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# pb4_11 - MPO 662 Problem Set 4 1 The Lax-Wendroff method...

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MPO 662 – Problem Set 4 1. The Lax-Wendroff method replaces time-derivatives with spatial derivatives and the lat- ter are approximated with centered differences. An alternate algorithm known as the Warming and Beam (Tannehill et al. , 1997; Durran, 1999) replaces the 1st and 2nd spatial derivatives with upstream differences: u n +1 j = u n j - μ u n j - u n j - 1 - μ (1 - μ ) 2 u n j - 2 u n j - 1 + u u j - 2 (1) where μ is the Courant number. Show that the truncation error of this scheme is O t 2 , Δ x 2 , Δ x Δ t ) and that it is stable for 0 μ 2. Hint: Derive the modified equation up to second order to show that second order nature of the scheme. 2. Determine the order of accuracy and the stability properties of the slant derivative ap- proximation to the advection equation: u n +1 j - u n j Δ t + c 2 u n +1 j - u n +1 j - 1 Δ x + u n j +1 - u n j Δ x ! = 0 (2) Hint: The expansions must be in space and time, be careful in your derivations 3. The advection operator does not produce dissipation at all, and for this reason it is often discretized with a centered difference scheme that is also free of numerical dissipation.

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pb4_11 - MPO 662 Problem Set 4 1 The Lax-Wendroff method...

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