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{ ******************************************************************
Numerical integration of a system of differential equations
by the Runge-Kutta-Fehlberg (RKF) method.
Adapted from a Fortran-90 program available at:
http://www.csit.fsu.edu/~burkardt/f_src/rkf45/rkf45.f90
****************************************************************** }
unit urkf;
interface
uses
utypes, uminmax;
procedure RKF45(F : TDiffEqs;
Neqn : Integer;
Y, Yp : PVector;
var T : Float;
Tout, RelErr, AbsErr : Float;
var Flag : Integer);
implementation
const
maxeqn : Integer = 0;
flag_save : Integer = -1000;
init : Integer = -1000;
kflag : Integer = -1000;
kop : Integer = -1;
nfe : Integer = -1;
relerr_save : Float = -1.0;
abserr_save : Float = -1.0;
h : Float = -1.0;
f1 : PVector = nil;
f2 : PVector = nil;
f3 : PVector = nil;
f4 : PVector = nil;
f5 : PVector = nil;
procedure Fehl(F : TDiffEqs;
Neqn : Integer;
Y : PVector;
T, H : Float;
Yp, F1, F2, F3, F4, F5, S : PVector);
{ ------------------------------------------------------------------
Fehl takes one Fehlberg fourth-fifth order step (double precision).
Discussion:
This routine integrates a system of Neqn first order ordinary
differential equations of the form
dY(i)/dT = F(T,Y(1:Neqn))
where the initial values Y and the initial derivatives
YP are specified at the starting point T.
The routine advances the solution over the fixed step H and returns
the fifth order (sixth order accurate locally) solution
approximation at T+H in array S.
The formulas have been grouped to control loss of significance.
The routine should be called with an H not smaller than 13 units of
roundoff in T so that the various independent arguments can be
distinguished.
Modified:
27 March 2004
Author:
H A Watts and L F Shampine,
Sandia Laboratories,
Albuquerque, New Mexico.
Reference:
E. Fehlberg,
Low-order Classical Runge-Kutta Formulas with Stepsize Control,
NASA Technical Report R-315.
L F Shampine, H A Watts, S Davenport,
Solving Non-stiff Ordinary Differential Equations - The State of the Art,
SIAM Review,
Volume 18, pages 376-411, 1976.
Parameters:
Input, external F, a user-supplied subroutine to evaluate the
derivatives Y'(T), of the form:
procedure(X : Float; Y, D : PVector);
Input, Neqn, the number of equations to be integrated.
Input, Y(Neqn), the current value of the
dependent variable.
Input, T, the current value of the independent
variable.
Input, H, the step size to take.
Input, YP(Neqn), the current value of the
derivative of the dependent variable.
Output, F1(Neqn), F2(Neqn), F3(Neqn), F4(Neqn), F5(Neqn),
derivative values needed for the computation.
Output, S(Neqn), the estimate of the solution at T+H.
------------------------------------------------------------------ }
const
C1 = 3.0 / 32.0;
C2 = 3.0 / 8.0;
C3 = 1.0 / 2197.0;
C4 = 12.0 / 13.0;
C5 = 1.0 / 4104.0;
C6 = 1.0 / 20520.0;
C7 = 1.0 / 7618050.0;
var
ch : Float;
i : Integer;
begin
ch := 0.25 * h;
for i := 1 to neqn do
f5^[i] := y^[i] + ch * yp^[i];
f(t + ch, f5, f1);
ch := C1 * h;
for i := 1 to neqn do
f5^[i] := y^[i] + ch * (yp^[i] + 3.0 * f1^[i]);
f(t + C2 * h, f5, f2);
ch := C3 * h;
for i := 1 to neqn do
f5^[i] := y^[i] + ch * (1932.0 * yp^[i] +
(7296.0 * f2^[i] - 7200.0 * f1^[i]));
f(t + C4 * h, f5, f3);
ch := C5 * h;
for i := 1 to neqn do
f5^[i] := y^[i] + ch * ((8341.0 * yp^[i] - 845.0 * f3^[i])
+ (29440.0 * f2^[i] - 32832.0 * f1^[i]));
f(t + h, f5, f4);
ch := C6 * h;
for i := 1 to neqn do
f1^[i] := y^[i] + ch * ((-6080.0 * yp^[i]
+ (9295.0 * f3^[i] - 5643.0 * f4^[i]))
+ (41040.0 * f1^[i] - 28352.0 * f2^[i]));
f(t + 0.5 * h, f1, f5);
{ Ready to compute the approximate solution at T+H. }
ch := C7 * h;
for i := 1 to neqn do
s^[i] := y^[i] + ch * ((902880.0 * yp^[i]
+ (3855735.0 * f3^[i] - 1371249.0 * f4^[i]))
+ (3953664.0 * f2^[i] + 277020.0 * f5^[i]));
end;
procedure ReDim_Arrays(neqn : Integer);
{ Redimensions global arrays if necessary }
begin
DelVector(f1, maxeqn);
DelVector(f2, maxeqn);
DelVector(f3, maxeqn);
DelVector(f4, maxeqn);
DelVector(f5, maxeqn);
maxeqn := neqn;
DimVector(f1, maxeqn);
DimVector(f2, maxeqn);
DimVector(f3, maxeqn);
DimVector(f4, maxeqn);
DimVector(f5, maxeqn);
end;
procedure RKF45(F : TDiffEqs;
Neqn : Integer;
Y, Yp : PVector;
var T : Float;
Tout, RelErr, AbsErr : Float;
var Flag : Integer);
{ ------------------------------------------------------------------
RKF45 carries out the Runge-Kutta-Fehlberg method (double precision).
