CS 205A class_19 notes

# Scientific Computing

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Lecture 19 Tuesday, December 4, 2007 Supplementary Reading: Osher and Fedkiw, § 18.3, § 23.1 1 Heat Equation The Heat Equation ∂T ∂t = ∇· ( k T ) is our model parabolic equation, and arises in several physical simulation applications. 1. We are commonly given initial values for T as T ( x, t = 0) = T 0 ( x ) and boundary conditions for t > 0. 2. In the case where the spatial component of T is one-dimensional, we have the equation T t = ( kT x ) x . 3. Parabolic equations approach a steady state . For example, in the heat equation we can take T t = 0, which gives ∇ · ( k T ) = 0, the Laplace equation. Derivation Starting from conservation of mass, momentum and energy one can derive ρe t + ρ ~ V · ∇ e + p ∇ · ~ V = ∇ · ( k T ) (1) where k : thermal conductivity T : temperature e : internal energy/unit mass ρe : internal energy/unit volume 1

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Ideal Material and Divergence-Free We first make the ideal material assumption, ie. e and T satisfy the relationship de = c v dT and that our domain is divergence-free ( ∇ · ~ V = 0) to simplify equation 1 to
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