Convergence characteristics of a domain decomposition scheme for approximation of quantum forces

• The convergence characteristics of a scheme for computing quantum forces is studied.
• The number of particles used in discretization has weak influence on the error.
• The overlap between sub-domains must be sufficiently large.
• The length scale of interaction among sub-domains controls the error in most cases.
• Refinement leads to convergence only with adequate control of truncation error.

We investigate the convergence characteristics of a domain decomposition scheme to approximately compute quantum forces in the context of semiclassical Bohmian mechanics. The study is empirical in nature. Errors in the approximate quantum forces are compiled while relevant parameters in the numerical scheme are systematically changed. The compiled errors are analyzed to extract underlying trends. Our analysis shows that the number of Bohmian particles used in discretization has relatively weak influence on the error, while the length scale of interaction among subdomains controls the error in most cases. More precisely, the overall numerical error decreases as the length scale of interaction among subdomains decreases, if the error due to the truncation of the tail of the probability density distribution is adequately controlled. Our results suggest that it may be necessary to develop an efficient method to effectively control the error due to the truncation of the tail. Further studies, especially rigorous mathematical ones, should follow to understand and improve the behavior of the scheme.