In her project, Yekaterina Pavlova applied nonlinear partial differential equations and numerical analysis to the problem of modeling shock formation, and her experience taught her how useful new mathematical techniques can be in simplifying difficult engineering problems. Yekaterina started graduate school at California Institute of Technology in Fall 2006, with an ultimate goal of being able to advance and revolutionize technology by applying new mathematical ideas to current problems. She quotes a favorite professor as saying, “It’s a very dangerous thing to believe you understand something,” and she credits her research experience with helping her learn to look beyond obvious answers to find more important questions.

In this paper we explore the use of the Leray α-regularization applied to the inviscid Burgers equation. In this regularization, an additional variable is introduced, which is a smoothed version of the original variable, and a vector system for the original and smoothed variable is solved simultaneously. We employ a hybrid algorithm combining a centered finite difference scheme for Burgers equation and a spectral method for the regularization. A parameter θ is introduced in order to conserve particular quantities associated with the solution of the regularized problem. For several values of θ, we compare the exact solutions to those of the regularized problem and investigate the dependence of the solutions on the regularization parameter α and on the mesh size. In particular, it is shown that under appropriate conditions and particular values of θ, the numerical Leray α-regularization scheme produces an approximate solution that appears to converge to the unique discontinuous (entropy) solution of Burgers equation as the mesh size h → 0, provided that the regularization parameter α and h are related to each other in a precise way. Interestingly, our results suggest that it is only the smoothed variable that converges to the entropy solution.

This research introduces a novel approach—inspired by recent developments in sub-grid scale models of turbulence—for computing hyperbolic conservation laws that arise in many problems including aircraft design, tsunami waves, and traffic bottlenecks. These equations are known to develop singularities in finite time, and capturing the physical solution beyond the singularity time has been a major challenge for the past half-century. Here, this novel approach is tested thoroughly using Burgers equation as a prototype, and the success and limitations of the approach are determined. Being involved in such state-of-the-art research provides undergraduates with a unique opportunity to bridge classroom mathematics experience and knowledge with real world applications.