4301ParabolicHigherD08

4301ParabolicHigherD08 - Lecture #3 (Part B) 17 September...

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Applied Mathematics 4301: Numerical Methods for PDEs Lecture #3 (Part B) 17 September 2008 Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2150@columbia.edu
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Lecture plan • Model parabolic problem (linear, homogeneous, Dirichlet) – with eigenfunction expansion solution • Explicit FD method – truncation error analysis – stability and convergence analysis • Implicit and Weighted FD methods – discretization and dimensional splittings – stability analysis • Generalizations of the model problem – complex geometry –an
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Pedagogical philosophy We will “spiral staircase” through the material Each sector of the staircase is subject matter (equation type, discretization method, solution algorithm, application set, etc.) Each time we reach a new level with one sector, we are ready to revisit the next sector on a higher level (tonight from 1D to higher D with parabolic problems) Good chance for review
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Higher dimensions – what’s new? 1. Dimensionality – the data structures (and the linear algebra associated with implicit methods) are more complex, since the map of the problem to a one- dimensional address space creates skips, which are “holes” in any matrix ordering. 2. Geometry – multidimensional domains may not be tensor-products of one-dimensional domains. 3. Boundary conditions – the boundary is no longer zero- dimensional (as in one dimension), and many special cases for discretization and coding may occur. 4. Coefficient anisotropy – physics can be different in different dimensions, and physics in different dimensions can couple together. Otherwise, we essentially repeat Lecture #2
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• Domain (“cylinder”) • Unknown –f i e l d • Governing equation – linear – constant coefficient – homogeneous – diffusion only • Initial condition – inhomogeneous • Boundary conditions – homogeneous – Dirichlet 2 2 2 2 y u x u t u + = ) , , ( t y x u ) , 0 [ , Ω t x ) , ( ) 0 , , ( 0 y x u y x u = Ω = ) , ( , 0 ) , ( y x y x
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“Reading” the multi-D Heat equation • This model problem corresponds to heat flow in a constant-property conducting domain, where u is the temperature (local internal energy), immersed at t=0 in a cold reservoir (BCs) • The equation states that the rate of change of temperature in time is proportional to the local curvature of temperature in space – Net curvature locally positive (at or near temperature minimum) increasing temperature – Net curvature locally negative (at or near temperature maximum) decreasing temperature
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“Reading” the multi-D Heat equation • Trivially, if u(x,y,t) is locally constant or locally linear, it is locally in equilibrium, u xx+ u yy =0 , and its temperature will not change
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This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.

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4301ParabolicHigherD08 - Lecture #3 (Part B) 17 September...

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