d02pr:: Ordinary Differential EquationsIntegrators for Stiff Ordinary Differential Systems (NAG Toolbox)

nag_ode_ivp_rkts_reset_tend (d02pr) and its associated functions (nag_ode_ivp_rkts_onestep (d02pf), nag_ode_ivp_rkts_setup (d02pq), nag_ode_ivp_rkts_interp (d02ps), nag_ode_ivp_rkts_diag (d02pt) and nag_ode_ivp_rkts_errass (d02pu)) solve the initial value problem for a first-order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate

y^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

nag_ode_ivp_rkts_reset_tend (d02pr) is used to reset the final value of the independent variable,

t_{f}

, when the integration is already underway. It can be used to extend or reduce the range of integration. The new value must be beyond the current value of the independent variable (as returned in tnow by nag_ode_ivp_rkts_onestep (d02pf)) in the current direction of integration. It is much more efficient to use nag_ode_ivp_rkts_reset_tend (d02pr) for this purpose than to use nag_ode_ivp_rkts_setup (d02pq) which involves the overhead of a complete restart of the integration.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

This example integrates a two body problem. The equations for the coordinates

(x (t), y (t))

of one body as functions of time

t

in a suitable frame of reference are

x^{''} = - \frac{x}{r^{3}}

y^{''} = - \frac{y}{r^{3}}, r = \sqrt{x^{2} + y^{2}} .

The initial conditions

\begin{array}{l} x (0) = 1 - ε, & x^{'} (0) = 0 \\ y (0) = 0, & y^{'} (0) = \sqrt{\frac{1 + ε}{1 - ε}} \end{array}

lead to elliptic motion with

0 < ε < 1

ε = 0.7

is selected and the system of ODEs is reposed as

\begin{array}{l} y_{1}^{'} = y_{3} \\ y_{2}^{'} = y_{4} \\ y_{3}^{'} = - \frac{y_{1}}{r^{3}} \\ y_{4}^{'} = - \frac{y_{2}}{r^{3}} \end{array}

over the range

[0, 6 π]

. Relative error control is used with threshold values of

1.0e−10

for each solution component and compute the solution at intervals of length

π

across the range using nag_ode_ivp_rkts_reset_tend (d02pr) to reset the end of the integration range. A high-order Runge–Kutta method (

method = - 3

) is also used with tolerances

tol = 1.0e−4

and

tol = 1.0e−5

in turn so that the solutions may be compared.

function d02pr_example


fprintf('d02pr example results\n\n');

% Set initial conditions and input
method = int64(-3);
tstart = 0;
tend   = 6*pi;
ecc    = 0.7;
yinit  = [1-ecc; 0; 0; sqrt((1+ecc)/(1-ecc))];
hstart = 0;
thresh = [1e-10; 1e-10; 1e-10; 1e-10];
n      = int64(4);
npts   = 96;
tol0   =  1.0E-4;
ysave  = zeros(npts+1, n);
tsave  = zeros(npts+1, 1);

% Set control for output
tol  = 10.0*tol0;
tinc = (tend-tstart)/npts;




% We run through the calculation twice with two tolerance values
for i = 1:2

  tol = tol*0.1;

  tendnu = tstart + tinc;

  % Call setup function
  [iwsav, rwsav, ifail] = d02pq(tstart, tendnu, yinit, tol, thresh, method);

  fprintf('\nCalculation with TOL = %8.1e\n\n', tol);
  fprintf('    t         y1        y2        y3        y4\n');
  fprintf(' %6.3f   %7.3f   %7.3f   %7.3f   %7.3f\n', tstart, yinit);

  tsave(1) = tstart;
  ysave(1, :) = yinit;
  tnow = tstart;
  j=1;
  while tnow < tend
    [tnow, ynow, ypnow, user, iwsav, rwsav, ifail] = ...
      d02pf(@f, n, iwsav, rwsav);

    if tnow >= tendnu
      % Just print some of the results
      if rem(j, 16) == 0
        fprintf(' %6.3f   %7.3f   %7.3f   %7.3f   %7.3f\n', tnow, ynow);
      end
      j = j+1;
      tsave(j) = tnow;
      ysave(j, :) = ynow;
      if tnow < tend
        tendnu = tendnu + tinc;
        [iwsav, rwsav, ifail] = d02pr(tendnu, iwsav, rwsav);
      end
    end
  end

  [fevals, stepcost, waste, stepsok, hnext, iwsav, ifail] = d02pt(iwsav, rwsav);
  fprintf('Cost of the integration in evaluations of f is %d\n', fevals);

end

% Plot results
fig1 = figure;
hold on;
xlabel('Orbit - x');
ylabel('Orbit - y');
plot(ysave(:,1), ysave(:, 2), '-xr');
% Plot errors with a different scale
ax1 = gca;
ax2 = axes('Position',get(ax1,'Position'),...
           'XAxisLocation','top',...
           'YAxisLocation','right',...
           'Color','none',...
           'XColor','k','YColor','k');
hold on;
axis([-0.25 0.05 -0.1 0.1]);
xlabel('x Deviation from True Ellipse');
ylabel('y Deviation from True Ellipse');
d = dev(ysave(:,1), ysave(:, 2));
plot(ax2, d.*cos(tsave), d.*sin(tsave),'-*b');
text(-0.05, 0, 'Deviation', 'Color', 'b');


function [yp, user] = f(t, n, y, user)
  r = sqrt(y(1)^2+y(2)^2);
  yp = [y(3); y(4); -y(1)/r^3; -y(2)/r^3];

function [d] = dev(x, y)
  ecc = 0.7;
  d =  abs((x+ecc).*(x+ecc)+y.*y/(ecc*ecc)-1);

d02pr example results


Calculation with TOL =  1.0e-04

    t         y1        y2        y3        y4
  0.000     0.300     0.000     0.000     2.380
  3.142    -1.700     0.000    -0.000    -0.420
  6.283     0.300    -0.000     0.000     2.380
  9.425    -1.700     0.000    -0.000    -0.420
 12.566     0.300    -0.000     0.000     2.380
 15.708    -1.700     0.000    -0.000    -0.420
 18.850     0.300    -0.000     0.001     2.380
Cost of the integration in evaluations of f is 1443

Calculation with TOL =  1.0e-05

    t         y1        y2        y3        y4
  0.000     0.300     0.000     0.000     2.380
  3.142    -1.700     0.000    -0.000    -0.420
  6.283     0.300    -0.000     0.000     2.380
  9.425    -1.700     0.000    -0.000    -0.420
 12.566     0.300    -0.000     0.000     2.380
 15.708    -1.700     0.000    -0.000    -0.420
 18.850     0.300    -0.000     0.000     2.380
Cost of the integration in evaluations of f is 1509

NAG Toolbox: nag_ode_ivp_rkts_reset_tend (d02pr)

▸▿ Contents

Purpose

Syntax

Description