Transcribed TextTranscribed Text

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

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

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)

By purchasing this solution you'll be able to access the following files:

for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available MATLAB for Mathematics Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Upload a file
Continue without uploading

We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats