A Cartesian Grid Embedded Boundary Method for the Heat Equation on Irregular Domains

Abstract

We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60-85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank-Nicolson time discretization is unstable, requiring us to use the L0-stable implicit Runge-Kutta method of Twizell, Gumel, and Arigu.

A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

“…However, Crank-Nicolson is only neutrally stable, and we found that it led to weak instabilities at coarse-fine interfaces, given the other choices we made in this algorithm. This behavior was similar to that noted at embedded boundaries in [16,20]. To eliminate this problem, we found it necessary to employ a different approach to computing the viscous terms in (1), using the L 0 scheme described in [30].…”

“…Previous semi-implicit methods have used the Crank-Nicolson scheme to compute the viscous terms in the update. However, for the discretizations used in this work, we found the neutrallystable Crank-Nicolson scheme suffered from weak instabilities at coarsefine interfaces, similar to the behavior noticed at embedded boundaries in [16,20]. To eliminate this problem, we employ a second-order semi-implicit Runge-Kutta scheme based on the L 0 -stable scheme in [30].…”

A fictitious domain approach with spread interface for elliptic problems with general boundary conditions

“…The original boundary and the approximated one lie in the same R d−1 space. For example, the following methods can be found in this group: truncated domains methods [6,7], immersed interface methods (I.I.M.) [8,9], penalty methods [1,2,10,11,12,4], fictitious domain methods with Lagrange multipliers [13,14,15], an adapted Galerkin method proposed in [16], and recent work on a fictitious model with flux and solutions jumps for general embedded boundary conditions [17,18].…”

“…[1,2,8,10,15,6,24] and the references herein. Only a few studies have been devoted to embedded Fourier B.C.…”

Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

“…Cartesian grid embedded boundary methods introduced in [21] and [32] to increase the geometric flexibility of finite volume methods. Away from the boundaries of the computational domain, this approach uses traditional finite difference discretizations on a regular Cartesian grid.…”