Problem 1. Consider the Lax-Wendroff scheme to the 1-D wave equation in a uniform grid shown in Fig. 1: 2₁+cu₂ = 0 (0≤x≤

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Problem 1 Consider The Lax Wendroff Scheme To The 1 D Wave Equation In A Uniform Grid Shown In Fig 1 2 Cu 0 0 X 1 (254.08 KiB) Viewed 34 times

Problem 1. Consider the Lax-Wendroff scheme to the 1-D wave equation in a uniform grid shown in Fig. 1: 2₁+cu₂ = 0 (0≤x≤1) 1. Prove that the scheme is second-order accurate both in time and in space. 2. Use the von Neumann stability analysis to show that the scheme is conditionally stable with the stability condition (2) 3. Use the matrix method to analyze the effects of two boundary conditions at x = 1 on the stability of the Lax-Wendroff scheme for the case of c> 0, where . (x=1) Specifically, you need to compute and plot max |, | vs. v for the interval of -1.2 ≤v≤1.2. In you in plot, divide the interval in v by using 200 uniform panels. Based on this result, determine approximately the stability condition for the L-W scheme for each boundary condition at i= IL below: Case 1: First-order one-sided scheme: both space and time: 2=0 I v ≤1 or -15vs1 2 u(x=0)=0 IL=100 u-u At (²) 12 = cAt 2Ax Case 2: Second-order one-sided scheme in both space and time that is consistent with the interior L-W scheme: c² At 24x² u-u- Ax +C. U=U₁2 -(3u4u-1+U2-2) -=0 ·(a₁u² + a₂u²₁_₁ + a₂u²/12_2 + a₂u²₁2-3) MAE 250D, Computational Aerodynamics, Su22, Copyright © 2022 by Xiaolin Zhong, All Rights Reserved. where a are undetermined coefficients that you need to derive by yourself for a one-sided approximation of (u) of 2nd order accuracy at i = IL in the following form: ¸à¸‚µ‚_ + A₂U_₂_²_\ + A‚µ‚₁2-2+₂-¹+0(x²) Ax² to 4x==2 ii it (1) (x=1) IL-1 IL Fig. 1. Uniform grid for a 1-D linear wave equation. (3) X (5) (6)