Motion value problems, integrals

MD tracks the temporal evolution of a microscopic model system by integrating the equations of motion for all microscopic degrees of freedom. Numerical integration algorithms for initial value problems are used for this purpose, and their strengths and weaknesses have been discussed in simulation texts [104-106]. 

To integrate the equations of motion in a stable and reliable way, it is necessary that the fundamental time step is shorter than the shortest relevant timescale in the problem. The shortest events involving whole atoms are C-H vibrations, and therefore a typical value of the time step is 2fs (10-15s). This means that there are up to one million time steps necessary to reach (real-time) simulation times in the nanosecond range. The ns range is sufficient for conformational transitions of the lipid molecules. It is also sufficient to allow some lateral diffusion of molecules in the box. As an iteration time step is rather expensive, even a supercomputer will need of the order of 106 s (a week) of CPU time to reach the ns domain. 

Accdg to Dunkle (Ref 28), Brode (Ref 14), in order to solve detonation problems without recourse to empirical values derived from explosion measurements, integrated the hydrodynamical equations of motion (which constitute a set of nonlinear partial... 

When considering that all the dimensionless values are functions of the dimensionless values y, tu A where A (A=r/VTJt) is the only variable, Sedov stated that the gas motion is automodeling, and the problem of detg it is reduced to integration of ordinary differential equations... 

Let us solve problem (3.75), (3.76) now. The right equation (3.75) can be evidently solved providing the linear shear stress in the duct over the droplet layer EPR (its value on the interface Th = t(5 + 0) = 1 - 8 contributes to the internal motion along with the pressure gradient p = 1) and the parabolic velocity profile in the duct s section free of EPR. The left system of equations admits the first integral... 

The value of the boundary-integral method is particularly evident if we consider problems in which one or more of the boundaries is a fluid interface. Here, for simplicity, we consider the generic problem of a drop in an unbounded fluid that is undergoing some mean motion that causes the drop to deform in shape. This type of problem is particularly difficult because the shape of the interface is unknown and is often changing with time. We shall see that the boundary-integral formulation provides a powerful basis to attack this class of problems, and in fact, is largely responsible for much of the considerable theoretical progress that has... 

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