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4.2.6 Numerov's Method

Mostly used for a certain kind of initial value problems (IVPs) that arise when boundary value problems (BVPs) are rewritten as IVPs. Many BVP has the general form with given boundary values and .

Example: one-dimensional Poisson equation with given at and . Thus and .

Let us assume we have, instead of and , the full set of initial values and .

Divide the interval into sub-intervals of length and at each intermediate point expand into a power series. Adding the Taylor formulae for and we find Inserting we have The fourth derivative is approximated by      In this way we find Numerov's formula    EXERCISE: Write a code that permits to solve a given second-order equation of motion by various algorithms. Apply the program to problems of point mechanics and explore the stabilities and accuracies of the diverse techniques.   Next: 4.3 Boundary Value Problems Up: 4.2 Initial Value Problems Previous: 4.2.5 Symplectic Algorithms
Franz J. Vesely Oct 2005