| Russian Mathematical Surveys |
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Brief Communications On estimating the local error of a numerical solution of a parametrized Cauchy problem E. B. Kuznetsova, S. S. Leonovab a Moscow Aviation Institute b RUDN University
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     We consider the Cauchy problem for a normal system of $n$ ordinary differential equations of the form
$$
\begin{equation}
\frac{d\mathbf{y}}{dt}=\mathbf{F}(\mathbf{y},t), \quad \mathbf{y}(t_0)=\mathbf{y}_0, \quad t \in [t_0;T], \quad t_0 < T < +\infty,
\end{equation}
\tag{1}
$$
where $\mathbf{y}\colon \mathbb{R} \to \mathbb{R}^n$, $\mathbf{F}\colon \mathbb{R}^{n+1} \to \mathbb{R}^n$, $t_0$ is the initial point, $\mathbf{y}_0$ is the vector of values of the function $\mathbf{y}(t)$ at $t_0$, $T$ is the right-hand endpoint of the interval of values of $t$.
One of the most efficient ways to solve numerically a problem of the form (1) in the case when it is ill conditioned is the method of best parametrization (the arc-length method) [1], Chs. 2 and 3. It consists in introducing the argument $\lambda$ which is the arc length of the integral curve of (1). The differential of $\lambda$ satisfies the relation where $d\mathbf{y}=(dy_1(t),\dots,dy_n(t))^\top$ is the differential of the vector-valued function $\mathbf{y}(t)$.When reduced to the argument $\lambda$ with differential (2), the system (1) takes the form
$$
\begin{equation}
\frac{d\mathbf{y}}{d\lambda}=\frac{1}{\sqrt{Q(\mathbf{y},t)}}\, \mathbf{F}(\mathbf{y},t), \quad \frac{dt}{d\lambda}=\frac{1}{\sqrt{Q(\mathbf{y},t)}}\,, \quad \mathbf{y}(0)=\mathbf{y}_0, \quad t(0)=t_0, \quad \lambda \in [0;\Lambda],
\end{equation}
\tag{3}
$$
where $\Lambda$ is the right-hand endpoint of the interval of values of $\lambda$ and $Q(\mathbf{y},t)=1+\mathbf{F}^\top\,\mathbf{F}$.
Apart from the monograph [1], Chs. 2 and 3, mentioned above, the fact that $\lambda$ can efficiently be used in the solution of problems of the form (1) was shown in [2]–[4], for instance. In those papers the authors obtained a number of important results, which however did not concern theoretical estimates for the local error of a numerical solution of transformed problems of the form (3). In the scalar case an estimate for the local error of a numerical solution in a neighbourhood of the limit singular point was obtained in [1], § 2.2, and [5]. Here we generalize this estimate to the vector-valued case and arbitrary points on the integral curve. Theorem. The local errors $\Delta_t$ and $\Delta_{\lambda}$ of numerical solutions of the problems (1) and (3), respectively, which are obtained by means of the Euler method satisfy the inequality where $E$ is the identity matric of size $n$ and $\|\mathbf{a}\|_2=(a_1^2+\cdots+a_n^2)^{1/2}$ for $\mathbf{a}=(a_1,\ldots,a_n)$. Proof. Assuming that $\mathbf{y} (t)$ is a sufficiently smooth function we use for it Taylor’s formula with linear terms and Lagrange remainder in a neighbourhood of a point $t_j$:
$$
\begin{equation}
\mathbf{y}(t_j+\tau)=\mathbf{y}(t_j)+\tau\,\frac{d\mathbf{y}}{dt}(t_j)+ \frac{\tau^2}{2}\,\frac{d^2\mathbf{y}}{dt^2}(t_j+\theta_1\tau),\qquad \theta_1 \in (0;1),
\end{equation}
\tag{5}
$$
where $\tau$ is the step size of integration with respect to $t$. From (5) we can find the local error of the explicit Euler method for the problem (1):
Using similar reasoning we obtain an estimate for the local error of the Euler method for the problem (3) reduced the best parametrization:
$$
\begin{equation}
\Delta_{\lambda} \leqslant \frac{l^2}{2} \biggl(\biggl\|\frac{d^2\mathbf{y}} {d\lambda^2}(\lambda_k+\theta_2 l)\biggr\|_2+ \biggl\|\frac{d^2 t}{d\lambda^2} (\lambda_k+\theta_2 l)\biggr\|_2\biggr), \qquad \theta_2 \in (0;1),
\end{equation}
\tag{6}
$$
where $l$ is the step size of integration with respect to $\lambda$.
Using these results we obtain an estimate for the local error of the parametrized problem (3) for the explicit Euler method. To do this we find the second derivatives of the vector-valued function $\mathbf{y}$ and function $t$ with respect to $\lambda$:
$$
\begin{equation}
\frac{d^2\mathbf{y}}{d\lambda^2}=\frac{1}{Q^2}(QE-\mathbf{F}\mathbf{F}^\top) \frac{d^2\mathbf{y}}{dt^2}(\lambda), \qquad \frac{d^2t}{d\lambda^2}=-\frac{1}{Q^2}\mathbf{F}^\top\, \frac{d^2\mathbf{y}}{dt^2}(\lambda).
\end{equation}
\tag{7}
$$
Using (7) and the relation $l=\tau\,\sqrt{Q(\mathbf{y}(\lambda_k),t(\lambda_k))}$ , which links the steps of integration $\tau$ and $l$, we can write (6) as
$$
\begin{equation}
\Delta_{\lambda} \leqslant \frac{1}{Q}\bigl(\|Q E-\mathbf{F}\mathbf{F}^\top\|_2+ \|\mathbf{F}^\top\|_2\bigr)\,\frac{\tau^2}{2}\, \biggl\|\frac{d^2 \mathbf{y}}{d t^2}(\lambda_k+\theta_2 l)\biggr\|_2.
\end{equation}
\tag{8}
$$
It remains to show that the point $\lambda_k+\theta_2 l$ corresponds to $t_k+\theta_1\tau$. As the values of $\lambda$ and $t$ are in a one-to-one correspondence, $\lambda_k$ corresponds to $t_{\lambda,k}=t_k$, while $\lambda_k+\theta_2 l$ corresponds to $t_{\lambda,k}+\theta_1\tau$, where $\theta_1$ and $\theta_2$ are related by $\theta_1=\theta_2\sqrt{Q(\mathbf{y}(\lambda_k),t(\lambda_k))}$ . Using this correspondence we can write the estimate (8) for the local error of the parametrized problem (3) in the form (4). $\Box$ The inequality (4) yields results obtained in [1], § 2.2, and [5]. Another consequence of the above theorem is a generalization of an estimate for the local error of the Cauchy problem (3) reduced to the best argument for a system of ordinary differential equations in a neighbourhood of a limit singular point. Corollary. In a neighbrourhood of a limit singular point the error of the numerical solution of the transformed problem (3) by means of the explicit Euler method is no greater than the local error of the solution of the original problem (1). 
Citation in format AMSBIB
\Bibitem{KuzLeo22} |
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