4301EllipticWFirstOrder

# 4301EllipticWFirstOrder - Lecture#8(Part B 22 October 2008...

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Applied Mathematics 4301: Numerical Methods for PDEs Elliptic problems w/ 1 st -order terms, Lecture #8 Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 [email protected] Yan Yan, teaching assistant [email protected] Lecture #8 (Part B) 22 October 2008

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Equations with first-order terms • Many problems in fluid mechanics, chemical engineering, optoelectronics, etc. have the abstract form: 1 0 where ) ( ) , ( ) ( ) , ( ) ( ) 0 , ( ] , 0 [ ], , [ for 0 ) , , , ( << < = = = = + ε β α γ t t b u t t a u x x u t t b a x t x u u F u F x xx t
Equations with first-order terms • Burger’s equation (below) is one highly studied model problem • Though nonlinear, it has analytical solutions for comparison for some BCs: ) ( ) , ( ) ( ) , ( ) ( ) 0 , ( ] , 0 [ ], , [ for 0 t t b u t t a u x x u t t b a x uu u F x xx t β α γ ε = = = = +

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Equations with first-order terms • A further specialization, useful for our purposes in APMA 4301 is the linear, steady (formally elliptic) case • This equation develops a “boundary layer” or “internal layer” as ε becomes small 0 ) ( , 0 ) ( , 1 0 where ( 0 ) , 1 ( , 0 ) , 0 ( ] 1 , 0 [ for ) ( ) ( ) ( 0 > << < ) = = = + + x q p x p t u t u x x f u x q u x p u x xx In a classic 1968 paper published in the journal (now) called Studies in Applied Mathematics , C. E. Pearson studies over 100 problems of form (*) numerically, by adaptively refining the mesh to control error; see also W. Wasow’s classic: The Capriciousness of Singular Perturbations
Eight formulations of (*) ] 1 , 0 [ , ] ) ( [ ] 1 , 0 [ ] 1 , 0 [ , ) ( ] 1 , 0 [ ] [ { min ] 2 ) ( )[ ( { min 0 / / 1 0 ' ' 2 )' ' ( ' ' ' 1 0 1 0 ' ' ' 1 0 0 0 1 0 2 0 2 1 0 2 1 0 2 ' 2 1 ' 2 1 H v dx v f qu pu v u H u C v dx v f u C u dx f u u dx f q x u f v v p q v v s rv v f qu u f qu pu u u + + = + = + = = + = + = + + ε φ ρ Q Q L L L 1) 2) 3) 4) 5) 6) 7) 8)

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Eight formulations of (*) 1) Orig. PDE BVP 2) Self-adjoint form 3) Liouville form 4) System of ODEs 5) Variational form 6) Least-squares 7) Strong Galerkin 8) Weak Galerkin ) , ( ) ( ) , ( ] ) ( ) 2 / 1 ( exp[ ) ( ] ) ( ) / 1 ( exp[ ) ( ' ' ' 1 0 1 0 ' ' ' ' 2 1 v f dx v qu pu dx v u v u B u u v v dt t p u x v dt t p x f qu pu u u x x = + + = + + ∫∫ ε ρ L
Which formulation? • For our finite difference work, we will start with (1) • For finite element work, we would start with (8) • Each of the others has a realm of utility – e.g., (4) is the starting point of a “shooting” method •I f p(x)=0 , we could show that (5) and (6) are equivalent to (8), but we are interested in the case of large p (relative to ε ) here

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Equations with first-order terms • A further special case, takes q(x)=0 and p(x) constant, with f(x)=-p • Swapping the inhomogeneity from the source to the RHS (by replacing u with the shift u-x ), this is a convectively dominated
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4301EllipticWFirstOrder - Lecture#8(Part B 22 October 2008...

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