Ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field arises on smooth surfaces of the phase space, are the topic of this work. The main emphasis is the study of solutions close to the intersection of two discontinuity surfaces. There, the so-called hidden dynamics describes the smooth transition from ingoing to outgoing solution directi...

In this article, we analyze the fifth-order weighted essentially non-oscillatory(WENO-5) scheme and show that, at a transition point from smooth region to a discontinuity point or vice versa, the accuracy order of WENO-5 is decreased. A new method is proposed to overcome this drawback by introducing 4th-order fluxes combined with high order smoothness indicator. Numerical examples show that the...

We consider Filippov sliding motion on a co-dimension 2 discontinuity surface. We give conditions under which Σ is attractive through sliding which are sharper than those given in a previous paper of ours. Under these sharper conditions, we show that the sliding vector field considered in the same paper is still uniquely defined and varies smoothly in x ∈ Σ. A numerical example illustrate our r...

This paper presents ongoing research in improvement of analytical tools for the characterization of rock mass structure from oriented borehole discontinuity data. Using automated multivariate cluster analysis and the “three dimensional stereonet” provides for fast and objective characterization of oriented core discontinuity data. Improvements to the CYL software include three new methods of cl...

This article presents two regularization techniques for systems of state-dependent neutral delay differential equations which have a discontinuity in the derivative of the solution at the initial point. Such problems have a rich dynamics and besides classical solutions can have weak solutions in the sense of Utkin. Both of the presented techniques permit the numerical solution of such problems ...

The optimal control problem of adjusting the inflow, of piecewise constant type, to a supply chain in order to minimize the queue size and the quadratic difference between the outflow and the expected one is considered. The controls are represented by the duration of injections of different amounts of goods. The supply chain is modelled by a PDE-ODE: the conservation law describes the density o...