Some aspects of using IV software to solve BVP via a Riccati transformation

摘要

We consider a numerical method for the solution of linear boundary value problems (BVPs) with separated boundary conditions (BCs) for ordinary differential equations (ODEs). By using a particular linear transformation, the problem is reformulated as three initial value problems (IVPs), one of them being a Riccati differential equation. The nonlinearity of the Riccati DE makes it necessary to deal with the eventuality of modifying the structure of integration: so-called “multiple embedding” equations must thus be derived, and a marching algorithm is incorporated. A stability analysis of the three IVPs indicates clearly for what class of problems the method is suited and why “stiff” IV methods should then frequently be used. This leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the Riccati DE. We interpret this occurrence in terms of the original BVP, and the resulting analysis suggests some final considerations, including the possibility of corrective measures.