A verified inexact implicit Runge-Kutta method for nonsmooth ODEs

Sayed Mahmoud, Xiaojun Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)


Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge-Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes.
Original languageEnglish
Pages (from-to)275-290
Number of pages16
JournalNumerical Algorithms
Issue number3
Publication statusPublished - 1 Mar 2008
Externally publishedYes


  • IRK method
  • Nonsmooth ODEs
  • Oscillations
  • Structural pounding
  • Verification of solution

ASJC Scopus subject areas

  • Applied Mathematics


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