Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
A ROBUST SOLVER FOR DELAY DIFFERENTIAL EQUATIONS
Ivan N. Kirschner
Anteon Corporation  Engineering Technology Center
One Corporate Place, Middletown, RI 02842-6277 USA
ABSTRACT
The dynamics of supercavities and supercavitating high-speed bodies involve a delayed argument. Under certain flow
conditions, supercavitating high-speed bodies are subject to the hydrodynamical effect known as cavity auto-oscillation.
Paryshev developed a theory to predict such behavior for nonstationary, axisymmetric cavities based on Logvinovich’
theory of independence of cavity section expansion. Paryshev’s theory involves a transformation of coordinates from a
local system fixed to each cavity section to a global system fixed to the cavitator. The resulting mathematical model
involves a system of integro-differential equations with delayed argument. Similarly, the hydrodynamic forces acting on
supercavitating bodies involve a delay associated with the advection of disturbances downstream from the cavitator to the
afterbody or fins.
Delay differential equations, which are special cases of Volterra functional differential equations, are found across a
broad spectum of disciplines. Numerical integration of delay differential equations cannot be based simply on the mere
adaptation of some standard ordinary differential equation solver to the presence of delayed terms, since such standard
methods are subject to failure in both order (that is, a loss of accuracy) and stability, depending on the problem being
solved. This paper presents an approach to the numerical solution of delay differential equations that appears to be quite
robust. In the proposed approach, the time history of the solution is represented using piecewise cubic Hermite
interpolating polynomials, which preserve the monotonicity of the solution, eliminating spurious oscillation in the
continuous extension of the discrete solution history, thereby stabilizing the solution method.
In addition to a brief description of the numerical approach, the paper includes validation against a number of known
or previously-computed solutions.
Keywords: supercavitation, cavity auto-oscillation, supercavitating vehicle dynamics, delay differential equations, Hermite polynomial
interpolation, monotonicity-preserving
NOMENCLATURE
General nomenclature:
t
time
PCHIP function
P t 
y
general solution variable or vector

time delay
For Paryshev’s system, equations (1)-(14):
constant defined by conditions of cavity gas
C
at start of simulation
coordinate along trajectory in flow direction
h
coordinate along trajectory of cavitator
h0  t 
h t 
coordinate along trajectory of cavity closure
kout
 0.01, cavity gas leakage constant
cavity length
min
mout
n
cavity ventilation mass flow rate
t 
cavity gas loss mass flow rate
pc  t 
reciprocal of polytropic constant of cavity
gas
cavity pressure
pv
vapor pressure of ambient liquid
p  h, t 
q
Q t 
pressure at infinity
volumetric cavity gas leakage rate at cavity
closure conditions
cavity volume
R
S  h, t 
cavity gas constant
cavity cross section
S0 , S '0 , k
Sb  h 
constants determined by characteristics of
cavitator
body cross section
Sc  t 
cross section at cavity mid-point
t
time of formation of cavity section at
hh
absolute temperature of cavity gas
speed of cavitator
T
V t 

c

v
density of ambient water
density of cavity gas
 p  pc 
 p  pv 
1 2 V  , cavitation number
1 2 V  , vaporous
2
2
cavitation number
t  t , time delay
For the simplified dynamical supercavitating rigid body
model, equations (15)-(17):
effective damping coefficient
b
magnification factor
h
Heaviside function
H
effective spring constant
k
distance between cavitator and transom
m
effective mass
x
body position

p
distance from transom edge to cavity
boundary in undisturbed condition
displacement of cavity surface due to
 t 
motion of cavitator
For the Mackey-Glass equation, equation (19):
a
random white blood cell destruction rate
parameters
b, n
density of circulating white blood cells
y t 
For the chemostat equation, equations (20):
species-specific parameters
a, b, c
chemostat flow rate
D
species-specific per-capita nutrient uptake
p  S t 
rate
concentration of unconsumed nutrient in
S t 
growth vessel
concentration of growth-limiting nutrient
S0
biomass of micro-organism population
x t 

