High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

Modeling Electrostatics and Geometrical Quantities in Molecular Biophysics Using a Gaussian-Based Model of Atoms

Electrostatic and geometric factors are critical to modeling the interactions and solvation effects of biomolecules in the aqueous environments of biological cells as they respectively influence the polar and non-polar components of the associated free energies. Conventional protocols use a hard-sphere model of atoms to devise and study the underlying thermodynamics. But this traditional model tends to overlook some of the important biophysical aspects at the cost of oversimplification of the representation of the solute-solvent environments. Here an alternative and physically appealing model of atoms – a Gaussian-based model, is presented which replaces the hard-sphere model with a smooth density-based description of atoms. This dissertation explains the derivation of a physically appealing dielectric distribution from the Gaussian schematic to model the electrostatics of biomolecules using the implicit-solvent/Poisson-Boltzmann (PB) formalism. It also demonstrates the advantages of using it for computing geometric properties of a molecule such as its volume and surface area (SA) for estimating non-polar portions of the free energy. While highlighting the qualitative importance of the Gaussian-based model, it offers conceptual proofs towards its validity through computational investigations of explicit solvent simulations. It also reports the key features of the Gaussian-based model, which impart to it the capacity of accurately capturing the crucial biophysical factors that characterize biomolecular properties, namely – the effect of intrinsic conformational flexibility and salt distribution. The non-triviality of these factors and their portrayal through the Gaussian models are meticulously discussed. A major theme of this work is the implementation of the Gaussian model of dielectric distribution and volume/SA estimation into the PB solver package called Delphi. These developments illustrate the manner in which the utility of Delphi has been expanded and its reputation as a popular tool for modeling solvation effects with appreciable time-efficacy and accuracy has been enhanced