Recall the heat equation which we solved numerically T= = DTxx There we implemented an explicit...

      

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Recall the heat equation which we solved numerically T₁= = DTxx There we implemented an explicit numerical scheme (FTCS) which led to a conditionally stable solution - meaning that for certain time step values our solution would be unstable. The recent lecture introduced an implicit numerical scheme known as the Backwards in Time Central in Space (BTCS) method, which is expressed below: (1+20) T+¹oT+1-oT!+1 DAt where o = This scheme is unconditionally stable, meaning we can choose any arbitrarily large step size and the solution is always stable. (Δ.x)2 In the tutorial you began by using the brute force solution method, which involves solving a matrix equation each timestep. The Thomas algorithm (pseudocode provided more efficient approach at solving the same system of equations. = Th is a much The Thomas algorithm splits the lower, major and upper diagonal into 3 one dimensional arrays a, b and c, respectively. It uses these vectors in a forward elimination phase and backward substitution phase to determine Tnp1, which is the temperature at the next timestep. Your MATLAB function should take as input dx, dt, nt (space discretization step, timestep size in seconds, and number of timepoints), and output Tn, a, b, c, m (the final temperature distribution, and the intermediate variables used in the Thomas algorithm). You MUST use the Thomas algorithm and ensure you include the variables a, b, c, and m in your code, which match the pseudocode described in the lecture slide. Recall the heat equation which we solved numerically T₁= = DTxx There we implemented an explicit numerical scheme (FTCS) which led to a conditionally stable solution - meaning that for certain time step values our solution would be unstable. The recent lecture introduced an implicit numerical scheme known as the Backwards in Time Central in Space (BTCS) method, which is expressed below: (1+20) T+¹oT+1-oT!+1 DAt where o = This scheme is unconditionally stable, meaning we can choose any arbitrarily large step size and the solution is always stable. (Δ.x)2 In the tutorial you began by using the brute force solution method, which involves solving a matrix equation each timestep. The Thomas algorithm (pseudocode provided more efficient approach at solving the same system of equations. = Th is a much The Thomas algorithm splits the lower, major and upper diagonal into 3 one dimensional arrays a, b and c, respectively. It uses these vectors in a forward elimination phase and backward substitution phase to determine Tnp1, which is the temperature at the next timestep. Your MATLAB function should take as input dx, dt, nt (space discretization step, timestep size in seconds, and number of timepoints), and output Tn, a, b, c, m (the final temperature distribution, and the intermediate variables used in the Thomas algorithm). You MUST use the Thomas algorithm and ensure you include the variables a, b, c, and m in your code, which match the pseudocode described in the lecture slide.

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