##
**A fast adaptive numerical method for stiff two-point boundary value problems.**
*(English)*
Zbl 0882.65066

An efficient adaptive algorithm is derived for approximation to the solution of a two point boundary problem for the inhomogeneous second-order linear ordinary differential equation with separated boundary conditions
\[
ru''(x)+ p(x)u'(x)+ q(x)u(x)= f(x),\quad r>0,\quad a\leq x\leq c,\tag{i}
\]

\[ a_{10}u(a)+ a_{11}u'(a)= c_1,\quad a_{21}u(c)+ a_{22}u(c)= c_2.\tag{ii} \] When \(r\) is a small parameter the system is said to be singularly perturbed. The program approximates a Green function \(G(x,t)\) and a weight function \(\sigma(x)\) associated with (i), (ii) such that \(\int^c_a G(x,t)f(t)\sigma(t)dt= u(x)\). The authors prove that the approximate solution \(u(x)\) of (i), (ii) on \([a,c]\) may be found by approximating \(\sigma(x)\) on a sequence of partitions of the interval in which successive partitions are refined by addition of one or more points.

After the computed solution reaches a prescribed degree of convergence inside a particular interval, the interval is not further subdivided [cf. L. Greengard and V. Rokhlin, Commun. Pure Appl. Math. 44, No. 4, 419-452 (1991; Zbl 0727.65068)]. The algorithm is especially effective for problems in which the solution has subdomains in which it is slowly varying and also rapidly varying.

Details of numerical results obtained when the method was applied to a package of equations with stiff behavior are presented. These include Bessel’s equation, an equation for viscous shock, a potential barrier problem and equations whose solutions have a turning point and a cusp. A detailed description of the algorithm is given. The authors also present very briefly an adaptive method to approximate eigenvalues and eigenfunctions for second-order ordinary differential equations.

\[ a_{10}u(a)+ a_{11}u'(a)= c_1,\quad a_{21}u(c)+ a_{22}u(c)= c_2.\tag{ii} \] When \(r\) is a small parameter the system is said to be singularly perturbed. The program approximates a Green function \(G(x,t)\) and a weight function \(\sigma(x)\) associated with (i), (ii) such that \(\int^c_a G(x,t)f(t)\sigma(t)dt= u(x)\). The authors prove that the approximate solution \(u(x)\) of (i), (ii) on \([a,c]\) may be found by approximating \(\sigma(x)\) on a sequence of partitions of the interval in which successive partitions are refined by addition of one or more points.

After the computed solution reaches a prescribed degree of convergence inside a particular interval, the interval is not further subdivided [cf. L. Greengard and V. Rokhlin, Commun. Pure Appl. Math. 44, No. 4, 419-452 (1991; Zbl 0727.65068)]. The algorithm is especially effective for problems in which the solution has subdomains in which it is slowly varying and also rapidly varying.

Details of numerical results obtained when the method was applied to a package of equations with stiff behavior are presented. These include Bessel’s equation, an equation for viscous shock, a potential barrier problem and equations whose solutions have a turning point and a cusp. A detailed description of the algorithm is given. The authors also present very briefly an adaptive method to approximate eigenvalues and eigenfunctions for second-order ordinary differential equations.

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

### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

34E13 | Multiple scale methods for ordinary differential equations |

34E15 | Singular perturbations for ordinary differential equations |

34B27 | Green’s functions for ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |