A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids |
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Authors: | Han Chen Chohong Min Frédéric Gibou |
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Affiliation: | (1) Computer Science Department, University of California, Santa Barbara, CA 93106, USA;(2) Department of Mathematics, KyungHee University, 130–701 Seoul, Korea;(3) Mechanical Engineering Department and Computer Science Department, University of California, Santa Barbara, CA 93106, USA |
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Abstract: | We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains
with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for
which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the
irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell
vertices (nodes) and use quadtree (in 2D) or octree (in 3D) data structures as efficient means to represent the grids. The
boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the
interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the
Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use
directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied
in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability
of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical
results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the
L
1 and L
∞ norms. |
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Keywords: | Poisson equation heat equation irregular domains supra-convergence non-graded adaptive Cartesian grids quadtrees octrees |
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