lec20 (1)

lec20 (1) - Introduction to Simulation - Lecture 20...

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Introduction to Simulation - Lecture 20 Finite-Difference Methods for Boundary Value Problems Jacob White Thanks to Jaime Peraire
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SMA-HPC ©2003 MIT Outline • Informal Finite Difference Methods – Heat Conducting Bar • More Formal Analysis of Finite-Difference Methods – Heat Equation – Consistency + Stability yields Convergence
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Heat Flow 1-D Example Discrete Representation 1) Cut the bar into short sections 1 T 2 T N T 1 N T 2) Assign each cut a temperature (1) T (0) T
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Heat Flow 1-D Example Equation Formulation 1, , 1 ii i s i h h h x +− =− ∆ 1 i T i T 1 i T + h + ,1 h x Incoming Heat ( ) s h Heat in from left Heat out from right Incoming heat per unit length 1 1, heat flow TT h x κ + + == Limit as the sections become vanishingly small 0 () lim ( ) xs hx Tx xxx ∆→ ∂∂
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1-D Example Normalized 1-D Equation Heat Flow Normalized Equation 2 2 () s Tx ux hf x xx x κ ∂∂ =− ⇒− = ux f x −=
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SMA-HPC ©2003 MIT
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lec20 (1) - Introduction to Simulation - Lecture 20...

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