Study of Limiting Methods for Computation of Conservation Laws and Other Hyperbolic Problems

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Many natural systems are modeled by differential equations of so-called hyperbolic type. For example, models used for weather forecasting, aircraft design, and study of ocean currents and biofluids are all hyperbolic differential equations. This research project concerns the development of techniques to approximate the solutions of hyperbolic differential equations. More precisely, the project continues the development of numerical methods for hyperbolic equations with non-smooth solutions (such as is the case when the terms appearing in the model itself are subject to the influence of non-smoothness; say, because of non-smooth boundaries, or interfaces). In such cases, effective methods must be able to remove approximation artifacts introduced by the non-smoothness, while maintaining as much as possible a behavior close to that of the underlying expected true solution. The impact of the project on applied domains in the sciences and engineering will be in introducing techniques to guarantee more accurate and efficient solutions based on numerical methods. Additional broader impacts include mentoring and collaborating with a graduate student and disseminating the research findings to the larger scientific community through seminars and public lectures. The project studies numerical methods for hyperbolic differential equations, in particular for problems that do not have smooth solutions, where accurate computations largely depend on nonlinear limiting methods. The PI will develop several techniques to improve existing numerical methods and also to develop new methods. One goal of the project is the development of new limiting methods for the "back and forth error compensation and correction method." A second goal of the project is to improve on the Courant-Friedrichs-Lewy (CFL) numbers of Runge-Kutta discontinuous Galerkin methods for hyperbolic conservation laws. Here, the main idea of the PI and collaborators is to use extra conservation constraints as penalty for the variational energy functional, and thereby achieve CFL numbers several times larger without increasing the complexity or reducing order of accuracy.