The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge--Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.

Fernando and Parlett observed that the dqds algorithm for singular values can be made extremely efficient with Rutishauser's choice of shift; in particular it enjoys ``local'' (or one-step) cubic convergence at the final stage of iteration, where a certain condition is to be satisfied. Their analysis is, however, rather heuristic and what has been shown is not sufficient to ensure asymptotic cubic convergence in the strict sense of the word. The objective of this paper is to specify a concrete procedure for the shift strategy and to prove with mathematical rigor that the algorithm with this shift strategy always reaches the ``final stage'' and enjoys asymptotic cubic convergence.

We study an extension of the Gale--Shapley marriage model and the Shapley--Shubik assignment model by considering linear valuations and bounded side payments. Our model includes the Eriksson--Karlander hybrid model as a special case. We propose a polynomial-time algorithm which finds a pairwise-stable outcome.

We consider the asymptotic behavior of the complete system of equations governing a heat-conductive, reactive, compressible viscous gas bounded by two infinite parallel plates. The motion is proved to tend towards the corresponding constant state, as time tends to infinity. Moreover, the decay rate is investigated.

In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.