We develop in this paper efficient and robust numerical methods for solving anisotropic Cahn-Hilliard systems. We construct energy stable schemes for the time discretization of the highly nonlinear anisotropic Cahn-Hilliard systems by using a stabilization technique. At each time step, these schemes lead to a sequence of linear coupled elliptic equations with constant coefficients that can be efficiently solved by using a spectral-Galerkin method. We present numerical results that are consistent with earlier work on this topic, and also carry out various simulations, such as the linear bi-Laplacian regularization and the nonlinear Willmore regularization, to demonstrate the efficiency and robustness of the new schemes.

The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod-eling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete and norms. We also present an efficient, practical nonlinear multigrid method . comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part . for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.

We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework. Utilizing an adaptive finite element implementation with effective gradient recovery techniques, we discuss how the Euler number can be accurately computed directly from the nu-merically solved phase field functions or order parameters. Numerical examples and applications to the topological analysis of point clouds are also presented.

We formulate a one-dimensional time-dependent non-linear mathematical model for some types of physiological fluid flow in collapsible tubes with discontinuous material properties. The resulting 6 × 6 hyperbolic system is analysed and the associated Riemann problem is solved exactly. Although the solution algorithm deals with idealised cases, it is nonetheless uniquely well-suited for assessing the performance of numerical methods intended for simulating more general situations. Moreover, our model may be a useful starting point for numerical calculations of realistic flows involving rapid and discontinuous material property variations. One important example in mind is the simulation of blood flow in medium-to-large veins in humans. Finally, we also discuss some peculiarities of the model regarding the loss of strict hyperbolicity and uniqueness. In particular we show an example in which the solution of the Riemann problem is non unique.

In event-driven algorithms for simulation of diffusing, colliding, and reacting particles, new positions and events are sampled from the cumulative distribution function (CDF) of a probability distribution. The distribution is sampled frequently and it is important for the efficiency of the algorithm that the sampling is fast. The CDF is known analytically or computed numerically. Analytical formulas are sometimes rather complicated making them difficult to evaluate. The CDF may be stored in a table for interpolation or computed directly when it is needed. Different alternatives are compared for chemically reacting molecules moving by Brownian diffusion in two and three dimensions. The best strategy depends on the dimension of the problem, the length of the time interval, the density of the particles, and the number of different reactions.

The Poisson-Boltzmann equation is widely used for modeling the electro-statics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. The default choice for the dielectric boundary has been the molecular surface, but the use of the van der Waals surface has also been advocated. Here we review recent studies in which the two choices are tested against experimental results and explicit-solvent calculations. The assignment of the solvent high dielectric constant to interstitial voids in the solute is often used as a criticism against the van der Waals surface. However, this assignment may not be as unrealistic as previously thought, since hydrogen exchange and other NMR experiments have firmly established that all interior parts of proteins are transiently accessible to the solvent.

We consider multi-physics computations where the Navier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain. The different subdomains are separated by well-defined interfaces. We consider time accurate computations resolving all time scales. For such computations, explicit time stepping is very efficient. We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability, or to a significant reduction of the stable time step size. Finding such interface conditions is non-trivial.

We discretize the problem with high order centered difference approximations with summation by parts boundary closure. We derive L2 stable interface conditions for the linearized one dimensional discretized problem. Furthermore, we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem. The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sjögreen in [8] as a special case.

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions, where the wave propagation is governed by the Helmholtz equation. The scattering problem is modeled as a boundary value problem over a bounded domain. Based on the Dirichlet-to-Neumann (DtN) operator, a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle. An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition. Numerical experiments are included to compare with the perfectly matched layer (PML) method to illustrate the competitive behavior of the proposed adaptive method.

In this paper, we highlight the benefits resulting from imposing energy-conserving equilibria in entropic lattice Boltzmann models for isothermal flows. The advantages are documented through a series of numerical simulations, such as Taylor-Green vortices, cavity flow and flow past a sphere.

