Numerical convergence

Likewise, efficient interface reconstruction algorithms and mixed cell thermodynamics routines have been developed to make three-dimensional Eulerian calculations much more affordable. In general, however, computer speed and memory limitations still prevent the analyst from doing routine three-dimensional calculations with the resolution required to be assured of numerically converged solutions. As an example. Fig. 9.29 shows the setup for a test involving the oblique impact of a copper ball on a hardened steel target... 

A common feature in the models reviewed above was to calculate pressure and temperature distributions in a sequential procedure so that the interactions between temperature and other variables were ignored. It is therefore desirable to develop a numerical model that couples the solutions of pressure and temperature. The absence of such a model is mainly due to the excessive work required by the coupling computations and the difficulties in handling the numerical convergence problem. Wang et al. [27] combined the isothermal model proposed by Hu and Zhu [16,17] with the method proposed by Lai et al. for thermal analysis and presented a transient thermal mixed lubrication model. Pressure and temperature distributions are solved iteratively in a iterative loop so that the interactions between pressure and temperature can be examined. 

So far, the only approximation in our description of the FMS method has been the use of a finite basis set. When we test for numerical convergence (small model systems and empirical PESs), we often do not make any other approximations but for large systems and/or ab //i/Y/o-determined PESs (AIMS), additional approximations have to be made. These approximations are discussed in this subsection in chronological order (i.e., we begin with the initial basis set and proceed with propagation and analysis of the results). 

When the FMS method was first introduced, a series of test calculations were performed using analytical PESs. These calculations tested the numerical convergence with respect to the parameters that define the nuclear basis set (number of basis functions and their width) and the spawning algorithm (e.g., Xo and MULTISPAWN). These studies were used to validate the method, and therefore we refrained from making any approximations beyond the use of a... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

Adoption of a many-body theory that is systematically improvable (at least in principle) and testable for its numerical convergence ... 

Overall, we seem to find reasons to be hopeful about the possibilities of the RQDO formalism for predicting spectral properties of interest in astrophysics and plasma physics. These reasons rest on the correctness of the resuits so far achieved for F/ and Cl I [20,21], and the low computational expense and avoidance of the numerous convergence probiems which are common in the multiconfigurational approaches. 

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