The static or time-independent Hamilton-Jacobi equation -- an example of which is the Eikonal equation -- is a fully degenerate nonlinear elliptic PDE. The viscosity solution of this PDE is useful in applications such as minimum time or cost optimal path planning or control to a target set. Fast label setting algorithms are available for approximating the viscosity solution of some classes of these equations; most notably the Fast Marching Method (which is a continuous version of Dijkstra's algorithm for shortest path through a discrete graph) applies to some forms of the Eikonal equation. The Ordered Upwind Method introduced by Sethian & Vladimirsky [SINUM 2003] expanded the class of equations to which these fast algorithms can be applied. The Monotone Acceptance Ordered Upwind Method (MAOUM) solves the same class of problems as does Sethian's & Vladimirsky's algorithm, but it does so with a precomputed stencil that can adapt to local grid spacing. Consequently, MAOUM is able to guarantee that nodes are accepted in order of their value and performs considerably better than the older algorithm when significant grid refinement is used to improve approximation quality for problems with nonsmooth solutions.

[Alton & Mitchell, J. Scientific Computing, 2012]

The Monge-Ampère equation is a nonlinear second order partial differential equation, which arise in many areas such as differential geometry and other applications. In image registration, one is interested in transforming one image to align with another image. One approach is based on the Monge-Kantorovich mass transfer problem. The goal is to find the optimal mapping $M$ which minimizes the Kantorovich-Wasserstein distance. The optimal mapping can be written as $M =\nabla \psi$ where $\psi$ satisfies the following Monge-Ampère equation \[

det (D^2 \psi(x) )= \frac{I_1(x)}{I_2(\nabla \psi))}

\]where $I_1$ and $I_2$ are the given images. Here $ det( D^2 \psi(x) )$ denotes the determinant of the Hessian of $\psi$. In this talk, we will present a multigrid method for solving the Monge-Amp` ere equation. We will discuss the discretization of the nonlinear equation and the issues of viscosity solutions and monotone finite difference and finite element schemes. We will then present a relaxation scheme which is a very slow convergent method as a standalone solver but it is very effective for reducing high frequency errors. We will adopt it as a smoother for multigrid and demonstrate its smoothing properties. Finally, numerical results will be presented to illustrate the effectiveness of the method.

The reliable simulation of shockwaves is critical in the prediction and study of many phenomena, where abrupt changes in material properties due to shockwaves can greatly affect regions of interest and activate physical mechanisms. When a physical shockwave is formed, it moves through the flow with a certain speed, having some finite width determined by physical dissipation until it encounters some event in its path. For numerical shockwaves, however, a numerical width is enforced, often much greater than the physical width. With this numerical width comes the formation of intermediate states having no direct physical interpretation. Even as the mesh is refined, these intermediate states do not go away; they simply occupy less space. The existence of intermediate states does raise some doubt, however, about how closely a captured shockwave may emulate an ideal discontinuous shockwave, or a real physical one.

There are in fact several types of error associated with intermediate shock states such as errors in shock position, spurious waves, or unstable shock behavior. These errors can be classified as numerical shockwave anomalies; they are numerical artifacts formed due to the presence of captured shockwaves within the flow solution. Each numerical shockwave anomaly is directly related to the nonlinearity of the jump conditions and to a resulting ambiguity in sub-cell shock position in a stationary shock. Two new flux functions are developed that do not have this ambiguity. On all of the shock anomalies in one-dimension, both flux functions show improvement on existing methods without smearing or diffusing the shock.