apma4301_IntoToHyperbolic

apma4301_IntoToHyperbolic - Lecture #13 (Part B) 3 December...

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Applied Mathematics 4301: Numerical Methods for PDEs Lecture #13: Introduction to numerical methods for hyperbolic PDEs 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 #13 (Part B) 3 December 2008
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Plan of presentation • Review properties of hyperbolic equations, and compare and contrast to our well-studied parabolic – Classical second-order hyperbolic shares the spatial analysis of the Laplacian, but has different time-like term (makes it time-reversible) – Classical first-order hyperbolic is a subset of the general parabolic problem, without the regularizing Laplacian term (here pure advection, the non-self-adjoint part) xx tt u a u 2 = x t au u ± =
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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 of sophistication Good chance for review
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From whence do 1 st order equations come? •F l u x f(u) is au Conservation of u in a fixed interval [ x 1 ,x 2 ] For sufficiently smooth functions Interchanging limits •S i n c e [ x 1 ,x 2 ] is arbitrary, the integrand must vanish identically everywhere If velocity is constant )) , ( ( )) , ( ( ) , ( 2 1 2 1 t x u f t x u f dx t x u dt d x x = = 2 1 2 1 )) , ( ( ) , ( x x x x dx t x u f x dx t x u dt d 0 ))] , ( ( ) , ( [ 2 1 = + x x dx t x u f x t x u t Given a passive scalar u(x,t) (e.g., thermal energy) being convected in a steady 1D medium streaming with velocity a(x,t), positive to the right 0 )) , ( ) , ( ( ) , ( = + t x u t x a x t x u t 0 = + x t au u
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Advection equation is a “singularly perturbed” limit of the parabolic case ) , ( ) , ( ) , ( ) ) , ( ( t x f u t x r x u t x a x u t x d x t u + + + = • Diffusion coefficient d(x,t) is always nonnegative • Advection coefficient a(x,t) may be of either sign •I f | a |>> d , the equation is dominated by advection, with diffusion arising only in a boundary layer • We have studied the steady form of this equation in the unit on elliptic equations • Special attention is required (upwinding, Petrov-Galerkin) due to nonselfadjointness • When a(x,t) is a function of u (nonlinear), further attention is required; see lecture #11. • Reducing the regularizing part d(x,t) makes things worse still – but delightfully interesting
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From whence do 2 nd order equations come? •D e n s i t y ρ , velocity u, pressure p, sound speed a Conservation of mass Conservation of momentum Adiabatic equation of state Cross-differentiate both PDEs Substitute second PDE in first Substitute from state eqn x t p u = ) ( Given a pair of such coupled first-order advection equations, we can derive a single second-order equation. Considering only the dominant terms (acoustics): 0 ) ( = + x t u d a dp 2 = , 0 ) ( = + xt tt u xx tx p u = ) ( , 0 = xx tt p 0 2 = xx tt a
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Canonical 1 st order hyperbolic •D
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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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apma4301_IntoToHyperbolic - Lecture #13 (Part B) 3 December...

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