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Laboratories

The following is a naming convention used throughout these laboratories.

aThe left-hand end point of an interval $[a, b]$ in space.
bThe right-hand end point of an interval $[a, b]$ in space.
cAs a scalar, a multiplicative constant.
As a vector, the coefficients of either a polynomial or a linear ordinary-differential equation where c(1) is the coefficient of the highest-order term and c(end) is the coefficient of the constant term or the undifferentiated term, respectively. For example, $c_1 x^2 + c_2 x + c_3$ or $c_1 y^{(2)}(t) + c_2 y^{(1)}(t) + c_3 y(t) = 0$.
fThe function $f(t, x)$, ${\bf f}(t, {\bf x})$ that defines the right-hand side of a first-order ordinary-differential equation.
fThe function $g(x)$ or $g(t)$ that defines the forcing functipn; that is, right-hand side of a second-order linear ordinary-differential equation or Poisson's equation.
hA step-size in space or time.
iAn index in space, often followed by the space variable, e.g., ix, iy, and iz.
jThe square root of -1.
kAn index in time.
mThe row-dimension of a matrix or size of the corresponding vector ${\bf b}$ in the matrix-vector problem ${\bf Mx} = {\bf b}$.
M-
nThe column-dimension of a matrix or size of the corresponding vector ${\bf x}$ in the matrix-vector problem ${\bf Mx} = {\bf b}$, or
the number of points we will break an interval $[a, b]$ or $[t_0, t_f]$ into.
N-
pAn arbitrary function or polynomial.
qAn arbitrary function or polynomial.
rAn arbitrary function or polynomial.
sA scaling factor.
tA prefix for a variable related to time.
uA prefix for a variable related to the solution of an ODE in space or space and time.
v-
w-
xA prefix for a variable related to space.
yEither a prefix for a variable with respect to space or the solution of an ODE in time.
zA prefix for a variable related to spaces.

There are a number of common suffices:

0An initial value for an initial-value problem; e.g., t0 or y0.
fA final point usually in time; e.g., tf.
_outAn output variable name.
_intrInterior points.
_bndryBoundary conditions.
_rngA range usually over time (an interval).

The reader will note that the topics covered in the laboratories focus on those topics covered in the first third of the lecture material. The in-class instruction for this course covers the algebraic mechanisms for solving these problems and the laboratories will focus on the numercial algorithms for approximating the solutions to these problems.

The first hour of each laboratory will be devoted to a presentation on the numerical methods in question. The second two hours will allow time for the students to do the laboratories and to get help from the teaching assistants.