constant function of constant delay
INTRODUCTION
Memory effects are prominent in supercavitation, resulting
in such behavior as cavity auto-oscillation [1] and
complicated body dynamics (see, for example, [2,3]).
Delay differential equations (DDEs) are found across
a broad spectrum of disciplines, including mechanics,
physics, biology, medicine, and economics, and may be
used to model such diverse phenomena as polymer
crystallization, relativistic dynamics, distributed networks,
ship course stabilization, the epidemiology of HIV, and the
chaotic release of mature cells into the blood stream of
leukemia patients (see, for example, [4]). In contrast to the
Cauchy problem for ordinary differential equations
(ODEs), which requires an initial condition at only a single
point, DDEs require an initial condition – the initial data –
over a domain that extends far enough into the past that the
delay can be computed at each point in the domain of the
solution. DDEs are often categorized according to the
form of the non-negative delayed arguments,  i : constant
 i   0i , constant  ; a variable function of
t ,  i   i  t   ; or dependent on the solution state,

i
time,
y t  ,

  i  t,y  t   , with the latter comprising the most
general case. Paryshev’s theory of cavity stability is
linearized about a nominal stationary condition. However,
when accounting for acceleration of a cavitator coupled to
a body that moves according to Newton’s second law,
Paryshev’s general nonlinear system of equations involves
a state-dependent delay.
As discussed in Bellen and Zennaro [5], numerical
integration of DDEs cannot be based simply on the mere
adaptation of some standard ODE solver to the presence of
delayed terms, since such standard methods are subject to
both order and stability failure, depending on the problem
being solved. This paper presents an approach to the
numerical solution of DDEs that appears to be quite robust.
In the proposed approach, the time history of the solution
is represented using piecewise cubic Hermite interpolating
polynomials (PCHIPs), which preserve the monotonicity of
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
the solution. This monotonicity-preserving property of the
PCHIP fit, and the associated elimination of spurious
oscillation, makes it attractive as a means of providing a
continuous extension of the discrete solution over the
history of the simulation up to and including the most
recent time step.
In order to avoid both re-fitting of the entire solution
at each time step and subsequent costly functional
evaluation, a scheme has been constructed wherein the
time history of the solution is covered with overlapping
PCHIP representations. The resulting method allows for
straightforward application of any pre-existing ODE
solver, and is immediately compatible with adaptive timestepping algorithms, for example, those designed to solve
stiff systems.
1. PROBLEM STATEMENT
1. 1 Statement of two problems of interest
Paryshev’s theory describing the dynamics of a cavitator
and its ventilated cavity was developed based on the
underlying principle of independence of the expansion of
cavity sections, as formulated by Logvinovich [6]. The
resulting system of equations may be summarized as
follows [1]:
Q

h
k
 h
p  h, t  dh 
0
 S '0 V  V  Sb 
2
k
pc  t   h

  S0   VS 'b  

S h ,t 
t
,
d 
S h ,t  
dt t
 S ' V  t   
 0
 V  t   
t
 
k  p  h , t   

 
 
p  h , v  dv  h
  V  t   
h
t 
 

k

(1)
(2)
p  h , t  ,

k
S  h , t   S '0 V  t      p  h , v  dv,
t
 t 
t

S h ,t 

h
 V t   
S '0 1 

 V  t    

(3)
(4)
t u

k  p  h , t   



p  h , v  dv du  ,



  V  t   
h
t  t 


S  h , t   VS 'b 

h  t

S  h , t   S 'b 
h


,
(5)
h  h0 ,
(6)
 h V ,
(7)
  1
h
V t  
p  h, t 
(8)
p  h, t   pc t  ,
Sb  S b  x  ,
m  mout  CQp
pc  in
,
nCQpcn-1
p  h
RT
 bp  H   t ,   1  H   t ,  x  t   hx  t   

 kp  H   t ,   1  H   t ,    x  t   hx  t    

,


q  kout ScV  v  1 ,


and
d
V  F V , h0 , pc , h ,... .
dt
(11)
(12)
(15)

