Mathematical & Computational Physics

This book aims to provide an overview of the recent research devoted to numerical simulations of physical systems and processes by discussing the tools used in the analyses. It contains 17 chapters, organized into 3 sections: 1. Section 1 - Computational Fluid Dynamics, Magnetohydrodynamics, Solid State. 2. Section 2 - Computational Many-body problem, Condensed Matter, and Molecular Dynamics. 3. Section 3 - Computational Molecular Biophysics, Cellular Structure Prediction, and Reaction–Diffusion Systems. Section 1 starts with a review of particle-based multiscale and hybrid methods applied to fluid mechanics, e.g. molecular dynamics, direct simulation Monte Carlo, lattice Boltzmann method, dissipative particle dynamics and smoothed-particle hydrodynamics. Thereupon, it will focus on numerical methods in the field of computational fluid dynamics, including the new Repeated Replacement Method (RRM) and iterative techniques for finite element Navier–Stokes approximations in incompressible fluid dynamics. Several methods and mathematical models are discussed in the encompassing context of magnetohydrodynamics modeling of the solar atmosphere, non-isothermal solidification of a metal alloy, electronic nature of compressibility in solids. This section ends with an illustration of solid state modeling applied to a papermaking process. Section 2 starts by reviewing of macroscopic transport equations in many-body systems which pose fundamental theoretical challenges in many domains ranging from inter and intra-cellular transport to diffusion in porous media. This section addresses various aspects of the many-body problem in molecular dynamics simulations and condensed matter. Section 3, starts by reviewing computational modeling in biology, covering a wide range of scales from organisms to electrons in atoms, also including the utility of molecular dynamics and quantum mechanical methods in biophysical and biochemical modeling. Next, overviews physics-based cellular structure prediction, focusing on a recently developed RNA statistical mechanical model (the Vfold model) and an application of Random Matrix Theory in the identification of gene expression. The last two chapters is devoted to new models for coupled systems of bulk-surface reaction–diffusion equations which have many applications in fluid dynamics and biological processes.