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1. Verify that the the backward-time central scheme for the heat equation is accurate of order (1,2) and the Crank-Nicolson scheme is (2,2). 2. For the forward-time central scheme for the heat equation, show that the scheme satisfies the maximum principle if only if 2bu < 1. 3. Use the method of Schur and von Neumann polynomial to verify that the scheme: 2A is unconditionally stable. 4. Consider the modified Crank-Nicolson scheme as -00+ = show that the above scheme is dissipative of order 4stableifoe<2. 5. (Coding problem) Impliment Crank-Nicolson scheme for the IVP: for - for - 1u(t,-1) = for OSISI. The exact solution is u(t, x) = sin T(I - t). Take A = 1, h = 1/10.1/20,1/40. For the numerical boundary condtion on . = 1, use = (a) In each time step, use the Thomas algorithm to solve the tridiagonal system. (b) Plot the exact solution and the numerical solution at t = 1. (c) Compute the error En = ||a - walls at time t = 1 for h = 1/10,1/20,1/40. The order of the accuracy of the numerical solution is approximated by looking at the number: order = Verify the order of accuracy of the solution by taking (hr,h2) = (1/10,1/20), (1/20,1/40) re- spectively. 1

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function q5Rc()

% wave running to the right

tend = 1; % time when to end integration, can be varied
% here are the three versions of integration:
[u1, x1] = CranckNic(1/10, tend);
[u2, x2] = CranckNic(1/20, tend);
[u3, x3] = CranckNic(1/40, tend);

% The three errors:
Eh1 = max(abs(u1 - sin(pi*(x1-tend))))
Eh2 = max(abs(u2 - sin(pi*(x2-tend))))
Eh3 = max(abs(u3 - sin(pi*(x3-tend))))

% requested at the end
order = [log(Eh1/Eh2)/log(2), log(Eh2/Eh3)/log(2)]

% Plots requested in 5(b)
plot(x1,u1,x2,u2,x3,u3,x3,sin(pi*(x3-tend)));
xlabel('x');
ylabel('u(x,t_end)')...

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