 p   H   t ,   1  H   t ,  
(10)
n
c
mout  q
(9)
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
mx  t   bf x  t   kf  t  x  t 
 mg  L
where
H   t , 
H  x  t   hx  t      p 
(16)
(13)
(14)
where V  V and the cavity ventilation rate, min  min t 
is specified. Here, equation (1) provides an expression for
the cavity volume, which has been derived from
Logvinovich’s differential equation for the expansion of a
transverse cavity section under the influence of a time- and
spatially-varying difference between the local ambient
pressure and that within the cavity. That equation
incorporates the effects of the cavitator drag and geometry.
Equation (6) can be considered a definition of cavity
length, and equation (7) is the kinematical relation among
its time rate of change, that of the coordinate of cavity
closure, and the cavitator speed. Equation (9) defines the
key pressure difference that governs supercavitating flows.
Equation (10) is a given function that defines the body
profile. Equations (11) and (12) are derived from basic
thermodynamics and considerations of conservation of
mass of the cavity gas. Equation (13) is Logvinovich’s
semi-empirical expression for gas leakage in the re-entrant
jet regime of cavitating flows. Equation (14) describing
the cavity dynamics is determined from the free-body
diagram of the cavitator and the body to which it is
attached; its functional form derives from the
hydrodynamical forces acting on the system.
The
remaining expressions result from applying the chain rule
to transform the expression for the evolution of a local
cavity transverse section to a global system with delay.
Delay is introduced into this system via the boxed
equations, (2), (3), (4), and (8); that the delay is statedependent can be seen in equation (8) via the velocity,
which is a state variable of the system.
The dynamics of supercavitating vehicles also
involves a time delay, associated with the advection of
disturbances at the cavitator downstream to the fins or
afterbody, where their effect is applied to affect the body
motion [3]. A simplified dynamical system suitable for
investigating heave or pitch motions of a supercavitating
body is presented in figure 1.
The equation of motion of such a system (in this case,
for a system without fins) is
Figure 1: A simplified dynamical systems model for
supercavitating high-speed bodies, from Kirschner,
et al, 2003.
and  p denotes the distance from the edge of the transom
to the cavity boundary in the undisturbed condition. Here
the time delay is approximated as the advection time from
the cavitator to the transom,   u . The Heaviside
function, H , captures the slope-discontinuous nature of
the forces as the transom contacts the cavity boundary.
The quantity h (here with a different meaning than in
Paryshev’s system, equations (1) through (14)) involves
the relationship between the motion of the mass and the
change in the shape of the cavity as transferred via the
cavitator. For example, heave motions represented by the
simplified model result in cavity displacement in the
direction of the mass displacement, whereas pitch motions
about a center located somewhere between the cavitator
and the transom have the opposite effect. Since pitch
motions also involve a weighting accounting for the
distance between the cavitator and the transom, in general
the quantity h is a magnification factor that can be
positive or negative, depending on the case of motion
considered and the vehicle geometry represented. With
this definition, the cavity disturbance is related to the mass
position by
(17)
 t   hx t   .
The left-hand side of the equation of motion represents
the effects of inertia (the first term), the fin forces
(damping and lift perturbation, as represented by the
second and third terms), and the afterbody planing forces
(damping and lift perturbation, as represented by the fourth
and fifth terms). The first term on the right-hand side
accounts for the effects of gravity (appropriate to motion in
the vertical plane; ignored for motion in the horizontal
plane). The final term represents the steady state lift in the
undisturbed condition. In this simplified system, the delay
is approximated as a constant or some specified function of
time; in a full six-degree-of-freedom simulation accounting
for body acceleration, the delay would be state-dependent.
As a system with delay, initial data must strictly be
specified on an initial interval, rather than at a single point.
It is expected that its solutions may be sensitive to the form
of this initial data, and the dynamical behavior of the
system may display certain of the peculiarities of DDEs.
Bellen and Zennaro [5] note several interesting
mathematical characteristics of DDEs, here summarized as
follows:
 In general, the solution, y , is not smoothly linked to
the initial function,   t  , at the point t0 , where only
C 0 -continuity can be assured. Such a derivative
jump discontinuity propagates from the initial point ,
t0 , along the integration interval, and gives rise to
subsequent discontinuity points where the solution is
smoothed out more and more. Thus, even if the
right-hand side, the delay, and the initial data are
C  -continuous, in general the solution is simply C1 continuous in t0 , t f  .

