P430 Flashcards
Algorithm
a set of steps used to solve a problem; frequently these are expressed
in terms of a programming language.
Analytical approach
finding an exact, algebraic solution to a differential equa-
tion.
Cell-center grid
a grid with a point in the center of each cell. For instance, a
cell-center grid ranging from 0 to 10 by steps of 1 would have points at 0.5,
1.5, . . . , 8.5, 9.5. Note that in this case, there are exactly as many points as
cells (10 in this case).
Cell-center grid with ghost points
the same as a regular cell-center grid, but
with one additional point at each end of the grid (with the same spacing as
the rest). A cell-center grid with ghost points ranging from 0 to 10 by steps
of 1 would have points at -0.5, 0.5, . . . , 9.5, 10.5. Note that in this case, there
are two more points than cells. This arrangement is useful when setting
conditions on the derivative at a boundary.
Cell-edge grid
a grid with a point at the edge of each cell. For instance, a cell-
edge grid ranging from 0 to 10 by steps of 1 would have points at 0, 1, . . . ,
9, 10. Note that in this case, there are actually N − 1 cells (where N is the
number of points: 11 in this case). The discrepancy between the number
of cell and number of points is commonly called the “fence post problem”
because in a straight-line fence there is one more post than spaces between
posts
Centered difference formula
calculates derivatives on grids whereby the slope
is calculated at a point centered between the two points where the function
is evaluated. This is typically the best formula for the first derivative that
uses only two values.
f ′(x) ≈ (f (x + h) − f (x − h))/
2h
Courant condition
also called the Courant-Friedrichs-Lewy condition or the
CFL condition, this condition gives the maximum time step for which an
explicit algorithm remains stable. In the case of the Staggered-Leap Frog
algorithm, the condition is τ > h/c.
Crank Nicolson method
an implicit algorithm for solving differential equations,
e.g. the diffusion equation, developed by Phyllis Nicolson and John Crank.
Dirichlet boundary conditions
boundary conditions where a value is specified
at each boundary, e.g. if 0 < x < 10 and the function value at x = 10 is forced
to be 40.
Dispersion
the spreading-out of waves
Eigenvalue problem
the linear algebra problem of finding a vector g that obeys
Ag = λg.
Explicit algorithm
an algorithm that explicitly use past and present values to
calculate future ones (often in an iterative process, where the future values
become present ones and the process is repeated). Explicit algorithms are
typically easier to implement, but more unstable than implicit algorithms
Extrapolation
approximating the value of a function past the known range of
values, using at least two nearby points.
Finite-difference method
method that approximates the solution to a differen-
tial equation by replacing derivatives with equivalent difference equations
Forward difference formula
a formula for calculating derivatives on grids whereby
the slope is calculated at one of the points where the function is evaluated.
This is typically less exact than the centered difference formula, although
the two both become exact as h goes to zero.
f ′(x) ≈ (f (x + h) − f (x))/
h
Gauss-Seidel method
a Successive Over-Relaxation technique with w = 1
Generalized eigenvalue problem
the problem of finding a vector g that obeys
Ag = λBg.
Implicit algorithm
an algorithm that use present and future values to calculate
future ones (often in a matrix equation, whereby all function points at a
given time level are calculated at once). Implicit algorithms are typically
more difficult to implement, but more stable than explicit algorithms.
Interpolation
approximating the value of a function somewhere between two
known points, possibly using additional surrounding points
Iteration
repeating a calculation a number of times, usually until the error is
smaller than a desired amount.
Korteweg-deVries Equation
a differential equation that describes the motion of
a soliton.
Lax-Wendroff algorithm
an algorithm that uses a Taylor series in time to obtain
a second-order accurate method
Neumann boundary conditions
boundary conditions where the derivative of
the function is specified at each boundary, e.g. if 0 < x < 10 and at x = 10
the derivative d x/d t is forced to be 13.
Resonance
the point at which a system has a large steady-state amplitude with
very little driving force
Roundoff
an error in accuracy that occurs when dealing with fixed-point arith-
metic on computers. This can introduce very large errors when subtracting
two numbers that are nearly equal
Second derivative formula
this is a centered formula for finding the second
derivative on a grid:
f ′′(x) ≈ (f (x + h) − 2 f (x) + f (x − h))/
h2
Staggered-Leap Frog
an algorithm for solving differential equations (e.g. the
wave equation) whereby the value of a function one time step into the
future is found current and past values of the function. It suffers from the
difficulty at the start of needing to know the function value prior to the
initial conditions.
Steady State
solutions to differential equations in which the system reaches an
equilibrium solution that continues on forever in the same fashion.
Successive Over-Relaxation
A way to shift the eigenvalues. More specifically, it
is using a multiplier w to shift the eigenvalues., with w > 1
Successive Over-Relaxation (SOR)
An algorithm for solving a linear system such
as Vnew = L × Vold + r by iterations, by shifting the eigenvalues. This can be
done via solving the equivalent problem of
Vnew = w × [RHS of previous equation] + (1 − w) × Vold .
For good choices of w, this can lead to much quicker convergence.
Transients
solutions to differential equations that are initially present but which
quickly die out (due to damping), leaving only the steady-state behavior
