A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

02/25/2022
by   Xiaobing Feng, et al.
0

This paper 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/12/2020

An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives

In this paper, we develop a new discontinuous Galerkin method for solvin...
research
04/30/2021

Spectral solutions of PDEs on networks

To solve linear PDEs on metric graphs with standard coupling conditions ...
research
06/05/2021

Second-order finite difference approximations of the upper-convected time derivative

In this work, new finite difference schemes are presented for dealing wi...
research
05/24/2020

Finite difference and numerical differentiation: General formulae from deferred corrections

This paper provides a new approach to derive various arbitrary high orde...
research
02/09/2021

Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations

We analyze the stability and functional superconvergence of discretizati...

Please sign up or login with your details

Forgot password? Click here to reset