l21 - Lecture 21: Unsteady Convection (Contd) Higher-Order...

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1 Lecture 21: Unsteady Convection (Cont’d) Higher-Order Schemes for Steady Convection
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2 Last Time… We looked more closely into the behavior of the convection operator by looking at unsteady convection Consider the stability of implicit and explicit time- differencing schemes for » CDS » UDS Consider the corresponding model equations and try to explain the observed behavior Looked briefly at Lax-Wendroff scheme
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3 This Time… Look more carefully at Lax-Wendroff scheme Start looking at higher-order schemes for the steady convection operator
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4 Unsteady Convection Equation For simplicity, consider linear wave equation Consider 1-D domain of length L=1. Initial condition: Solution: How well do our numerical schemes capture this? Convection-diffusion equation with =0, =1, u=constant Traveling wave
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5 Wave Forms Sine Wave Square Wave For u>0, profiles shift to the right by (ut) in time t. For u=1 and t=0.25, the shift is 0.25 units
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6 Explicit UDS (Cont’d) Start with sine or square wave at t=0. Choose mesh of 50 cells and time step commensurate with =0.5. Take 25 time steps and plot result.
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7 Model Equations Model equation for explicit UDS Model equation for explicit CDS: Model equation for implicit CDS: Note negative diffusion coefficient – responsible for instability Again, false diffusion term makes time-stepping diffusive Note false diffusion term
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8 Lax-Wendroff Scheme This is a classic scheme for the linear wave equation Recall explicit CDS model equation Try to make explicit CDS scheme stable by counteracting its negative diffusion coefficient Lax-Wendroff starts with:
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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.

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l21 - Lecture 21: Unsteady Convection (Contd) Higher-Order...

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