Realizability condition

Vedula, P., P. K. Yeung, and R. O. Fox (2001). Dynamics of scalar dissipation in isotropic turbulence A numerical and modeling study. Journal of Fluid Mechanics 433, 29-60. Verman, B., B. Geurts, and H. Kuertan (1994). Realizability conditions for the turbulent stress tensor in large-eddy simulations. Journal of Fluid Mechanics 278, 351-362. Vervisch, L. (1991). Prise en compte d effets de cinetique chimique dans lesflammes de diffusion turbulente par Tapproche fonction densite de probabilite. Ph. D. thesis, Universite de Rouen, France. 

The exponential increase in computer power and the development of highly efficient algorithms has distinctly expanded the range of structures that can be treated on a first-principle level. Using parallel computers, AIMD simulations of systems with few hundred atoms can be performed nowadays. This range already starts to approach the one relevant in biochemistry. Indeed, some simulations of entire biomolecules in laboratory-realizable conditions (such as crystals or aqueous solutions) have been performed recently [25-28]. For most applications however, the systems are still too large to be treated fully at the AIMD level. By combining AIMD simulations with a classical MD force field in a mixed quantum mechanical/molecular mechanical fashion (Hybrid-AIMD) the effects of the protein environment can be explicitly taken into account and the system size can be extended. 

AIMD simulation of the full system in laboratory-realizable conditions (e.g. in the crystal phase or in aqueous solution)... 

RNA and DNA are in general very difficult to model with force-field based approaches. One major difficulty is to reproduce the backbone conformation (crucial for any modeling of nucleic acids), as the corresponding torsional energy barriers are very small [29]. First results from AIMD are encouraging the calculated structure of a hydrated GpG RNA duplex in laboratory realizable conditions (that is, in the crystal phase)[25] showed excellent agreement with experiment and provided the H-bond network postulated by the crystallographers. 

The nonnegativity of hff) is guaranteed under the following realizability condition ... 

The reader should notice that this realizability condition on the time step At applies even when a high-order spatial reconstruction is used for the velocity. It is also worth noting that Eq. (8.26) applies even when the velocity is discontinuous (e.g. at a shock). 

According to the realizability condition, these NDE must both be nonnegative for all values of v. 

As before, the numerical NDE in Eq. (8.49) will be nonnegative only if we use QBMM and impose the following realizability conditions on the abscissas ... 

If the realizability condition in Eq. (8.52) is satisfied, found from Eq. (8.53) is guaranteed to be realizable. Using the method described above for the PBE (Eq. (8.35)), it is straightforward to extend Eq. (8.53) to second-order time-stepping. The extension to multiple velocity components v in multiple spatial dimensions is a bit more complicated. As described in Yuan Fox (2011) and in Section B.3 of Appendix B, when the CQMOM is used to constmct the multivariate quadratures all permutations must be used in a consistent manner in order to get the correct kinetic energy fiuxes. Nevertheless, for each permutation of the CQMOM, the basic time-stepping formula in Eq. (8.53) is used to update the transported moment set by modifying the definition of K , to include the multivariate moments in the optimal-moment set. Readers interested in more details on multidimensional free transport should consult Yuan Eox (2011) and Section B.3 of Appendix B. 

The realizability condition on A/ is thus the same as in Eq. (8.52). In summary, the four-stage time-stepping procedure starting from the transported moment set M is as follows ... 

When the first-order scheme is semi-implicit with 0 < a < 1, this realizability condition is a mix of those found for pure advection and pure diffusion. 

The first term on the right-hand side will be nonnegative when At satisfies the realizability condition... 

The constant 0 in the Stokes-Einstein diffusivity is added to treat the limit of vanishing size so that the diffusivity coefficient remains finite. The realizability condition for Eq. (8.75) is... 

The realizability condition for Eq. (8.78) is that all of the coefficients involving the weights must be nonnegative. This condition yields... 

Note that if = 0, then w l 2jkai T-i/2jkar realizability condition is... 

The global realizability condition (which determines the time step) is... 

The resulting realizability condition is analogous to Eq. (B.23), but with the velocity abscissas and appearing in place of and %i-i/2jk.r respectively. 

If the second realizability condition holds, then M is realizable. 

The treatment of the z direction proceeds in the same manner, starting from M . The third realizability condition for At is... 

The updated transported moment set M will be realizable provided that the global realizability condition is satisfied for all permutations of the CQMOM. ... 

In other words, the realizability condition for a given direction must be satisfied for both permutations of the CQMOM used in that direction. 

The final terms in Eqs. (B.50) and Eq. (B.51) are nonnegative, thus nfjl will be realizable if the following realizability condition for At holds ... 

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