Discussion:
This routine is primarily designed to solve non-stiff and mildly stiff
differential equations when derivative evaluations are inexpensive.
It should generally not be used when the user is demanding
high accuracy.
This routine integrates a system of Neqn first-order ordinary
differential equations of the form:
dY(i)/dT = F(T,Y(1),Y(2),...,Y(Neqn))
where the Y(1:Neqn) are given at T.
Typically the subroutine is used to integrate from T to TOUT but it
can be used as a one-step integrator to advance the solution a
single step in the direction of TOUT. On return, the parameters in
the call list are set for continuing the integration. The user has
only to call again (and perhaps define a new value for TOUT).
Before the first call, the user must
* supply the subroutine F(T,Y,YP) to evaluate the right hand side;
and declare F in an EXTERNAL statement;
* initialize the parameters:
Neqn, Y(1:Neqn), T, TOUT, RELERR, ABSERR, FLAG.
In particular, T should initially be the starting point for integration,
Y should be the value of the initial conditions, and FLAG should
normally be +1.
Normally, the user only sets the value of FLAG before the first call, and
thereafter, the program manages the value. On the first call, FLAG should
normally be +1 (or -1 for single step mode.) On normal return, FLAG will
have been reset by the program to the value of 2 (or -2 in single
step mode), and the user can continue to call the routine with that
value of FLAG.
(When the input magnitude of FLAG is 1, this indicates to the program
that it is necessary to do some initialization work. An input magnitude
of 2 lets the program know that that initialization can be skipped,
and that useful information was computed earlier.)
The routine returns with all the information needed to continue
the integration. If the integration reached TOUT, the user need only
define a new TOUT and call again. In the one-step integrator
mode, returning with FLAG = -2, the user must keep in mind that
each step taken is in the direction of the current TOUT. Upon
reaching TOUT, indicated by the output value of FLAG switching to 2,
the user must define a new TOUT and reset FLAG to -2 to continue
in the one-step integrator mode.
In some cases, an error or difficulty occurs during a call. In that case,
the output value of FLAG is used to indicate that there is a problem
that the user must address. These values include:
* 3, integration was not completed because the input value of RELERR, the
relative error tolerance, was too small. RELERR has been increased
appropriately for continuing. If the user accepts the output value of
RELERR, then simply reset FLAG to 2 and continue.
* 4, integration was not completed because more than MAXNFE derivative
evaluations were needed. This is approximately (MAXNFE/6) steps.
The user may continue by simply calling again. The function counter
will be reset to 0, and another MAXNFE function evaluations are allowed.
* 5, integration was not completed because the solution vanished,
making a pure relative error test impossible. The user must use
a non-zero ABSERR to continue. Using the one-step integration mode
for one step is a good way to proceed.
* 6, integration was not completed because the requested accuracy
could not be achieved, even using the smallest allowable stepsize.
The user must increase the error tolerances ABSERR or RELERR before
continuing. It is also necessary to reset FLAG to 2 (or -2 when
the one-step integration mode is being used). The occurrence of
FLAG = 6 indicates a trouble spot. The solution is changing
rapidly, or a singularity may be present. It often is inadvisable
to continue.
* 7, it is likely that this routine is inefficient for solving
this problem. Too much output is restricting the natural stepsize
choice. The user should use the one-step integration mode with
the stepsize determined by the code. If the user insists upon
continuing the integration, reset FLAG to 2 before calling
again. Otherwise, execution will be terminated.
* 8, invalid input parameters, indicates one of the following:
Neqn <= 0;
T = TOUT and |FLAG| /= 1;
RELERR < 0 or ABSERR < 0;
FLAG == 0 or FLAG < -2 or 8 < FLAG.
Modified:
27 March 2004
Author:
H A Watts and L F Shampine,
Sandia Laboratories,
Albuquerque, New Mexico.
