Showing papers by "Stanley Osher published in 1989"

The design of algorithms for hypersurfaces mowing with curvature-dependent speed

Abstract: The need to follow fronts moving with curvature-dependent speed arises in the modeling of a wide class of physical phenomena, such as crystal growth, flame propagation and secondary oil recovery. In this paper, we show how to design numerical algorithms to follow a closed, non-intersecting hypersurface propagating along its normal vector field with curvature-dependent speed. The essential idea is an Eulerian formulation of the equations of motion into a Hamilton- Jacobi equation with parabolic right-hand side. This is in contrast to marker particle methods, which are rely on Lagrangian discretizations of a moving parameterized front, and suffer from instabilities, excessively small time step requirements, and difficulty in handling topological changes in the propagating front. In our new Eulerian setting, the numerical algorithms for conservation laws of hyperbolic systems may be used to solve for the propagating front In this form, the entropy-satisfying algorithms naturally handle singularities in the propagating front, as well as complicated topological changes such as merging and breaking. We demonstrate the versatility of these new algorithms by computing the solutions of a wide variety of surface motion problems in two and three dimensions showing sharpening, breaking and merging.

Nonconvex Scalar Conservation Laws in One and Two Space Dimensions

Abstract: Consider the initial value problem for a scalar conservation law $${u_t} + \Delta .f\left( u \right) = 0,\quad t \geqslant 0,\quad \bar x \in {\mathbb{R}^n}$$ (1.1) $$u\left( {x,0} \right) = {u_0}\left( x \right)$$ (1.1)