Some examples of mathematical modelling in glass industry

The evolution of viscous gobs is considered as they appear in glass morphology. It is shown how this can be modelled by a Stokes equation. Two applications are considered in more details. One is the densification process through sintering and the other the pressing of a glass in a mould

Decoupling and stability of algorithms for boundary value problems

The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated

Local defect correction for glass flow simulation

Local defect correction for glass flow simulation b

Boundary value problems and dichotomic stability

Since the conditioning of a boundary value problem (BVP) is closely related to the existence of a dichotomic fundamental solution (i.e., where one set of modes is increasing and a complementary set is decreasing), it is important to have discretization methods that conserve this dichotomy property. The conditions this imposes on such a method are investigated in this paper. They are worked out in more detail for scalar second-order equations (the central difference scheme), and for linear first-order systems as well; for the latter type both one-step methods (including collocation) and multistep methods (those that may be used in multiple shooting) are examin