Unlike ODEs, there is no longer injectivity between
the initial data and the set of solutions. As an
example, Bellen and Zennaro [5] present a case for
which all solutions are identically unity for any initial
function defined on a certain interval such that its
value at t  0 is unity.
 For the case of state-dependent delay, irregularity of
the initial function,   t  , may cause a loss of
uniqueness of the solution or its termination after
some bounded interval.
 The addition of a delayed term may drastically
change the behavior of the solution by acting as a
stabilizer or a de-stabilizer of models governed by
ODEs.
 Perhaps most importantly for supercavitating
vehicles, whereas bounded solutions of autonomous
ODEs may oscillate only if the system has at least
two components, and may behave chaotically only if
there are at least three components (PoincaréBendixon theorem), DDEs may exhibit oscillatory
and even chaotic behavior in the scalar case.
Bellen and Zennaro [5] also discuss the fact that many
common numerical solution techniques for ODEs are
subject to order or stability failure when applied to DDEs.
2. NUMERICAL SIMULATION
2. 1 A PCHIP covering method
In order to solve problems in high-speed
hydrodynamics involving a time delay, such as the
Paryshev system presented in section 1, a DDE solver has
been developed. There were two primary objectives: (1) a
robust solver that avoids some of the numerical issues
described above; (2) a solver that is easily incorporated
into existing software employing MATLAB ODE solvers.
A fundamental process in numerical solution of DDEs
is providing a means of evaluating the solution at all
required points in the domain including the initial interval
and that of the solution up through and including the last
computed time step. In general, the solution must be
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
available at any point in this domain, not simply those
discrete points at which it was initially computed. Thus,
some continuous extension of the discrete solution must be
provided. It was found that the PCHIP function available
in the MATLAB library provides a useful method of fitting
the initial data and the solution.
The PCHIP function [7] fits a piecewise cubic Hermite
polynomial to a set of points  t j , y j  , j  1: n . The PCHIP
function finds values of an underlying interpolating
function, y  P(t ) , at intermediate points, such that: (1) on
each subinterval, P(t ) is the cubic Hermite interpolant to
the given values and certain slopes at the two endpoints;
(2) P(t ) interpolates y (t ) , i.e., P  t j   y j , and the first
derivative P ' t  is continuous; and (3) the slopes at the fit
data, t j , are chosen in such a way that P  t  preserves the
shape of the data and respects monotonicity – that is, on
intervals where the data are monotonic, so is P  t  , and at
points where the data has a local extremum, so does P  t  .
Note that the second derivative, P ''(t ) is probably not
continuous at the fit data points: there may be jumps at t j .
The difference between PCHIP and a cubic spline is
that the second derivatives at the nodes are not necessarily
continuous. A cubic spline will produce a smoother result
that will be more accurate if the data are values of a
smooth function. Conversely, PCHIP has no overshoots
and less oscillation if the data are not smooth, and is less
computationally costly to set up. The two fits are equally
expensive to evaluate. Comparison of the two fits for
arbitrary data sets are shown in figure 2. It is readily seen
that, if such a curve represented the solution history of a
simulation, the spurious oscillations in the cubic spline
curves in figure 2 could introduce subsequent oscillations
into the ongoing solution. Under many conditions, these
oscillations could be amplified as the solution proceeded,
leading to a loss of stability. Thus, the PCHIP fit is
proposed as a preferred approach to providing the required
continuous extension of the discrete solution history. (It is
