(Programming) Modify the given Backward-Difference and Crank-Nicolson codes to solve the heat equation Ut = 1 16uxx 0 0 u(0, t) = u(1, t) = 0, u(x, 0) = 2 sin 2πx, π² The exact solution is u(x, t) = 2e sin 2x. Stop computing at time t = 1. t> 0, 0 ≤ x ≤ 1. a² At (Δx)2 (a) Use Backward-Difference method, λ = number of subintervals in x direction). Observe the errors and error ratios, Can you determine if the method is of order Ax or (Ax)²? (b) Use Crank-Nicolson method, λ = 1, with m= 4, 8, 16, 32, 64, 128, and 256. (m is the number of subintervals in x direction). Observe the errors and error ratios, Can you determine if the method is of order Ax or (Ax)²? = 1, with m = 4, 8, 16, 32, 64, 128, and 256. (m is the
(Programming) Modify the given Backward-Difference and Crank-Nicolson codes to solve the heat equation Ut = 1 16uxx 0 0 u(0, t) = u(1, t) = 0, u(x, 0) = 2 sin 2πx, π² The exact solution is u(x, t) = 2e sin 2x. Stop computing at time t = 1. t> 0, 0 ≤ x ≤ 1. a² At (Δx)2 (a) Use Backward-Difference method, λ = number of subintervals in x direction). Observe the errors and error ratios, Can you determine if the method is of order Ax or (Ax)²? (b) Use Crank-Nicolson method, λ = 1, with m= 4, 8, 16, 32, 64, 128, and 256. (m is the number of subintervals in x direction). Observe the errors and error ratios, Can you determine if the method is of order Ax or (Ax)²? = 1, with m = 4, 8, 16, 32, 64, 128, and 256. (m is the
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. (Programming) Modify the given Backward-Difference and Crank-Nicolson codes to solve the heat
equation
Ut =
1
•Uxx¹
16
u(0, t) = u(1, t) = 0,
u(x, 0) = 2 sin 2πx,
π²
-t
The exact solution is u(x, t) = 2e¯¯4 sin 2πx.
Stop computing at time t
1.
=
0 < x < 1, t> 0
=
a² ▲t
(Ax)²
1, with m =
(a) Use Backward-Difference method,
4, 8, 16, 32, 64, 128, and 256. (m is the
number of subintervals in x direction). Observe the errors and error ratios, Can you determine if the
method is of order Ax or (Ax)²?
(b) Use Crank-Nicolson method, 1, with m = 4, 8, 16, 32, 64, 128, and 256. (m is the number of
subintervals in x direction). Observe the errors and error ratios, Can you determine if the method is
of order Ax or (Ax)²?
t> 0,
0 ≤ x ≤ 1.
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VIEWStep 2: (a) Finding the Backward-Difference iteration solution
VIEWStep 3: Creating the error plot using Python Code
VIEWStep 4: Code Explanation
VIEWStep 5: Code Output
VIEWStep 6: (b) Finding the Crank-Nicolson iteration solution
VIEWStep 7: Creating the error plot using Python Code
VIEWStep 8: Code Explanation
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