# A Cartesian grid embedded boundary method for the heat equation on irregular domains

TL;DR: An algorithm for solving the heat equation on irregular time-dependent domains is presented, based on the Cartesian grid embedded boundary algorithm of Johansen and Colella, combined with a second-order accurate discretization of the time derivative.

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 L{sub 0}-stable implicit Runge--Kutta method of Twizell, Gumel, and Arigu.

## Summary (1 min read)

### 1. INTRODUCTION

- For the time discretization, for the fixed-boundary problem the authors use either the Crank-Nicolson method or the method of Twizell, Gumel and Arigu (TGA) [10] .
- The authors algorithm is stable and achieves second-order accuracy both on problems with fixed domain and on problems with a time-dependent domain (t) with boundaries moving with constant velocities.

### FIG. 1.

- Centers of cells in (t old ) are shown with solid circles, and centers of cells in (tnew) -(t old ) are shown with unfilled circles.
- The authors solve (19) numerically on a rectangular domain with three elliptically-shaped holes, with boundary conditions computed using the exact solution (18).
- In the moving-boundary problem, the holes move with constant velocities.
- With both fixed and moving boundaries, the authors solve two separate problems with different boundary conditions: Dirichlet conditions on all boundaries; Dirichlet conditions on the fixed external boundaries, but Neumann conditions on the boundaries of the ellipses.

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##### Citations

697 citations

### Cites background from "A Cartesian grid embedded boundary ..."

...[31] extended this approach to the solution of the time-dependent heat equation....

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616 citations

322 citations

### Cites methods from "A Cartesian grid embedded boundary ..."

...Although both adaptive schemes employ projection methods to solve the incompressible Navier–Stokes equations, the present work employs a cell-centered projection method that makes use of an implicit L-stable discretization of the viscous terms [15,16] and a second order Godunov method for the explicit treatment of the nonlinear advection terms [17–20]....

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...[16] is used to integrate the viscous terms in time....

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285 citations

### Cites methods from "A Cartesian grid embedded boundary ..."

...projection method that makes use of an implicit L-stable discretization of the viscous terms [15,16] and a second order Godunov method for the explicit treatment of the nonlinear advection terms [17–19]....

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...[16] is used to integrate the viscous terms in time....

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193 citations

### Cites background from "A Cartesian grid embedded boundary ..."

...[22] to the solution of the timedependent heat equation....

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##### References

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36,037 citations

### Additional excerpts

...Similar approaches based on formally inconsistent discretizations at the irregular boundary have been used previously and observed to be stable [1, 9], so we expect that the extension to the more accurate boundary discretization should be straightforward....

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17,825 citations

7,325 citations

### "A Cartesian grid embedded boundary ..." refers background in this paper

...However, it is well known that, for any domain with smooth boundary, a smooth function can be extended to all of R with a bound on the relative increase in the C norms that depends only on the domain and (k; ) [5]....

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1,163 citations

### "A Cartesian grid embedded boundary ..." refers methods in this paper

...The method described here, together with that in [6] for elliptic PDEs and [8] for hyperbolic PDEs, provide the fundamental components required for developing second-order accurate methods for a broad range of continuum mechanics problems in irregular geometries based on the predictor–corrector approach in [2]....

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442 citations

### "A Cartesian grid embedded boundary ..." refers background or methods in this paper

...As in previous work on elliptic problems [6], our approach uses a finite-volume discretization, which embeds the domain in a regular Cartesian grid....

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...We follow the approach described in [6, 7]....

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...With Dirichlet boundary conditions as from (3), we compute an estimate of ∂ψ ∂n by interpolating from the grid values and the values at the boundaries; for details, see [6]....

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...The method described here, together with that in [6] for elliptic PDEs and [8] for hyperbolic PDEs, provide the fundamental components required for developing second-order accurate methods for a broad range of continuum mechanics problems in irregular geometries based on the predictor–corrector approach in [2]....

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...This is routine for the case in which the embedded boundary is contained in the finest level of refinement [6], but requires some additional discretization design when the embedded boundary crosses coarse–fine interfaces....

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