Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations

TL;DR

This paper introduces the vanishing moment method and moment solutions for second order fully nonlinear PDEs, enabling constructive, convergent numerical approximations that align with viscosity solutions when they exist.

Contribution

It proposes a new vanishing moment method to compute moment solutions for nonlinear PDEs, providing a practical alternative to viscosity solutions.

Findings

Numerical experiments confirm convergence of the vanishing moment method.

Moment solutions coincide with viscosity solutions when they exist.

The method is compatible with various numerical schemes.

Abstract

This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.