Lect09 - Discretization of PDEs and Tools for the Parallel...

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Unformatted text preview: Discretization of PDEs and Tools for the Parallel Solution of the Resulting Systems Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee Wednesday March 17, 2010 CS 594, 03-17-2010 CS 594, 03-17-2010 Outline Part I Partial Differential Equations Part II Mesh Generation and Load Balancing Part III Tools for Numerical Solution of PDEs CS 594, 03-17-2010 Part I Partial Differential Equations CS 594, 03-17-2010 Mathematical Modeling Mathematical Model: a representation of the essential aspects of an existing system which presents knowledge of that system in usable form (Eykhoff, 1974) Mathematical Modeling: Real world Model ←→ Navier-Stokes equations: ∇ · u = 0 ∂ u ∂ t =- ( u · ∇ ) u- 1 ρ ∇ p + ν ∇ 2 u + f B . C . , etc . CS 594, 03-17-2010 Mathematical Modeling We are interested in models that are Dynamic i.e. account for changes in time Heterogeneous i.e. account for heterogeneous systems Typically represented with Partial Differential Equations CS 594, 03-17-2010 Mathematical Modeling How can we model for e.g. Heat Transfer ? Heat * a form of energy (thermal) Heat Conduction * transfer of thermal energy from a region of higher temperature to a region of lower temperature Some notations Q : amount of heat k : material conductivity T : temperature A : area of cross-section CS 594, 03-17-2010 Heat Transfer The Law of Heat Conduction 4 Q 4 t = k A 4 T 4 x Change of heat is proportional to the gradient of the temperature and the area A of the cross-section. Q : amount of heat k : material conductivity T : temperature A : area of cross-section CS 594, 03-17-2010 Heat Transfer Consider 1-D heat transfer in a thin wire so thin that T is piecewise constant along the slides, i.e. T ( t ), T 1 ( t ), T 2 ( t ), etc. ideally insulated Let us write a balance for the temperature at T 1 for time t + 4 t T 1 ( t + 4 t ) =? CS 594, 03-17-2010 Heat Transfer T 1 ( t + 4 t ) ≈ T 1 ( t ) + k 4 t ( T 2 ( t )- T 1 ( t )) ( 4 x ) 2 + k 4 t ( T ( t )- T 1 ( t )) ( 4 x ) 2 = T 1 ( t ) + k 4 t T 2 ( t )- 2 T 1 ( t ) + T ( t ) ( 4 x ) 2 Take lim 4 x , 4 t → ⇒ ∂ T ∂ t = k ∂ 2 T ∂ x 2 ( Exercise ) CS 594, 03-17-2010 Heat Transfer...
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This note was uploaded on 04/01/2010 for the course COMPUTER S cs202 taught by Professor Jiuhui during the Spring '08 term at 東京国際大学.

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Lect09 - Discretization of PDEs and Tools for the Parallel...

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