Universal AMG Accelerated Embedded Boundary Method Without Small Cell Stiffness

TL;DR

This paper introduces a universal embedded boundary finite difference method that avoids small cell stiffness, ensuring symmetric positive definite systems and efficient solutions for wave, heat, and Poisson equations with potential extensions to complex, moving boundary problems.

Contribution

The paper presents a novel embedded boundary method that is universally applicable, avoids small cell stiffness, and is compatible with algebraic multigrid acceleration for fast, stable simulations.

Findings

Method is accurate and stable across wave, heat, and Poisson equations.

Efficiently solves large systems using conjugate gradient with algebraic multigrid.

Applicable to complex and moving boundary problems like Stefan and Navier-Stokes equations.

Abstract

We develop a universally applicable embedded boundary finite difference method, which results in a symmetric positive definite linear system and does not suffer from small cell stiffness. Our discretization is efficient for the wave, heat and Poisson's equation with Dirichlet boundary conditions. When the system needs to be inverted we can use the conjugate gradient method, accelerated by algebraic multigrid techniques. A series of numerical tests for the wave, heat and Poisson's equation and applications to shape optimization problems verify the accuracy, stability, and efficiency of our method. Our fast computational techniques can be extended to moving boundary problems (e.g. Stefan problem), to the Navier-Stokes equations, and to the Grad-Shafranov equations for which problems are posed on domains with complex geometry and fast simulations are very important.