A classic example of a diffusion problem with a time-varying boundary condition is the diffusion of...

 

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A classic example of a diffusion problem with a time-varying boundary condition is the diffusion of heat into the crust of the Earth, as surface temperature varies with the seasons. Suppose the mean daily temperature at a particular point on the surface varies as: 2t To(t) = A + B sin where T = 365 days, A = 10 -C and B = 12 -C. At a depth of 20m below the surface almost all annual temperature variation is ironed out and the temperature is, to a good approximation, a constant 11 -C (which is higher than the mean surface temperature of 10 -C-temperature increases with depth, due to heating from the hot core of the planet). The thermal diffusivity of the Earth's crust varies somewhat from place to place, but for our purposes we will treat it as constant with value D = 0.1 mday' Write a program to calculate the temperature profile of the crust as a function of depth up to 20m and time up to 10 years, using Forward Time Centered Space (FTCS). Instructions: Start with temperature everywhere equal to 10 -C, except at the surface and the deepest point. Choose values for the number of grid points and the time-step h, considering the length scale and time scale of this problem, and making sure to satisfy the stability condition for FTCS. Then run your program for the first nine simulated years, to allow it to settle down into whatever pattern it reaches. Then for the tenth and final year plot four temperature profiles taken at 3-month intervals on a single graph to illustrate how the temperature changes as a function of depth and time. Make sure, by comparing with a larger number of grid points and a smaller time-step h (differing by at least a factor of 2), that your choices for these give accurate results. A classic example of a diffusion problem with a time-varying boundary condition is the diffusion of heat into the crust of the Earth, as surface temperature varies with the seasons. Suppose the mean daily temperature at a particular point on the surface varies as: 2t To(t) = A + B sin where T = 365 days, A = 10 -C and B = 12 -C. At a depth of 20m below the surface almost all annual temperature variation is ironed out and the temperature is, to a good approximation, a constant 11 -C (which is higher than the mean surface temperature of 10 -C-temperature increases with depth, due to heating from the hot core of the planet). The thermal diffusivity of the Earth's crust varies somewhat from place to place, but for our purposes we will treat it as constant with value D = 0.1 mday' Write a program to calculate the temperature profile of the crust as a function of depth up to 20m and time up to 10 years, using Forward Time Centered Space (FTCS). Instructions: Start with temperature everywhere equal to 10 -C, except at the surface and the deepest point. Choose values for the number of grid points and the time-step h, considering the length scale and time scale of this problem, and making sure to satisfy the stability condition for FTCS. Then run your program for the first nine simulated years, to allow it to settle down into whatever pattern it reaches. Then for the tenth and final year plot four temperature profiles taken at 3-month intervals on a single graph to illustrate how the temperature changes as a function of depth and time. Make sure, by comparing with a larger number of grid points and a smaller time-step h (differing by at least a factor of 2), that your choices for these give accurate results.