2. (a) Solve the linear equation for t = 1 with h = 0.01 and CFL...

Transcribed Image Text:

2. (a) Solve the linear equation for t = 1 with h = 0.01 and CFL number v=k/h = 0.5, U₁ + u₂ = 0₁ u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, using the leap-frog scheme and the Lax-Wendroff scheme. (b) Solve the equation Ut+Uxxx = 0, u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, analytically in terms of the Airy function. (c) Compare the results of (a) and (b) with rescaling. Find the order of both schemes around discontinuity. 3. Use the nonconservative upwind scheme to solve the Riemann problem. u₁ +чux = 0. u(x, 0) = u₁, x < 0; u(x, 0) = ur, x≥ 0. (a) What is the exact shock speed for u₁ = 1 and 0 < u, < 1? What is the corresponding numerical shock speed as the CFL number approaches 0? (b) Do the same problem for the equation u++u²u₁ = 0. 4. Find the numerical solution at t = T to the nonlinear Riemann problem. u₁+f(u) = 0, f(u) = u²/2. u(x, 0) = u₁, x<0; u(x, 0) = ur, x≥0. Set dx =0.01, T = 1, dt = dx/4. (a) For u₁ = 2, u = -1, plot the solution obtained using upwind scheme, modified Lax-Friedrichs scheme, and two-step Lax-Wendroff scheme. Describe your observation. (b) For u₁ = −1, u, = 2, plot the solution obtained using upwind scheme, modified Lax-Friedrichs scheme, and two-step Lax-Wendroff scheme. Describe your observation. (c) Add the entropy fix to the upwind scheme. Plot the solution for u₁ = −1, u₁ = 2. 5. Find the numerical solution at t = T to the nonlinear Riemann problem. u₁+ ƒ(u)x = 0, ƒ(u) = u²/2. u(x, 0) = u₁, x<0; u(x, 0) = ur, x ≥ 0. Set dx =0.01, T = 1, dt = dx/4. (a) For u 2, u = -1, solve the equation using Lax-Wendroff scheme with minmod limiter. Compare with the solution obtained using the L-W scheme without limiter. (b) For u 1, ur = 1.2, plot the solution obtained using L-W scheme with limiter. Apply the entropy fix, and plot the solution. 2. (a) Solve the linear equation for t = 1 with h = 0.01 and CFL number v=k/h = 0.5, U₁ + u₂ = 0₁ u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, using the leap-frog scheme and the Lax-Wendroff scheme. (b) Solve the equation Ut+Uxxx = 0, u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, analytically in terms of the Airy function. (c) Compare the results of (a) and (b) with rescaling. Find the order of both schemes around discontinuity. 3. Use the nonconservative upwind scheme to solve the Riemann problem. u₁ +чux = 0. u(x, 0) = u₁, x < 0; u(x, 0) = ur, x≥ 0. (a) What is the exact shock speed for u₁ = 1 and 0 < u, < 1? What is the corresponding numerical shock speed as the CFL number approaches 0? (b) Do the same problem for the equation u++u²u₁ = 0. 4. Find the numerical solution at t = T to the nonlinear Riemann problem. u₁+f(u) = 0, f(u) = u²/2. u(x, 0) = u₁, x<0; u(x, 0) = ur, x≥0. Set dx =0.01, T = 1, dt = dx/4. (a) For u₁ = 2, u = -1, plot the solution obtained using upwind scheme, modified Lax-Friedrichs scheme, and two-step Lax-Wendroff scheme. Describe your observation. (b) For u₁ = −1, u, = 2, plot the solution obtained using upwind scheme, modified Lax-Friedrichs scheme, and two-step Lax-Wendroff scheme. Describe your observation. (c) Add the entropy fix to the upwind scheme. Plot the solution for u₁ = −1, u₁ = 2. 5. Find the numerical solution at t = T to the nonlinear Riemann problem. u₁+ ƒ(u)x = 0, ƒ(u) = u²/2. u(x, 0) = u₁, x<0; u(x, 0) = ur, x ≥ 0. Set dx =0.01, T = 1, dt = dx/4. (a) For u 2, u = -1, solve the equation using Lax-Wendroff scheme with minmod limiter. Compare with the solution obtained using the L-W scheme without limiter. (b) For u 1, ur = 1.2, plot the solution obtained using L-W scheme with limiter. Apply the entropy fix, and plot the solution.

Answer rating: 100% (QA)

I understand the need for concise explanations Heres a breakdown of the concepts involved in each problem focusing on key steps and avoiding unnecessa... View the full answer