noted, however, that other methods, such as B-splines, may
also be adequate to this task, and may have other
advantages. These have not been explored under the
current effort.)
Refitting the entire initial interval and the solution at
each time step is extremely costly from a computational
perspective. Therefore, a covering of the entire domain up
through the last computed time step is generated, wherein
each fit covers at most a fixed number of points and
overlaps with the subsequent fit on its last two intervals. A
schematic of this approach is presented in figure 3. When
the covering is to be evaluated on the overlapping portion
of two PCHIP fits, that fit which provides the most interior
interval for the given location is selected. So, for example,
point A in figure 3 would be evaluated using PCHIP #1,
whereas PCHIP #2 would be employed for point B . In
numerical experiments, covering the domain using this
scheme reduced computational times very significantly in
comparison with a refitting of the entire initial interval and
solution at each time step.
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5-4
-4
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
(2) the ability of each solver to optionally call a userspecified function upon the successful integration of each
successive time step. This latter feature was designed into
MATLAB to allow plotting of the solution as the
simulation proceeds, and is designated as an “output
function” in the MATLAB documentation [7]. In the
current procedure, this capability is recast to provide the
necessary continuous extension of the discrete solution
after each integration step. The structure of the resulting
simulation algorithm is shown in figure 4.
data
pchip
data
spline
pchip
data
spline
pchip
data
spline
pchip
spline
-2
-2
0
0
2
2
Simulator
4
4
a
data
pchip
data
spline
pchip
data
spline
pchip
data
spline
pchip
spline
1.1
1.1
1
1
Right-hand side update
0.9
0.9
0.8
1
2
3
0.8 0
b
0
1
2
3
Figure 2: Comparison of PCHIP and cubic spline
fits of two arbitrary data sets – a entire data set;
b close-up view showing overshoots in cubic spline
fit.
2
y
PCHIP #1
A
B
PCHIP #3
1
initial data
PCHIP #2
overlaps
most recent
solution point
0
-1
0
1
2
ODE solver
t
3
Figure 3: The PCHIP covering method for
continuous extension of the discrete DDE solution.
2. 2 MATLAB implementation
The PCHIP covering method was implemented in
MATLAB. It was desirable to allow the use of existing
MATLAB ODE solvers [7], which have been developed
and optimized for various problems, and include such
sophisticated features as automated adaptive time-stepping.
In order to back-fit the solution history, advantage was
taken of two features available in the standard MATLAB
ODE solvers, namely: (1) the ability to pass a userspecified function to update the right-hand side of the
system of ODEs through the solver argument list; and,
Output function
Figure 4: Structure of the simulation algorithm for
solution of DDEs using standard MATLAB ODE
solvers and the output function feature to allow
back-fitting of the solution history.
As will be shown below, this approach appears to be
suitable for solving a broad range of DDE problems. It
should be noted that it has not been tested for certain
classes of problems, such as those with vanishing delay, of
which the pantograph equation is an example. (For a
discussion of this equation, see, for example, [5]; its
stochastic form is discussed in [8].) It is expected that
modification would be required to apply this method to
such problems. The delay does not vanish for the Paryshev
system except when the cavity length is zero, a case that is
not especially interesting for real problems in high-speed
hydrodynamics.
2. 3 Computational results
The PCHIP covering method was applied in comparison
with various known results. Selected examples are
presented in figures 5 through 9.
The stability failure that can occur for DDEs is
exemplified by a class of constant coefficient linear test
equations:
4

