3. Use the matrix method to analyze the effects of two boundary conditions at x=1 on the stability of the Lax-Wendroff s

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3 Use The Matrix Method To Analyze The Effects Of Two Boundary Conditions At X 1 On The Stability Of The Lax Wendroff S 1 (93.88 KiB) Viewed 22 times

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 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 in both space and time: * o (x=1) u(x=0)=0 IL = 100 0=¹-n_n³ + "n_ In ΔΙ • Case 2: Second-order one-sided scheme in both space and time that is consistent with the interior L-W scheme: 2=0 | 2 c^t 2Ax Ar n+1 u = u -(3u-4u-1 + U₁-2) IL-2 c² At² + 24+² (a‚¹²1 + a₂¹²-1 + ª²u²1-2+²-3) -(au +azu AX= = {2 2-1 i 2+1 (x=1) IL-1 IL Fig. 1. Uniform grid for a 1-D linear wave equation. 8 (3) 26 (4) (5)