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SPRING SEMESTER 2021-2022

MATH 4 064 - COURSEWORK 2

1. Problem set 1: Parabolic PDEs

Consider the following system of Partial Differential Equations (PDEs)

⎧ ⎪ , ) = ᵆ + ( , ),

with Neumann boundary conditions

in = [0, ] × (0, ] in = [0, ] × (0, ]

= 0 | =0, ,

where ᵆ and are the diffusion coefficients ( ≫ ᵆ ). The forcing term is given by

( , ) ≔ ( 0 + ) − ,

[30 marks]

(1)

(2)

(3)

where 0 = 0.067.

(a) Write down the Crank– Nicolson scheme for this system of PDEs using first-order finite difference

approximations (using the two ending grid points) for the Neumann boundary conditions.

(b) Treating the forcing term explicitly, implement and solve part (a) for = 1 and = 200 with the initial

condition be given by

( , 0) = {0.1 for 0 ≤ ≤ 0.9,

( , 0) =

(4)

(5)

where = 2.26, = 1/10, ᵆ = 0.01 , using ℎ = /99 and = /999. Plot your solution (two heatmaps, for and , on the plane ( , ), and one plot for and , versus at final simulation time). [check https://www.tutorialspoint.com/matplotlib/matplotlib_contour_plot.htm for plotting heatmaps.]

(c) Let ℐ1 be the index set of all nodes (grid points) for a discretization with ℎ 1 = /99, ℐ2 for ℎ2 = ℎ 1 /2, ℐ3 for ℎ3 = ℎ2 /2, and ℐ4 for ℎ4 = ℎ3 /2. Compute

where and are given functions. Your code should use central finite differences to approximate the Laplacian operator on a uniform grid with + 1 nodal points in each coordinate direction. Note, your code will need to solve the linear system of size ( − 1) 2

= , (8)

where is the vector of approximate nodal values, and should include separate routines to find and given , and .

(b) Suppose ( , ) = cos(4 ) cos(4 ) solves the above Poisson’s equation on = [0, 1]2 . Find

and and hence test your code by selecting = 16, 32, 64, 128, 256, 512, 1024 and computing

of the approximate solution when = 64. [Hint: You will need to use a sparse matrix format for , otherwise the system will not fit in memory for large N. Check https://www.geeksforgeeks.org/ how-to-create-a-sparse-matrix-in-python/ and https://docs.scipy.org/doc/scipy/referen ce/ generated/scipy.sparse.linalg.spsolve.html ]

[40 marks]

Consider the hyperbolic PDE (scalar transport)

(9)

(10)

(d) (two-dimensional) Using the solver implemented in (b), solve the scalar transport equation with the following parameters: = (1, 0.5), 0 = (1 − cos )(1 − cos ), = = 2 /20, and = 1/10. Compare (plotting) your solution with the analytical one and compute the discrete 2 using the grid norm function for the final simulation time.