t  0,
 y '  t    y  t    y  t  1 ,
(18)
5

 y  t   1,
t  0,

whose solutions as presented in reference [5] and here
depicted in figure 5a for various values of  , are
asymptotically stable for any   0 .
Surprisingly, the asymptotic stability of these
solutions are not reflected in the results of certain
numerical solutions that are stable for the case without
delay. The comparison using the PCHIP covering method
is presented in figure 5b. It can be seen that the
comparison is excellent, in contrast with those produced
using the midpoint rule and shown in figure 5c. The
PCHIP covering method also compares favorably with the
trapezoidal rule.
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
  1
1
  5
y
1.5
0.8
y
0.6
y'
y
y
1
y
0.4
0.2
0.5
0
-5
1
0.8
0
0.8
0.6
-0.05
0.6
5
t
  25
5
t 0.4
5
t
y
1
0
10
0.8
-1
0.8
-1
  y1
0.2
-4
0.2
-4
0
-5-5
0
0
-5
-5
0
0
00.2
0.2
0.4
1
1
0
0
0.8 0.8
5
t
10
15
a
5
5
0.40.6
t
t
y
y
0
-500
0.1
0.05
0.2
0
0.4
5
0.6
t
y
1
y0
10
0.8
15
1
  100
y
0
-0.05
yp
yp y
y
0.6
-0.4
0.4
-0.6
0.2 0.2
-0.15 -0.15
-0.1
0.2
-0.8
0
0
-5
0
-5
0
-0.2 -0.2
0 0.2 0.20.4
0
5
5
t
t 0.6
0.4
y
y
0
15
10 10
15-1-5
0.60.8 0.8 1 0 1
0
0.2
0.4
0.2
0.4
5
t 0.6
y
10
0.8
0.6
0.8
15
1
b
5
0
-1
0
-2
-2
-5
yp
0
-1
-3
-3
-10
-4
-4
-15
-5
-5
-20
01
0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1
y
y
yp
-0.1
0
-500 0
0 500
t
0.4
-10
  25
y
1.5
-0.05
0.6
-5
0
-20-515
10
15
10
0.6
0.8 1 0 1
0.8
1
0
0.5
  y5
0.8
-0.2
0.4
0.4 -0.1
-0.1
yp
y
0.5
0.2
-15
-0.05 -0.05
0.6 0.6
0
0
1
5
y
0.8
0
ypy
y
yp
ypy
0.4
-3
y
0
15 -5 15
0.8
1
10
0.6
-2
0.4
-3
yp
10
0.6
0.05
y'
y
1
y
c
Figure 5: Comparison of solutions of equation (18)
– a actual solution; b numerical solution as
produced using the PCHIP covering method;
c numerical instability demonstrated by the
midpoint rule (   50 ).
Figures 5a and c
reproduced from Bellen and Zennaro [5] by
permission of Oxford University Press.
Another comparison was performed for the MackeyGlass equation governing the release of mature blood cells
into the blood stream:
by  t   
(19)
y ' t  
 ay  t  .
n
1   y  t   
This equation describes the time rate of change of the
density of the circulating white blood cells in terms of the
current flux (the first term on the right-hand side) of new
white blood cells into the system in response to the
demand created at a time  in the past and their random
destruction rate (the last term). For certain values of the
parameters and delay, the solution is oscillatory, and can
be chaotic, as in the case of patients with leukemia. By
way of validation, the results of such a case as computed
using the PCHIP covering method are compared with those
presented in [5] in figure 6.
1000
500
0 1000
0.5
t
y
y
b
Figure
6: Comparison of chaotic solutions of the
0.1
Mackey-Glass equation, equation (19), with
0.05
a  0.1 , b  0.2 , n  10 , and   20 – a as
presented
by Bellen and Zennaro [5] by permission
0
of Oxford University Press; b as produced using the
-0.05
PCHIP covering method.
In order to validate the PCHIP covering method for a
-0.1
0.5 1
1 1.5
1.5
system0 0.5
of DDEs
– rather
than a scalar
DDE, the code was
y
y
applied to the transient oscillations induced by delayed
growth response in a chemostat, and the results compared
with those of Xia, et al, [9] who explored the Hopf
bifurcations that occur as the parameter measuring delay
passes through a certain critical values. The single-species
chemostat equations govern the dynamics of the
concentration of unconsumed nutrient, S  t  , and the
t
yp
0
0.2
a
y
1
1
yp
0
1000
0.1
y
y
0.2
500
1.5
y
0.4
0.2
-0.15
0
t
1.5
0.6
0.2
y
0
-500
t
y
0.8
0.4
0.6
-2
  100
1
y
y
yp
y
15
0.4
-0.1
0
0
-5 -0.2-5
0
yp
10
y
1
0
0
biomass of the population of micro-organisms, x  t  . The
governing equations are
S ' t    S 0  S t  D  p  S t  x t  ,
x '  t    Dx  t    p  S  t     x  t    ,
(20)
where   exp  D  , S 0 is the concentration of growthlimiting nutrient, and D is the chemostat flow rate. The
value D  0.2079 was selected by the cited researchers.
The inhibitory response function is defined by
aS
pS   2
.
(21)
S  bS  c
The values of the other various parameters were selected
[9] as a  34.711 , b  0.25 , c  0.04 . The delay was
selected as   19.725 .
Xia, et al, [9] showed that equations (20) display
varying degrees of transient oscillatory behavior that can
be controlled by step changes in the initial data. Their
solutions for three nearly identical sets of initial data
display markedly different behavior. These results are
presented in figure 7, where they are compared with those
of the PCHIP covering method. The initial data for these
three cases were defined by step functions:

