SUMMARYA methodology for squeezing the most out of massively parallel processors when solving partial di erential evolution equations by implicit schemes is presented. Its key components include a preferred implicit time-integrator, a decomposition of the time-domain into time-slices, independent time-integrations in each time-slice of the semi-discrete equations, and Newton-type iterations on a coarse time-grid. Hence, this methodology parallelizes the time-loop of a time-dependent partial di erential equation solver without interfering with its sequential or parallel space-computations. It is particularly interesting for timedependent problems with a few degrees of freedom such as those arising in robotics and protein folding applications, where the opportunities for parallelization over the degrees of freedom are limited. Error and stability analyses of the proposed parallel methodology are performed for ÿrst-and second-order hyperbolic problems. Its feasibility and impact on reducing the solution time below what is attainable by methods which address only parallelism in the space-domain are highlighted for uid, structure, and coupled uid-structure model problems.

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The velocity boundary condition that must be imposed at an interface between a porous medium and a free fluid is investigated. A heterogeneous transition zone characterized by rapidly varying properties is introduced between the two homogeneous porous and free fluid regions. The problem is solved using the method of matched asymptotic expansions and boundary conditions between the two homogeneous regions are obtained. The continuity of the velocity is recovered and a jump in the stress built using the viscosity (and not the effective viscosity) appears. This result also provides an explicit dependence of the stress jump coefficient to the internal structure of the transition zone and its sensitivity to this micro structure is recovered.