Xiaobing Feng, Thomas Lewis, Kellie Ward
Abstract:
This article develops a unified general framework for designing convergent
finite difference and discontinuous Galerkin methods
for approximating viscosity and regular solutions of fully nonlinear second order PDEs.
Unlike the well-known monotone (finite difference) framework,
the proposed new framework allows for the use of narrow stencils and unstructured grids
which makes it possible to construct high order methods.
The general framework is based on the concepts of consistency and g-monotonicity
which are both defined in terms of various numerical derivative operators.
Specific methods that satisfy the framework are constructed using numerical moments.
Admissibility, stability, and convergence properties are proved,
and numerical experiments are provided along with some computer implementation details.
Published August 25, 2022.
Math Subject Classifications: 65N06, 65N12.
Key Words: Fully nonlinear PDEs; viscosity solutions; Monge-Ampere equation;
Hamilton-Jacobi-Bellman equation; narrow-stencil; generalized monotonicity;
g-monotonicity; numerical operators; numerical moment.
DOI: https://doi.org/10.58997/ejde.conf.26.f1
Show me the PDF file (710 K), TEX file for this article.
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Xiaobing Feng Department of Mathematics The University of Tennessee Knoxville, TN 37996, USA email: xfeng@utk.edu |
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Thomas Lewis Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA email: tllewis3@uncg.edu |
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Kellie Ward Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA email: kmward7@uncg.edu |
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