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

该项目将开发新的计算方法，用于模拟天体物理、流体力学、图像处理、波传播、几何光学、生物学和燃烧理论中的各种应用。该项目将专注于如何可靠和有效地近似解决一类抽象问题，这些问题可以用来模拟与应用相关的各种现象。这些方法将被证明能产生准确的答案，而且实施起来也很简单。该项目将包括指导和广泛培训研究生的活动，以便他们为在工业或学术界的职业生涯做好准备。该项目将制定、分析和测试新的窄模板有限差分和间断Galerkin方法，用于逼近完全非线性偏微分方程组的粘性解，如Monge-Ampère方程、Hamilton-Jacobi-Bellman方程和定常的Hamilton-Jacobi方程，以及非散度形式的二阶椭圆型偏微分方程组的解。该项目将探索和推广PI最近开发的新的分析技术，以证明定常Hamilton-Jacobi-Bellman方程的一种简单的非单调窄模板有限差分方法的可容许性、稳定性和收敛。另一个目标是基于广义单调性而不是标准单调性的概念来形式化抽象收敛框架，因为新方法不需要使用宽模板。当偏微分方程组中存在一阶项时，新的窄模板方法易于描述和实现，并且比单调方法具有更高的截断误差。该项目的另一个目标是使用完全非线性的想法来激发新的分析技术来近似非线性反应扩散方程的正解；这将有助于在近似完全非线性问题时消除比较原理假设的需要。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.