辅导案例-MAT 362

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Project 5 MAT 362
Due Sunday November 15th, Midnight.
The goal of this project is to investigate the heat and wave partial differential
equations (PDE), as well as a nonlinear elliptic ordinary BVP.
(1) Code and test explicit, implicit, Crank-Nichols, and method of lines (MOL)
solvers for the heat equation
ut = cuxx, t > 0, x ∈ (0, 1)
with 0-D BC and initial temperature distribution u(0, x) = f(x). Test with
initial temperatures given by f(x) = sin(pix), sin(pix) + 12 sin(3pix), and
something more interesting like a bump function. Compare approximate
solutions to actual solutions. Try different values of c > 0, dx, and dt.
Note the conditional or unconditional stabilities of the algorithms. Try 0-
Neumann BC at one or both ends.
(2) Code and test a MOL solver for the wave equation
utt = c
2uxx, t > 0, x ∈ (0, 1)
with 0-D BC and initial displacement and velocity given by u(0, x) = f(x)
and ut(0, x) = g(x), respectively. Test with initial displacements given by
f(x) = sin(pix) and the piecewise defined f(x) = (x− 14 )( 34 −x), x ∈ [ 14 , 34 ],
f(x) = 0 otherwiswe, assuming g = 0. Compare approximate solutions to
actual solutions obtained by the general solution or D’Alembert’s formula.
Try different values of c, dx, and dt. Run for a long time an observe sta-
bility, phase drift, or amplitude dissipation. Try 0-Neumann BC at one or
both ends.
(3) Solve y′′ + sy + y3 with 0-D BC using Newton’s method. Vary s and draw
some bifurcation branches in the ||y||∞ vs s plane.
Final Project ideas:
(1) Repeat problems 1), 2), or 3) on the square (0, 1)2.
(2) Write a general region code to produce the D2 matrix for a subregion of
(0, 1)2. Apply to solve an elliptic, hyperbolic, or parabolic problem on that
region.
(3) Experiment with 1), 2), or 3) adding other terms like convection or damp-
ing, and/or for other boundary conditions.

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