 x , t  t1   , t2 
(22)
x t    1
,
t   t2 , 0

 x2 ,
where t2  17.753 and x1  0.03 were selected by Xia,
et al, [9]. The values of the initial data on either side of the
jump at t  t2 were selected [9] in order to demonstrate the
dramatic sensitivity of the solution to minor changes in the
magnitude of the step increase. These are given in table 1.
In comparing the results of the PCHIP covering method, it
was found that slightly different values of the step increase
were required to produce the curves presented in [9].
These are also presented in table 1.
x2  0.0024
x2  0.002359125
x2  0.0021
a
x2  0.00275
x2  0.0027158
x2  0.0025
b
Figure 7: Comparison of solutions for delayed
growth response in a chemostat, equation (20) – a as
presented by Xia, et al, [9]; b as produced using the
PCHIP covering method.
Table 1: Comparison of parameters defining the
initial data as selected by Xia, et al, [9] and as
determined via numerical experimentation to
produce similar results using the PCHIP covering
method.
Initial Data Value after Jump, x2
Case
Xia, et al, [9]
PCHIP covering
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
equations (20) was solved in the current effort. This more
general approach undoubtedly introduced some minor
numerical differences into the solution.
Secondly, Xia, et al, [9] used Euler’s method with a
constant step size of t  0.001 , whereas the PCHIP
covering method was exercised using a standard MATLAB
variable-order, multi-step solver that provides automated
adaptive time-stepping. This routine, ODE15S, employs
an algorithm that is based on numerical differentiation
formulas. It is specifically designed for stiff problems.
This particular routine was employed in all of the tests
presented herein, since it has been found to perform well
for an existing simulation tool for the dynamics of
supercavitating high-speed bodies, one that is not treated
explicitly as a DDE solution, but is planned for upgrading
using the current method. These fundamental differences
between ODE15S and the Euler method with constant time
step size used in [9] also would be expected to lead to
numerical differences in the two solutions. In fact,
considering the more sophisticated numerical approaches
embedded in the methods used to test the PCHIP covering
method, it seems likely that those results are more accurate
than those presented in [9].
Each of the previous examples involved constant
delays. In order to validate the PCHIP covering method
for systems with state-dependent delay as is found in the
Paryshev system, equations (1), consider the following
examples cited in [5]:



1
t
 y '  t   y  t   y  y  t     log 
 , 1  t  5,
t

 log  t   1 

 y 1  1,
(25)
which has the solution
(26)
y  t   log  t   1;
and the Feldstein-Neves equation:
1 1

y y t   2  1 ,
1  t  3,
 y ' t   2
(27)
t

 y 1  1,
t

1,

the solution of which has a jump discontinuity in its second
derivative at t  2 :


3
0.0021
0.0025
Some discussion of the minor numerical differences
presented in table 1 is warranted. Firstly, Xia, et al, [9]
restricted their solutions to that subset satisfying
(23)
S  t    eD x t   S 0 ,
 t,
1  t  2,

(28)
y t    t 1 
2
2  t  3.
 t
   1 
2 
4 2 
The solutions of equation (25) and equation (27) are
plotted and compared with their analytical solutions in
figures 8 and 9, respectively.
It can see that the
comparison is excellent.
which allowed them to reduce the governing system,
equation (20), to the following scalar equation:
CONCLUSIONS
1
0.0024
0.00275
2
0.002359125
0.0027158
x '  t    Dx  t   e  D x  t    p  S 0  e D x  t   .
(24)
In exercising the PCHIP covering method, it was
specifically desired to check that the code was working for
a system of equations. Thus, although equation (23) was
used to determine the initial data for S  t  corresponding
to that for x  t  , condition (22), the full system of
A DDE solver has been developed based on a
covering of the computed solution using piecewise cubic
Hermite interpolating polynomials (PCHIPs). Tests of the
method for several examples found in the DDE-related
literature suggest that it is stable and accurate for problems
involving both constant and state-dependent delay. The
stability is attributed to the monotonicity-preserving
properties of the PCHIP fit. The method has been tested
successfully for both scalar equations and systems of
equations. It was found that the covering method is much
faster than an approach wherein the entire solution is re-fit
at each succeeding time-step.
3
PCHIP covering method
y
2
y
1
Third International Summer Scientific School
«High Speed Hydrodynamics and Numerical Simulation»,
June 2006, Kemerovo, Russia
The author is grateful to the Office of Naval Research, who
supported this effort under Navy contract N00014-05-C0054. Thanks are in order to Professors Alfredo Bellen
and
Marino Zennaro
(Dipartimento
di
Scienze
Matermatiche, Università di Trieste, Italy) and their
publisher, Oxford University Press, for providing
permission to reproduce several of the figures used for
purposes of comparison in this article, as indicated in the
captions.
Finally, the author wishes to thank
Dr. James S. Uhlman and Mr. Bart Burkewitz (Anteon
Corporation) for their review and suggestions.
y  t   log  t   1
0
REFERENCES
-1
-2
0
1
2
3
4
5
t
Figure 8: Comparison of the analytical solution of
equation
(25) with that produced using the PCHIP
10
covering method.
yp
8
2
PCHIP covering method
6
1.5
y
y
4
1
2
0.5
0
-2
0
0
 t,

