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**Solving nonstiff higher order ODEs directly by the direct integration method.**
*(English)*
Zbl 0689.65047

For solving initial value problems of a system of higher order nonstiff ordinary differential equations a direct method is proposed. If an equation is of dth order the values of the dth derivatives are interpolated and then d times integrated. This leads to a predictor and to a corrector formula depending whether known values are interpolated or not. These formulas have been implemented in a variable order variable stepsize code, which has been tested on a set of 10 problems ranging from a second order up to a 6th order equation. The test problems have the following features: 1 pure integration, 7 linear with constant coefficients with a possible non-constant inhomogeneity, 1 linear with nonconstant coefficients.

The only truly nonlinear problem is Newton’s equation of motion for the two body problem. The problems are nonstiff. The code has been compared with applying an Adams code to the first order system obtained by standard reduction of the higher order system. The results indicate that with respect to actual computing time the direct integration is faster than solving the reduced system by Adams method. A scheme for solving directly stiff second order differential equations is presented too.

The only truly nonlinear problem is Newton’s equation of motion for the two body problem. The problems are nonstiff. The code has been compared with applying an Adams code to the first order system obtained by standard reduction of the higher order system. The results indicate that with respect to actual computing time the direct integration is faster than solving the reduced system by Adams method. A scheme for solving directly stiff second order differential equations is presented too.

Reviewer: R.Jeltsch

### MSC:

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

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

34A34 | Nonlinear ordinary differential equations and systems |

70F05 | Two-body problems |

### Keywords:

reduction to first order; comparison of methods; higher order nonstiff ordinary differential equations; direct method; predictor; corrector; variable order variable stepsize code; test problems; Newton’s equation of motion; two body problem; Adams code
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\textit{M. b. Suleiman}, Appl. Math. Comput. 33, No. 3, 197--219 (1989; Zbl 0689.65047)

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