4. Scientific Computation

A major component of all the research projects I am involved in is the development of accurate and efficient numerical methods for systems of nonlinear (primarily parabolic) PDEs. I work mostly with finite volume schemes which have been applied in the context of problems in flow in porous media and pollutant transport. Some other examples of related methods I have studied are:

  1. High resolution Godunov-type methods for hyperbolic PDEs: in which an approximate analytical solution called a Riemann solver is incorporated into a finite volume scheme to improve the treatment of discontinuities. I have applied these methods in the study of a variety of problems, including:
    Density from the Euler equations on a moving grid.
    I have made extensive use of CLAWPACK in this work, which is a general and robust publicly-available code for solving hyperbolic conservation laws.

  2. Moving mesh methods: in which the mesh points evolve according to a parabolic moving mesh PDE that concentrates points in regions where the solution values or gradients are large. Examples include:

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