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6.2 CubicInterpolated Propagation (CIP)
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Introduced a decade ago by Yabe and Aoki [34], a whole family of
schemes have been proposed along the same lines, relying on different
interpolations to propagate the solution along the characteristics.
Using a cubicHermite polynomial, the discretized function and its first
derivative
is approximated in a continuous
manner with
Both
satisfy the master advection equation

(1) 
and each is split into an advection (top) and nonadvection phase (bottom)

(2) 
For the advection phase, the solution can be integrated analytically simply
by shifting the cubic polynomials
along the characteristics

(3) 
without any restriction on the size or the sign of the CourantFriedrichLewy
(CFL) number
(exercise 6.01).
For pedagogical reasons, the scheme implemented in JBONE
assumes that
so that both quantities
can be interpolated from only one piece of the piecewise continuous
polynomial, namely
defined in the interval
.
After an initialization where the
function is discretized
with cubicHermite polynomials by sampling on a grid and the
derivative are calculated
with centered finite differences, the CIP scheme is
implemented as
double alpha=timeStep*diffusCo/(dx[0]*dx[0]); //These are only constant
double beta =timeStep*velocity/(dx[0]); // if the problem and the
int n=f.length1; // mesh are homogeneous
for (int j=0; j<n; j++) {
a=dx[0]*(df[j]+ df[j+1])2*(f[j+1]f[j]);
b=dx[0]*(df[j]+2*df[j+1])3*(f[j+1]f[j]);
fp[j+1]= f[j+1] beta*(dx[0]*df[j+1]beta*(bbeta*a));
dfp[j+1]= df[j+1] beta/dx[0]*(2*b3*beta*a);
}
a=dx[0]*(df[n]+ df[0])2*(f[0]f[n]);
b=dx[0]*(df[n]+2*df[0])3*(f[0]f[n]);
fp[0]= f[0] beta*(dx[0]*df[0]beta*(bbeta*a));
dfp[0]= df[0] beta/dx[0]*(2*b3*beta*a);
The applet below
illustrates the high quality of this approach, which combines a low level
of dispersion with low damping.
JBONE applet: press Start/Stop
to simulate the advection of a box function and to test how cubic
Hermite FEM interpolated propagation (CIP) affects the dispersion
and the damping of short and long wavelengths superposed in a box
function.

Some additional bookkeeping is of course necessary in a code that is intended
for
: exercise 6.01 deals with exactly this problem and can be
implemented in a similar manner as illustrated with the
CubicSpline scheme.
SYLLABUS Previous: 6.1 Splitting advection from
Up: 6 LAGRANGIAN METHOD
Next: 6.3 NonLinear equations with
