Final Projects Math104 CChoose one problem1. The cellar. Neglecting the curvature of the Earth and the diurnal (daily) variation oftemperature, the distribution of temperature T (x, t) at a depth x and a time t is givenby the Heat equation:∂T∂t= κ∂2T∂x2. (1)Here κ is thermometric diffusivity of soil whose value is approximately κ = 2 ×10−3cm2/sec (the fundamental time scale is a year, 3.15 × 107sec). Assume that thetemperature f(t) at the surface of the Earth (x = 0) has only two values, a “summer”value for half of the year and a “winter” value for the other half, and that this patternis repeated every year (i.e. at x = 0 the temperature is periodic with a period of ayear). The temperature T should decay to zero as x → ∞.a) Show that the backward-difference scheme for (1) is consistent and unconditionallystable. What is the order of the scheme?b) Implement the backward-difference scheme to find a numerical approximation to(1). Consider the initial condition u0(x) = f(t)e−q1x, where q1= 0.71m−1. Foryour computational spatial domain take a sufficiently long interval so that theright-end boundary condition u = 0 can be use d. Select k and h small enough toresolve well the numerical solution. Plot the numerical solution at several times.c) From your numerical solution, find the depth xcat which the temperature isopposite in phase to the surface temperature, i.e, it is summer at xcwhen iswinter at the surface. Note that the temperature variation at xcis much smallerthan that at the surface. This makes the depth xcideal for a wine cellar orvegetable storage.2. A simple model for air quality control. An air pollutant gets advected by the windand at the same time diffuses as it travels. The time evolution of the concentrationu(x, y, t) of the pollutant at position (x, y) and at time t can be modelled by theadvection diffusion equationut+ Uwux+ Vwuy= D(uxx+ uyy), (2)where (Uw, Vw) are the components of the wind velocity and D > 0 is the diffusivitycoefficient (assumed small) of the p ollutant in the air.a) The one-dimensional case of (2) isut+ Uwux= Duxx. (3)(If D = 0, this is the simple one-way wave equation (also called advection equa-tion) we have seen in class). Assuming Uw< 0 (and constant) find the stabilitycondition for the schemeun+1m− unmk+ Uwunm+1− unmh= Dun+1m+1− 2un+1m+ un+1m−1h2. (4)1b) Write down an algorithm (doesn’t have to be a code but it should be c lear andunambiguous as to how to proceed) to compute the solution to (3) at each timestep.c) If D = 0, one gets an “upwind” scheme for the one-way way equation. Find themodified (model) equation for the “upwind” scheme and show that the schemeis dissipative. How do you have to take your numerical parameters to guaranteethat your numerical diffusion is much less than the “real” diffusion when D 6= 0?.Explaind) Write a consistent and stable scheme for (2).3. Acoustic waves. The air pressure p(x, t) in an organ pipe is governed by the waveequation∂2p∂t2= c2∂2p∂x20 < x < l, t > 0, (5)where l is the length of the pipe and c is a constant. If the pipe is open, the boundarycinditions are given byp(0, t) = p0and p(l, t) = p0. (6)If the pipe is closed at the end x = l the boundary conditions arep(0, t) = p0and∂p∂x(l , t) = 0. (7)Assume that c = 1, l = 1, and the initial conditions arep(x, 0) = p0cos 2πx, and∂p∂t(x, 0) = 0 0 ≤ x ≤ 1. (8)a) Write down an explicit finite difference method for (5) and give stability conditionsand the order of the method.b) Implement your method given in a) for the open pipe with p0= 0.9, and withstep sizes k = h = 0.05. Plot your numerical solution at t = 0.5 and t = 1.0.c) Implement your method given in a) for the closed pipe at x = l with p0= 0.9,and with step sizes k = h = 0.05. Plot your numerical solution at t = 0.5 andt = 1.0.d) Repeat b) for k = h = 0.025. C onstruct a higher order approximation by extrapo-lating your numerical solutions corresponding to k = h = 0.05 and k = h = 0.025.What’s the order of the new