d02puc provides details about global error assessment computed during an integration with either
d02pec,
d02pfc or
d02pgc.
d02puc and its associated functions (
d02pec,
d02pfc,
d02pgc,
d02phc,
d02pjc,
d02pqc,
d02prc,
d02psc and
d02ptc) 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
where
$y$ is the vector of
$\mathit{n}$ solution components and
$t$ is the independent variable.
After a call to
d02pec,
d02pfc or
d02pgc,
d02puc can be called for information about error assessment, if this assessment was specified in the setup function
d02pqc. A more accurate ‘true’ solution
$\hat{y}$ is computed in a secondary integration. The error is measured as specified in
d02pqc for local error control. At each step in the primary integration, an average magnitude
${\mu}_{i}$ of component
${y}_{i}$ is computed, and the error in the component is
It is difficult to estimate reliably the true error at a single point. For this reason the RMS (root-mean-square) average of the estimated global error in each solution component is computed. This average is taken over all steps from the beginning of the integration through to the current integration point. If all has gone well, the average errors reported will be comparable to
tol (see
d02pqc). The maximum error seen in any component in the integration so far and the point where the maximum error first occurred are also reported.
Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University
Not applicable.
If the integration has proceeded ‘well’ and the problem is smooth enough, stable and not too difficult then the values returned in the arguments
rmserr and
errmax should be comparable to the value of
tol specified in the prior call to
d02pqc.
This example integrates a two body problem. The equations for the coordinates
$(x\left(t\right),y\left(t\right))$ of one body as functions of time
$t$ in a suitable frame of reference are
The initial conditions
lead to elliptic motion with
$0<\epsilon <1$.
$\epsilon =0.7$ is selected and the system of ODEs is reposed as
over the range
$[0,3\pi ]$. Relative error control is used with threshold values of
$\text{1.0e\u221210}$ for each solution component and a high-order Runge–Kutta method (
${\mathbf{method}}=\mathrm{Nag\_RK\_7\_8}$) with tolerance
${\mathbf{tol}}=\text{1.0e\u22126}$.
Note that for illustration purposes since it is not necessary for this problem, this example integrates to the end of the range regardless of efficiency concerns (i.e., returns from
d02pec with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_RK_POINTS,
NE_STIFF_PROBLEM or
NW_RK_TOO_MANY).