newmark(f, t0, t1, y0, ..., N = 100, zeta = 0.25, theta = 0.5)
y0
needs to be a vector of length 2, the first component
representing y(t0)
, the second dy/dt(t0)
.t
for grid (or `time') points between t0
and t1
, and y
an n-by-2 matrix with solution variables in
columns, i.e. each row contains one time stamp.N
steps. Function f
must return a vector, whose elements hold the evaluation
of f(t,y)
, of the same dimension as y0
. Each row in the
solution array Y corresponds to a time returned in t
.
The method is `implicit' unless zeta=theta=0
, second order if
theta=1/2
and first order accurate if theta!=1/2
.
theta>=1/2
ensures stability.
The condition set theta=1/2; zeta=1/4
(the defaults) is a popular
approach that is unconditionally stable, but introduces oscillatory
spurious solutions on long time intervals.
(For these simulations it is preferable to use theta>1/2
and
zeta>(theta+1/2)^(1/2)
.)
No attempt is made to catch any errors in the root finding functions.
ode23
, cranknic
# Mathematical pendulum m l y'' + m g sin(y) = 0
pendel <- function(t, y) -sin(y[1])
sol <- newmark(pendel, 0, 4*pi, c(pi/4, 0))
plot(sol$t, sol$y[, 1], type="l", col="blue",
xlab="Time", ylab="Elongation/Speed", main="Mathematical Pendulum")
lines(sol$t, sol$y[, 2], col="darkgreen")
grid()
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