l23 - Lecture 23: Higher-Order Schemes (Contd) 1 Last Time...

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1 Lecture 23: Higher- Order Schemes (Cont’d)
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2 Last Time… We looked at: Higher-order schemes for the steady convection operator based on Taylor series Added dissipation schemes
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3 This Time… We will start considering how to limit spatial oscillations. We will consider Monotonicity, monotonicity preservation concepts Godunov theorem Total variation, total variation diminishing (TVD) concepts Non-linear schemes using limiters
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4 Monotonicity/Monotonicity Preservation Concepts For elliptic diffusion equation and for parabolic unsteady diffusion equation without source our discrete equations gave us This ensures bounded solutions For hyperbolic equations, boundedness is not a useful concept ; 0 P nb nb nb a a a 
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5 Wave Transport P Time t Time t+ t • As wave passes through point P, value can exceed both old time value as well as current neighbor values • We have to think about what properties about our solution we want to impose on our numerical scheme
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6 Monotonicity Time t Time t+ t 0 1 0 1 0 1 0 1 If ( , ) ( , ), then ( , ) ( , ) x t x t x t t x t t    x x
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7 Monotonicity Preservation Time t Time t+ t x x If (x,t) is monotonic in x, then (x,t+ t) is also monotonic in x No new maxima or minima are created 
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8 Godunov Theorem
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This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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l23 - Lecture 23: Higher-Order Schemes (Contd) 1 Last Time...

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