My own research, which has been carried out at the Aerospace Computing Laboratory at Stanford University in collaboration with Professor ZJ Wang at Iowa State University, has concentrated on the following contributions:
GasKinetic schemes for the NavierStokes equationsConventional techniques for the NavierStokes equations necessitate the use of different discretization stecils for the inviscid and viscous flux components. While these techniques often have a physically sound motivation, algorithmically it can be quite challenging to discretize the NavierStokes equations on general meshes, i.e. arbitrary polyhedra.GasKinetic finite volume schemes offer an attractive alternative in that they allow the discretization of the NavierStokes equations on a simple and universal nextneighbor stencil. This is accomplished by applying the discretization to the gaskinetic distribution function, rather than the macroscopic variables. The connection between the mesoscopic and macroscopic level is established by the governing gaskinetic equations, i.e. the Boltzmann or BGK equations and related framework from kinetic gas theory, such as the ChapmanEnskog expansion. The discretization of the Navier Stokes equations can be accomplished by first computing gradients of the macroscopic variables, and subsequent discretization of a reconstructed gaskinetic distribution function, which is a function of the macroscopic variables and their gradients, and must be consistent to the NavierStokes order of accuracy in ChapmanEnskog expansion. Written in terms of the macroscopic variables this amounts to a nonlinear reconstruction and averaging of flow variables and their gradients to approximate convective and viscous fluxes. Algorithms for general unstructured meshesI am interested in the software engineering aspects related to creating flexible computational tools for modeling physics on general unstructured meshes. There are numerous challenges related to computational geometry, algorithmic details (such as multigrid techniques) for solvers that operate on arbitrary polyhedral meshes. This is true even for loworder approximations, and for any type of physics problem. In this context I have been involved in the design and developed of a computational architecture for modeling compressible fluid flow on arbitrary polyhedral meshes, designated flo3xx.Publications