Reference:
E. Fehlberg,
Low-order Classical Runge-Kutta Formulas with Stepsize Control,
NASA Technical Report R-315.
L F Shampine, H A Watts, S Davenport,
Solving Non-stiff Ordinary Differential Equations - The State of the Art,
SIAM Review,
Volume 18, pages 376-411, 1976.
Parameters:
Input, external F, a user-supplied subroutine to evaluate the
derivatives Y (T), of the form:
sub f ( t as double, y() as double, yp() as double )
Input, Neqn, the number of equations to be integrated.
Input/output, Y(Neqn), the current solution vector at T.
Input/output, YP(Neqn), the current value of the
derivative of the dependent variable. The user should not set or alter
this information
Input/output, T, the current value of the independent
variable.
Input, TOUT, the output point at which solution is
desired. TOUT = T is allowed on the first call only, in which case
the routine returns with FLAG = 2 if continuation is possible.
Input, RELERR, ABSERR, the relative and absolute
error tolerances for the local error test. At each step the code
requires:
abs ( local error ) <= RELERR * abs ( Y ) + ABSERR
for each component of the local error and the solution vector Y.
RELERR cannot be "too small". If the routine believes RELERR has been
set too small, it will reset RELERR to an acceptable value and return
immediately for user action.
Input/output, FLAG, indicator for status of integration.
On the first call, set FLAG to +1 for normal use, or to -1 for single
step mode. On return, a value of 2 or -2 indicates normal progress,
while any other value indicates a problem that should be addressed.
------------------------------------------------------------------ }
const
remin = 1.0E-12;
maxnfe = 3000;
var
k, mflag : Integer;
ae, dt, ee, eeoet, esttol, et : Float;
hmin, relerr_min, s, scale, tol, toln, ypk : Float;
hfaild, outp : Boolean;
label
Cont, Done;
begin
{ Check the input parameters. }
if (neqn < 1) or (relerr < 0) or (abserr < 0) or
((flag = 0) or (flag > 8) or (flag < -2)) then
begin
flag := 8;
exit;
end;
mflag := abs(flag);
{ Is this a continuation call? }
if mflag <> 1 then
begin
if (t = tout) and (kflag <> 3) then
begin
flag := 8;
exit;
end;
if mflag = 2 then
begin
if kflag = 3 then
begin
flag := flag_save;
mflag := abs(flag)
end
else if init = 0 then
flag := flag_save
else if kflag = 4 then
nfe := 0
else if (kflag = 5) and (abserr = 0) then
exit
else if (kflag = 6) and (relerr <= relerr_save) and (abserr <= abserr_save) then
exit;
end
else { FLAG = 3, 4, 5, 6, 7 or 8. }
begin
if flag = 3 then
begin
flag := flag_save;
if kflag = 3 then mflag := abs(flag)
end
else if flag = 4 then
begin
nfe := 0;
flag := flag_save;
if kflag = 3 then mflag := abs(flag)
end
else if (flag = 5) and (abserr > 0) then
begin
flag := flag_save;
if kflag = 3 then mflag := abs(flag)
end
else { Integration cannot be continued because the user did not }
exit; { respond to the instructions pertaining to FLAG = 5,6,7,8 }
end;
end;
{ Save the input value of FLAG. }
{ Set the continuation flag KFLAG for subsequent input checking. }
flag_save := flag;
kflag := 0;
{ Save RELERR and ABSERR for checking input on subsequent calls. }
relerr_save := relerr;
abserr_save := abserr;
{ Restrict the relative error tolerance to be at least
2 * EPS + REMIN
to avoid limiting precision difficulties arising from impossible
accuracy requests. }
relerr_min := 2 * MachEp + remin;
{ Is the relative error tolerance too small? }
if relerr < relerr_min then
begin
relerr := relerr_min;
flag := 3;
kflag := 3;
exit
end;
dt := tout - t;
{ Initialization:
Set the initialization completion indicator, INIT;
set the indicator for too many output points, KOP;
evaluate the initial derivatives;
set the counter for function evaluations, NFE;
estimate the starting stepsize. }
if mflag = 1 then
begin
init := 0;
kop := 0;
f(t, y, yp);
nfe := 1;
if t = tout then
begin
flag := 2;
exit;
end;
end;
if init = 0 then
begin
init := 1;
h := abs(dt);
toln := 0;
for k := 1 to neqn do
begin
tol := relerr * abs (y^[k]) + abserr;
if tol > 0 then
begin
toln := tol;
ypk := abs(yp^[k]);
if tol < ypk * h * h * h * h * h then
h := Exp(0.2 * Ln(tol / ypk));
end
end;
if toln <= 0 then h := 0;
h := FMax(h, 26 * MachEp * FMax(abs(t), abs(dt)));
flag_save := sgn(flag) * 2
end;
{ Set the stepsize for integration in the direction from T to TOUT. }
h := sgn(dt) * abs(h);
{ Test to see if too may output points are being requested. }
if 2 * abs(dt) <= abs(h) then kop := kop + 1;
{ Unnecessary frequency of output. }
if kop = 100 then
begin
kop := 0;
flag := 7;
exit
end;
{ If we are too close to the output point, then simply extrapolate and return. }
if abs(dt) <= 26 * MachEp * abs(t) then
begin
t := tout;
for k := 1 to neqn do
y^[k] := y^[k] + dt * yp^[k];
f(t, y, yp);
nfe := nfe + 1;
flag := 2;
exit
end;
{ Initialize the output point indicator. }
outp := False;
{ To avoid premature underflow in the error tolerance function,
scale the error tolerances. }
scale := 2 / relerr;
ae := scale * abserr;
{ Redimension global arrays if necessary }
if neqn > maxeqn then ReDim_Arrays(neqn);
{ Step by step integration. }
repeat
hfaild := False;
{ Set the smallest allowable stepsize. }
hmin := 26 * MachEp * abs(t);
{ Adjust the stepsize if necessary to hit the output point.
