# Interdisciplinary Applied Mathematics

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Un+1 — и At

~(VUx)n+^(VUx)

oUnxX+

5

~(VUxT- (14.2a) (14.2b)

where At is the time step, Un denotes the value of U at time t, and Un+1 denotes the value of U at time t + At. The intermediate (predicted) state is denoted by U. Therefore, this splitting scheme corresponds to obtaining a predicted field (U) first by advection, and then correcting it with diffusion. For enhanced accuracy and stability of the advection step we have used a third-order Adams-Bashforth scheme. For the diffusion step we have used the Euler backward scheme. This can easily be extended to higher-order time-accurate schemes.

This time-splitting procedure accommodates the implementation of the elemental interface connectivity conditions properly and efficiently. Specifically, the elemental interface connectivity conditions are handled by a characteristic decomposition for the advection substep, and a direct stiffness summation for the diffusion substep. The characteristic treatment is stable for elemental interfaces and inflow/outflow conditions for hyperbolic problems (such as the Euler equations), and the direct stiffness summation procedure is widely used for elemental interface conditions in parabolic problems.

It has been demonstrated in earlier chapters that the majority of microflows (with the exception of micronozzles, Section 6.6) are in the low Reynolds number regime (Re < O(1)) due to the small characteristic dimensions. In some cases, the inertial terms in the Navier-Stokes equations can be neglected, but the continuity equation for density must still be solved

in the characteristic form. Thus, use of time-splitting is inevitable for simulating microflows with our algorithm. For creeping Stokes flow, with the velocity rather than the characteristic dimension being small, the Mach number becomes very small, and use of compressible algorithms becomes inefficient. This is due to the fact that the wave speeds are dominated by the speed of sound cs, and u ^ cs. In such a case we use the incompressible version of p,Flow.

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