A Monotone, Second Order Accurate Scheme for Curvature Motion

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that represent interfaces implicitly. The novelty is monotonicity: it is possible to preserve the comparison principle of the exact evolution while achieving second order in time consistency. As a consequence of monotonicity, convergence to the viscosity solution of curve shortening is ensured by existing theory

Statistical exponential formulas for homogeneous diffusion

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula $$u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \,$$ where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by $$\left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by $$\left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly

Algorithms for Area Preserving Flows

We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening

An efficient threshold dynamics method for topology optimization for fluids

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.Comment: 23 pages, 24 figure