Narrow-Stencil Numerical Methods for Approximating Nonlinear Elliptic Partial Differential Equations

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The project will develop new computational methods for simulating various applications in astrophysics, fluid mechanics, image processing, wave propagation, geometric optics, biology, and combustion theory. The project will focus on how to reliably and efficiently approximate solutions to a class of abstract problems that can be used to model various phenomena relevant to the applications. The methods will be proven to yield accurate answers and will also be simple to implement. The project will involve activities towards mentoring and broadly training graduate students so that they are prepared for both an industrial career or a career in academia. The project will formulate, analyze, and test new narrow-stencil finite difference and discontinuous Galerkin methods for approximating viscosity solutions of fully nonlinear PDEs such as the Monge-Ampère equation, the Hamilton-Jacobi-Bellman equation, and the stationary Hamilton-Jacobi equation as well as solutions of second order elliptic PDEs in non-divergence form. The project will explore and extend the novel analytic techniques the PI recently developed to prove the admissibility, stability, and convergence of a simple non-monotone narrow-stencil finite difference method for stationary Hamilton-Jacobi-Bellman equations. Another objective is to formalize an abstract convergence framework based on the notion of generalized monotonicity rather than standard monotonicity, as the new methods do not require the use of wide-stencils. The new narrow-stencil methods are easy to formulate and implement and have higher-order truncation errors than monotone methods when first-order terms are present in the PDE. Another goal of the project is to use fully nonlinear ideas to motivate new analytic techniques for approximating positive solutions of nonlinear reaction diffusion equations; these will help eliminate the need for a comparison principle assumption when approximating fully nonlinear problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.