y t    t 1 
2
-1   0  1  1  t 2

2 
 4 2 y
1
2
1 t  2
2  3t  3
3
t
yp
Figure 9: Comparison of the analytical solution of
equation (27) with that produced using the PCHIP
2
covering method.
Based on the success of these tests, the method is
considered
1.5 ready for application to DDE problems of
interest in high-speed hydrodynamics, specifically,
solution of Paryshev’s system of equations for the
dynamics
of supercavities and another set of equations
1
governing the dynamics of supercavitating bodies. Such
application will be the subject of one or more future
articles.
0.5
The prospect will also be explored of increasing the
computational speed of the PCHIP covering method by
applying the Woodbury formula (Uhlman,[10]; Press, et al,
0
[11]) for 0computing
0.5the inverse
1 of “small”
1.5 changes
2 to a
matrix, the inverse of which isy already known. The intent
of such an approach is rapid re-fitting of the current PCHIP
fit in a covering as each new point is appended to the DDE
solution.
ACNOWLEGEMENTS
1. Paryshev, E.V., (1978) “A System of Nonlinear
Differential Equations with a Time Delay, Describing the
Dynamics of Non-Stationary, Axially Symmetric
Cavities,” Trudy, TsAGI, No. 1907, Moscow, Russia;
translated from the Russian.
2. Kirschner, I.N., D.C. Kring, A.W. Stokes, N.E. Fine,
and J.S. Uhlman (2002) “Control Strategies for
Supercavitating Vehicles,” J. of Vibration and Control,
8 2, Sage Publications, London, England, UK. (Also
presented at the Eighth International Symposium on
Nonlinear Dynamical Systems, Blacksburg, VA.)
3. Kirschner, I.N., B.J. Rosenthal, and J.S. Uhlman (2003)
“Simplified
Dynamical
Systems
Analysis
of
Supercavitating High-Speed Bodies,” Cav03-OS-7-005,
Proceedings of the Fifth International Symposium on
Cavitation (CAV2003), Osaka, Japan.
4. Kolmanovskii, V., and A. Myshkis (1992) Applied
Theory of Functional Differential Equations, Kluwer
Academic Publishers, Dordrecht, The Netherlands.
5. Bellen, A., and M. Zennaro (2003) Numerical Methods
for Delay Differential Equations, Oxford University Press,
Oxford, UK.
6. Logvinovich, G.V., (1969) Hydrodynamics of FreeBoundary Flows, Naukova Dumka, Kiev, Ukraine;
translated from the Russian by the Israel Program for
Scientific Translations, Jerusalem (1972).
7. MathWorks
(2001)
MATLAB
release 12.1
documentation, The MathWorks, Natick, Massachusetts,
USA.
8. Baker, C.T.H., and E. Buckwar (2000) “Continuous  Methods for the Stochastic Pantograph Equation,”
Electronic Transactions on Numerical Analysis, Kent State
University, Kent, Ohio, USA.
9. Xia, H., G.S.K. Wolkowicz, and L. Wang (2005)
“Transient Oscillations Induced by Delayed Growth
Response in the Chemostat,” J. Math. Biol., 50, pp 489530.
10. Uhlman, J.S., (2006) Private communication.
11. Press, W.H., B.P. Flannery, S.A. Teukolsky, and
W.T. Vetterling (1989) Numerical Recipes: The Art of
Scientific Computing, Cambridge University Press, New
York, New York, USA.