Direct event location techniques based on Adams multistep methods for discontinuous ODEs

Abstract In this paper we consider numerical techniques to locate the event points of the differential system x ′ = f ( x ) , where f is a discontinuous vector field along an event surface Σ = { x ∈ R n | h ( x ) = 0 } splitting the state space into two different regions R 1 and R 2 and f ( x ) = f i ( x ) when x ∈ R i , for i = 1 , 2 while f 1 ( x ) ≠ f 2 ( x ) when x ∈ Σ . Methods based on Adams multistep schemes which approach the event surface Σ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f 1 (respectively, f 2 ) in the region R 2 (respectively R 1 ) and are based on the computation–at each step–of a new time step τ reducing the value of the event function h ( x ) by a fixed quantity.

[1]  Luca Dieci,et al.  Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side , 2013 .

[2]  Luca Dieci,et al.  Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field , 2011, Numerische Mathematik.

[3]  Luca Dieci,et al.  One-sided direct event location techniques in the numerical solution of discontinuous differential systems , 2015 .

[4]  Marius-F. Danca,et al.  Suppressing chaos in discontinuous systems of fractional order by active control , 2014, Appl. Math. Comput..

[5]  H. D. Jong,et al.  Qualitative simulation of genetic regulatory networks using piecewise-linear models , 2004, Bulletin of mathematical biology.

[6]  Luca Dieci,et al.  A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis☆ , 2013 .

[7]  Nicoletta Del Buono,et al.  On the Equivalence between the Sigmoidal Approach and Utkin's Approach for Piecewise-Linear Models of Gene Regulatory Networks , 2014, SIAM J. Appl. Dyn. Syst..

[8]  Roberto Garrappa,et al.  On some generalizations of the implicit Euler method for discontinuous fractional differential equations , 2014, Math. Comput. Simul..

[9]  L. Shampine,et al.  Event location for ordinary differential equations , 2000 .

[10]  C. Budd,et al.  Review of ”Piecewise-Smooth Dynamical Systems: Theory and Applications by M. di Bernardo, C. Budd, A. Champneys and P. 2008” , 2020 .

[11]  Luciano Lopez,et al.  On the continuous extension of Adams-Bashforth methods and the event location in discontinuous ODEs , 2012, Appl. Math. Lett..

[12]  Luca Dieci,et al.  Fundamental matrix solutions of piecewise smooth differential systems , 2011, Math. Comput. Simul..

[13]  Vijay Kumar,et al.  A state event detection algorithm for numerically simulating hybrid systems with model singularities , 2007, TOMC.

[14]  Huaiqin Wu,et al.  Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations , 2009 .

[15]  Bolesław Kacewicz,et al.  Optimal adaptive solution of piecewise regular systems of IVPs with unknown switching hypersurface , 2014, Appl. Math. Comput..

[16]  Roger B. Sidje,et al.  On the equivalence of the continuous Adams-Bashforth method and Nordsieck's technique for changing the step size , 2013, Appl. Math. Lett..

[17]  N. Del Buono,et al.  Geometric Integration on Manifold of Square Oblique Rotation Matrices , 2001, SIAM J. Matrix Anal. Appl..

[18]  Lawrence F. Shampine,et al.  Reliable solution of special event location problems for ODEs , 1991, TOMS.

[19]  Reinhold Mannshardt.  One-step methods of any order for ordinary differential equations with discontinuous right-hand sides , 1978 .

[20]  Pawel Przybylowicz,et al.  Optimal solution of a class of non-autonomous initial-value problems with unknown singularities , 2014, J. Comput. Appl. Math..

[21]  K. T. Chau,et al.  Experimental and theoretical simulations of seismic poundings between two adjacent structures , 2003 .

[22]  Vijay Kumar,et al.  An asynchronous integration and event detection algorithm for simulating multi-agent hybrid systems , 2004, TOMC.