Many molecular events are associated with small or large conformational changes occurring in the corresponding proteins. Modeling such changes is a challenge and requires significant amount of computing time. From point of view of electrostatics, these changes can be viewed as a reorganization of local charges and dipoles in response to the changes of the electrostatic field, if the cause is insertion or deletion of a charged amino acid. Here we report a large scale investigation of modeling the changes of the folding energy due to single mutations involving charged group. This allows the changes of the folding energy to be considered mostly electrostatics in origin and to be calculated with DelPhi assigning residue-specific value of the internal dielectric constant of protein. The predicted energy changes are benchmarked against experimentally measured changes of the folding energy on a set of 257 single mutations. The best fit between experimental values and predicted changes is used to find out the effective value of the internal dielectric constant for each type of amino acid. The predicted folding free energy changes with the optimal, amino acid specific, dielectric constants are within RMSD=0.86 kcal/mol from experimentally measured changes.

A general analysis of the hydrodynamic limit of multi-relaxation time lattice Boltzmann models is presented. We examine multi-relaxation time BGK collision operators that are constructed similarly to those for the MRT case, however, without explicitly moving into a moment space representation. The corresponding ‘moments’ are derived as left eigenvectors of said collision operator in velocity space. Consequently we can, in a representation independent of the chosen base velocity set, generate the conservation equations. We find a significant degree of freedom in the choice of the collision matrix and the associated basis which leaves the collision operator invariant. We explain why MRT implementations in the literature reproduce identical hydrodynamics despite being based on different orthogonalization relations. More importantly, however, we outline a minimal set of requirements on the moment base necessary to maintain the validity of the hydrodynamic equations. This is particularly useful in the context of position and time-dependent moments such as those used in the context of peculiar velocities and some implementations of fluctuations in a lattice-Boltzmann simulation.

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the dif-fusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

The additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements (TSEM). The method is a generalization of the corresponding method in tensorial quadrilateral spectral elements (QSEM). The proposed preconditioners are based on partitioning the domain into overlapping subdomains, solving local problems on these subdomains and solving an additional coarse problem associated with the subdomain mesh. The results of numerical experiments show that the proposed preconditioner are robust with respect to the number of elements and are more efficient than the preconditioners with generous overlaps.

This is a brief review of the computational modeling of protein-ligand interactions using a recently developed fully polarizable continuum model (FPCM) and rational drug design. Computational modeling has become a powerful tool in understanding detailed protein-ligand interactions at molecular level and in rational drug design. To study the binding of a protein with multiple molecular species of a ligand, one must accurately determine both the relative free energies of all of the molecular species in solution and the corresponding microscopic binding free energies for all of the molecular species binding with the protein. In this paper, we aim to provide a brief overview of the recent development in computational modeling of the solvent effects on the detailed protein-ligand interactions involving multiple molecular species of a ligand related to rational drug design. In particular, we first briefly discuss the main challenges in computational modeling of the detailed protein-ligand interactions involving the multiple molecular species and then focus on the FPCM model and its applications. The FPCM method allows accurate determination of the solvent effects in the first-principles quantum mechanism (QM) calculations on molecules in solution. The combined use of the FPCM-based QM calculations and other computational modeling and simulations enables us to accurately account for a protein binding with multiple molecular species of a ligand in solution. Based on the computational modeling of the detailed protein-ligand interactions, possible new drugs may be designed rationally as either small-molecule ligands of the protein or engineered proteins that bind/metabolize the ligand. The computational drug design has successfully led to discovery and development of promising drugs.

In their tubelike phase, nanowire-adsorbed polymers exhibit strong structural similarities to morphologies known from single-walled carbon (hexagonal) and boron (triangular) nanotubes. Since boron/boron nitride tubes require some disorder for stability the triangular polymer tubes provide a closer analog to the carbon tubes. By means of computer simulations of both two and three dimensional versions of a coarse-grained bead-spring model for the polymers, we investigate their structural properties and make a detailed comparison with structures of carbon nanotubes.