Compact Scheme for Harmonic Oscillator Equation with Coulomb Friction
by
Chaoqun Liu
Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA
Coauthors: James L. Thompson (Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA)

An implicit predictor-corrector scheme is used to solve ordinary differential equations for the harmonic oscillator problem with Coulomb friction. The scheme is combination of Adams-Moulton method and the compact scheme. The differential equations for the harmonic oscillator problems in this paper are of the type

x''(t) = f(t, x(t), x'(t)), x(t=0)=xo, x'(t=0) = vo,

(1)

where the function f(t, x(t), x'(t)) is discontinuous on a set of measure zero in the variable t. The computational problems caused by these discontinuities is avoided by detecting the zero crossing points of x’(t), calculating x(t) at the crossing point, and restarting the computation. The predictor-corrector scheme uses a sixth order three-step explicit method for the predictor, which uses Hermite rather than the Lagrange interpolation used by the Adams-Bashforth method. The corrector is a fifth order two-step implicit method with Hermite interpolation. This compact method has shown fifth order accuracy with good stability, and is easily extended to general ODEs with two or three steps.

Date received: October 17, 2002