4301Parabolic1D08

4301Parabolic1D08 - Applied Mathematics 4301: Numerical...

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Unformatted text preview: Applied Mathematics 4301: Numerical Methods for PDEs Parabolic Systems in 1D 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 S. W. Mudd Bldg. 287 yy2250@columbia.edu Lecture #2 10 September 2008 Lecture plan Model 1D parabolic problem (linear, constant coefficient, homogeneous, Dirichlet) with product-series analytical solution Explicit FD method discretization & marching solution truncation error stability and convergence Implicit FD method discretization & linear algebraic solution (Thomas) stability and convergence Weighted FD method (Crank-Nicolson) discretization new issues Generalizations of the model problem boundary conditions beyond Dirichlet variable coefficients nonlinearity Text basis Chapter 2 of Numerical Solution of Partial Differential Equations : An Introduction K. W. Morton & D. E. Mayers Cambridge University Press, 2005 concentrates on finite differences and standard model problems (easiest to analyze and understand) we add finite elements and spectral methods tonight (and in P.S. #2) we quote from Richtmyer and Morton (1967) General setting for M&M, Chap 2 Domain unbounded bounded semi-bounded Unknown continuous field Governing equation quasilinear linear constant coefficient ) , , ( ) , , ( ) , , ( ) ) , , ( ( ) , , ( t x u d u t x u c x u t x u b x u t x u a x t u t x u e + + + = ) , ( ) , ( ) , ( ) ) , ( ( ) , ( t x d u t x c x u t x b x u t x a x t u t x e + + + = d cu x u b x u a t u + + + = 2 2 ] , [ ], , [ F R L t t x x x = = = R L x x , ) , ( t x u 1 , = = R L x x = = R L x x , General setting for M&M, Chap 2 Initial condition Boundary conditions general Dirichlet Neumann homogeneous ) ( ) , ( x u x u = R L i t c t x x u t b t x u t a i i i i i , ), ( ) , ( ) ( ) , ( ) ( = = + ) ( ) , ( t c t x x u i i = ) ( ) , ( t c t x u i i = ) ( = t c i FD setting for M&M, Chap 2 Spatial Discretization Temporal Discretization Unknown discrete gridfunction Governing equation several, following Initial condition Boundary conditions J x x h jh x x x x x x J j x L R j R J L j = + = = = , , , , , , 1 , L ) , ( k j k j t x u U N t k nk t t t t n t F n F N n = = = = , , , , 2 , 1 , L ) ( j j x u U = R L i t c h U U t b U t a k i k i k i k i k i k i , ), ( * ) ( ) ( * = = + Model Problem for M&M, Chap 2 D o m a i n Unknown field Governing equation linear constant coefficient homogeneous diffusion only Initial condition inhomogeneous Boundary conditions homogeneous Dirichlet 2 2 x u t u = ) , ( t x u ) , [ ], 1 , [ t x ) ( ) , ( x u x u = ) , 1 ( ) , ( = = t u t u 1 x t Reading the 1D Heat equation...
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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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4301Parabolic1D08 - Applied Mathematics 4301: Numerical...

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