Suppose you are interested in the steady state temperature profile T(x, y) of a square surface with a known spatially dependent thermal diffusivity α(x, y) subject to Dirichlet boundary conditions. The time-dependent PDE governing this system is

and the boundary conditions are given by Fig. 7.16.

(a) List the most efficient combination of methods discussed in this book that you should use to solve the steady problem. In addition, using scaling arguments, estimate the time it would take to solve the problem with n_x = n_y = 100 nodes. Assume that if you discretize the x-domain using n_x = 10 nodes and the y-domain using n_y = 10 nodes, you get a program that runs in 10⁻¹ seconds.

(b) Suppose the unsteady problem must reach a time t = 10 in order to reach equilibrium. List the combination of methods you would use to solve the unsteady problem. If one time step requires 10 ms, estimate the minimum number of seconds it will take to reach the steady state using a stable, explicit time integration method. Assume that max(λ_i) = 10², max[α(x, y)] = 1 and that the discretization is n_x = n_y = 101.

Transcribed Image Text:

ат at = ах[«с»*]+[«»7] a(x,y) x(x,y).