Rate of convergence of multistep codes started by variation of order and stepsize.

*(English)*Zbl 0727.65067An analysis is made of optimal general procedures for programming variable step variable formula codes for the approximate solution of ordinary differential equations (VSVF algorithms). Given the system \((i) y'=f(x,y),\) \(a\leq x\leq b\), \(y(a)=y_ 0\), the authors consider a set of algorithms of the form
\[
(ii)\quad y_{n+1}=\sum^{s}_{i=0}a_{n,i}y_{n-i}+b_ nh_ ny'_{n+1}+\sum^{r}_{i=0}b_{n,i}h_{n-1-i}y'_{n-i}
\]
y\({}'_ n=f(x_ n,y_ n)\), where \(a=x_ 0<x_ 1<...<x_ N=b,\) \(h_ n=(x_{n+1}-x_ n),\) \(y_ n\approx y(a+nh_ n)\), \(n=0,1,...,N-1\). f is assumed to satisfy the Lipschitz condition and a stability condition.

The problem considered is given a certain tolerance \(\tau\) how best to choose stepsizes \(h_ n\) and formulas such that \((y(x_ n)-y_ n)=O(\tau).\) The recursive computation of \(y_ n\) starting from \(y_ 0\) is separated into an initial and final phase. The first phase employs a formula from (ii) having a low order m with step sizes less or equal to \(h=(\tau)^{1/m+1}.\) This phase stops after a fixed number of steps q, \(q>m\). The final phase starts with \(y_ q\) and a formula with order p, \(m<p\leq q\), with steps which increase in step size from \(h=(\tau)^{1/m+1}\) to \(H=(\tau)^{1/p+1}\) where the ratio \(h_ i/h_{i-1}=r_ i\), \(r_ i\geq r_{i-1}\), \(1\leq r_ i\leq R\), \(i=q+1,...,N\). The existence of R follows from the stability assumption. The total error in the computation is proved to be \(y(x_ n)-y_ n=O(H^{p+1})\) which establishes that using a formula with low order of accuracy during the initial phase does not limit the total error in the computation. Numerical results obtained with \(f=-y\), \(0\leq x\leq 5\), \(m=2\), \(p=6\), \(r_ i/r_{i-1}=2\), confirm the applicability of the analysis.

The problem considered is given a certain tolerance \(\tau\) how best to choose stepsizes \(h_ n\) and formulas such that \((y(x_ n)-y_ n)=O(\tau).\) The recursive computation of \(y_ n\) starting from \(y_ 0\) is separated into an initial and final phase. The first phase employs a formula from (ii) having a low order m with step sizes less or equal to \(h=(\tau)^{1/m+1}.\) This phase stops after a fixed number of steps q, \(q>m\). The final phase starts with \(y_ q\) and a formula with order p, \(m<p\leq q\), with steps which increase in step size from \(h=(\tau)^{1/m+1}\) to \(H=(\tau)^{1/p+1}\) where the ratio \(h_ i/h_{i-1}=r_ i\), \(r_ i\geq r_{i-1}\), \(1\leq r_ i\leq R\), \(i=q+1,...,N\). The existence of R follows from the stability assumption. The total error in the computation is proved to be \(y(x_ n)-y_ n=O(H^{p+1})\) which establishes that using a formula with low order of accuracy during the initial phase does not limit the total error in the computation. Numerical results obtained with \(f=-y\), \(0\leq x\leq 5\), \(m=2\), \(p=6\), \(r_ i/r_{i-1}=2\), confirm the applicability of the analysis.

Reviewer: J.B.Butler jun.(Portland)

##### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |