Building convergent difference schemes for fully nonlinear equations is a challenge.
In the case of first order equations, schemes have been built based on ideas from Conservation Laws.
Recently several authors have tried adapting techniques from fluid mechanics or finite elements to solve second order equations.
In this talk we will discuss wide stencil schemes for functions of the eigenvalues of the hessian.
These PDEs include: the Monge-Ampere equation, the Pucci Maximal and Minimal equations, and the equation for the convex envelope.
After discretizing, the resulting algebraic equations are nonlinear and non-differentiable,
which makes building implicit solvers difficult.
As a result, solution methods have been iterative, and consequently suffer from CFL-type time step limitations.
Recent work has explored techniques to improve accuracy and solution speed.
In this direction we will discuss:
- hybrid schemes for better accuracy,
- implicit solvers for fully nonlinear equations.