A Second Order Accurate Finite Difference Scheme for the Heat Equation on Irregular Domains and Adaptive Grids
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0910-A05-07
A Second Order Accurate Finite Difference Scheme for the Heat Equation on Irregular Domains and Adaptive Grids Han Chen1, Chohong Min2, and Frederic Gibou3 1 Department of Computer Science, Univesity of California, Santa Barbara, CA, 93106 2 Department of Mathematics, University of California, Santa Barbara, CA, 93106 3 Department of Mechanical Engineering & Department of Computer Science, University of California, Santa Barbara, CA, 93106
ABSTRACT We present a finite difference scheme for solving the variable coefficient heat equations with Dirichlet boundary conditions on irregular domains. A quadtree data structure is used to represent the non-graded adaptive Cartesian grids, and the interface is represented by the zero value points of the level set function. Numerical results in two spatial dimensions demonstrate second order accuracy for both the solution and its gradient in the L1 and L∞ norms. INTRODUCTION The Stefan problem is a model at the core of diffusion dominated phenomena and is used for example in the study of thin films grown by molecular beam epitaxy or crystals grown from a melt. The solution of the heat equation on irregular domains plays an important role in determining the final accuracy of the solution of the Stefan problem, and the moving interface which may undergo complex topological changes needs to be tracked accurately and efficiently. Implicit representation of the interface between the two material phases, such as the level set method [5] or the phase-field method [3] has been proven to be advantageous over the explicit approaches, e.g., front tracking [2]. With implicit representations topological changes can be handled naturally. In this paper, we follow the work of Chen et al. [1] and present a finite difference scheme for solving the heat equation with Dirichlet boundary conditions on irregular domains. The level set method is employed to represent the interface. Solutions are sampled at vertices of a nongraded Cartesian grid, which is represented by the quadtree data structure. We use the CrankNicolson scheme in the discretization of the heat equation to avoid the stringent time step restrictions imposed by explicit schemes. Computational results demonstrate second order accuracy for both the solution and its gradient in the L1 and L∞ norms. EQUATIONS AND NUMERICAL METHODS Poisson Equation Consider a Cartesian computational domain, Ω , with exterior boundary, ∂Ω , and a lower dimensional interface, Γ , which divides the computational domain into disjoint pieces, the
interior region Ω − and the exterior region Ω + . The variable coefficient Poisson equation is written as r r r ∇ ⋅ (ρ x∇u ( x )) = f ( x ), (1)
where
∇=(
∂ ∂ , ) ∂x ∂y
is the gradient operator. On ∂Ω and Γ , Dirichlet boundary conditions are
specified. Thus, equation (1) decouples into two equations, and the solutions can be obtained independently. In the following examples, we solve equations only in Ω − . Heat Equation The standard heat equation is ρc v Tt = ∇ ⋅ ( k∇T ), (2) where T is the tempe
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