Look ahead two steps to avoid drastic changes in the stepsize and
thus lessen the impact of output points on the code. }
dt := tout - t;
if 2.0 * abs(h) > abs(dt) then
begin
{ Will the next successful step complete the integration to the output point? }
if abs(dt) <= abs(h) then
begin
outp := True;
h := dt
end
else
h := 0.5 * dt;
end;
{ Here begins the core integrator for taking a single step.
The tolerances have been scaled to avoid premature underflow in
computing the error tolerance function ET.
To avoid problems with zero crossings, relative error is measured
using the average of the magnitudes of the solution at the
beginning and end of a step.
The error estimate formula has been grouped to control loss of
significance.
To distinguish the various arguments, H is not permitted
to become smaller than 26 units of roundoff in T.
Practical limits on the change in the stepsize are enforced to
smooth the stepsize selection process and to avoid excessive
chattering on problems having discontinuities.
To prevent unnecessary failures, the code uses 9/10 the stepsize
it estimates will succeed.
After a step failure, the stepsize is not allowed to increase for
the next attempted step. This makes the code more efficient on
problems having discontinuities and more effective in general
since local extrapolation is being used and extra caution seems
warranted.
Test the number of derivative function evaluations.
If okay, try to advance the integration from T to T+H. }
repeat
{ Have we done too much work? }
if maxnfe < nfe then
begin
flag := 4;
kflag := 4;
exit
end;
{ Advance an approximate solution over one step of length H. }
Fehl(f, neqn, y, t, h, yp, f1, f2, f3, f4, f5, f1);
nfe := nfe + 5;
{ Compute and test allowable tolerances versus local error estimates
and remove scaling of tolerances. The relative error is
measured with respect to the average of the magnitudes of the
solution at the beginning and end of the step. }
eeoet := 0;
for k := 1 to neqn do
begin
et := abs(y^[k]) + abs(f1^[k]) + ae;
if et <= 0 then
begin
flag := 5;
exit
end;
ee := abs((-2090.0 * yp^[k] + (21970.0 * f3^[k] - 15048.0 * f4^[k]))
+ (22528.0 * f2^[k] - 27360.0 * f5^[k]));
eeoet := FMax(eeoet, ee / et);
end;
esttol := abs(h) * eeoet * scale / 752400.0;
if esttol <= 1 then goto Cont;
{ Unsuccessful step. Reduce the stepsize, try again.
The decrease is limited to a factor of 1/10. }
hfaild := True;
outp := False;
if esttol < 59049.0 then
s := 0.9 / Exp(0.2 * Ln(esttol))
else
s := 0.1;
h := s * h;
if abs(h) < hmin then
begin
flag := 6;
kflag := 6;
exit;
end;
until False;
{ We exited the loop because we took a successful step.
Store the solution for T+H, and evaluate the derivative there. }
Cont:
t := t + h;
for k := 1 to neqn do
y^[k] := f1^[k];
f(t, y, yp);
nfe := nfe + 1;
{ Choose the next stepsize. The increase is limited to a factor of 5.
If the step failed, the next stepsize is not allowed to increase. }
if 0.0001889568 < esttol then
s := 0.9 / Exp(0.2 * Ln(esttol))
else
s := 5.0;
if hfaild then s := FMin(s, 1.0);
h := sgn(h) * FMax(s * abs(h), hmin);
{ End of core integrator
Should we take another step? }
if outp then
begin
t := tout;
flag := 2;
exit
end;
if flag <= 0 then goto Done;
until False;
{ One step integration mode. }
Done:
flag := -2;
end;
end.
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