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final_proj - in space and write a program for the scheme(ii...

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MATH 692: Final Project (revised! see the note below) Due date: May 4 1. Consider Burgers Equation: u t - νu xx + uu x = 0 , x ( - 1 , 1); u ( ± 1 , t ) = 0 , with the initial condition u ( x, 0) = u 0 ( x ). (i) Write down a scheme which is second-order semi-implicit in time and Legendre-Galerkin in space; and write a program for the scheme. (ii) Take u ( x, 0) = - sin πx , and ν = 0 . 01. Use N = 129 and a time step sufficiently small for the scheme to be stable. Plot the approximation solution at t = 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 , 1 . 2. 2. Consider Allen-Cahn Equation: Consider the Allen-Cahn equation: u t - Δ u + 1 ε 2 ( u 3 - u ) = 0 , ( x, y ) ( - 1 , 1) 2 ; ∂u ∂n | Ω = 0 , with the initial condition u ( x, y, 0) = u 0 ( x, y ). (i) Write down a scheme which is first-order semi-implicit in time and Chebyshev-collocation
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Unformatted text preview: in space; and write a program for the scheme. (ii) Take u ( x,y ) = tanh ± √ x 2 + y 2-. 5 2 ε ² with ε = 0 . 04. Use 65 × 65 points and Δ t sufficiently small for the scheme to be stable. Plot the levelset u ( x,y,t ) = 0 for t = 0 , 1 , 5 , 10 , 50 , 100. Note: If you find it difficult to treat the Neumann boundary condition here, you may replace the Neumann boundary condition by the Dirichlet boundary condition: u | ∂ Ω = 1. 1...
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