The structural and functional description of protein-protein complexes and their comprehension is a key concept, not only to increase the scientific knowledge in basic terms but also for the application to the biomedical and pharmaceutical industry. The binding association between proteins is nowadays attribute to a few key residues at the interface – the hot-spots. The complex between the RNase inhibitor (RI) and RNaseA protein provides an excellent system to study the role of the functional epitope as it is essential in various molecular recognition processes and constitute one of the tightest complexes known. An energetic pattern of the interface is accomplished by computational alanine scanning mutagenesis and a dynamical characterization is accomplished by a detailed study of the molecular dynamical simulations. A special emphasis is given to the role of solvation across the interface and the shielding of warm-and hot-spots from water.

Discontinuous Galerkin (DG) and matrix-free finite element methods with a novel projective pressure estimation are combined to enable the numerical modeling of magma dynamics in 2D and 3D using the library deal.II. The physical model is an advection-reaction type system consisting of two hyperbolic equations to evolve porosity and soluble mineral abundance at local chemical equilibrium and one elliptic equation to recover global pressure. A combination of a discontinuous Galerkin method for the advection equations and a finite element method for the elliptic equation provide a robust and efficient solution to the channel regime problems of the physical system in 3D. A projective and adaptively applied pressure estimation is employed to significantly reduce the computational wall time without impacting the overall physical reliability in the modeling of important features of melt segregation, such as melt channel bifurcation in 2D and 3D time dependent simulations.

The lattice Boltzmann method is applied to the investigations of the diffusivity and the permeability in the gas diffusion layer (GDL) of the polymer electrolyte fuel cell (PEFC). The effects of the configuration of water droplets, the porosity of the GDL, the viscosity ratio of water to air, and the surface wettability of the GDL are investigated. From the simulations under the PEFC operating conditions, it is found that the heterogeneous water network and the high porosity improve the diffusivity and the permeability, and the hydrophobic surface decreases the permeability.

We present an accelerated method for stochastically simulating the dynamics of heterogeneous cell populations. The algorithm combines a Monte Carlo approach for simulating the biochemical kinetics in single cells with a constant-number Monte Carlo method for simulating the reproductive fitness and the statistical characteristics of growing cell populations. To benchmark accuracy and performance, we compare simulation results with those generated from a previously validated population dynamics algorithm. The comparison demonstrates that the accelerated method accurately simulates population dynamics with significant reductions in runtime under commonly invoked steady-state and symmetric cell division assumptions. Considering the increasing complexity of cell population models, the method is an important addition to the arsenal of existing algorithms for simulating cellular and population dynamics that enables efficient, coarse-grained exploration of parameter space.

We present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1D models. The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented to the maximal order p=18 in the time of propagation (Monte Carlo time step), which substantially improves the convergence of discretized amplitudes to their exact continuum values. Symbolic derivation of higher-order effective actions is implemented in SPEEDUP Mathematica codes, using the recursive Schrödinger equation approach. In addition to the general 1D quantum theory, developed Mathematica codes are capable of calculating effective actions for specific models, for general 2D and 3D potentials, as well as for a general many-body theory in arbitrary number of spatial dimensions.

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numericalmodel leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ = (1), and a special initial guess for λ ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ = (1) and λ ≪ 1. The (m + 1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.

In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve the boundary value problem for the stress field associated with dislocations in a finite medium. Numerical examples are presented to examine the influence from material boundaries on dislocations.

This study aims to investigate turbulent plasma flow using the lattice Boltzmann (LB) method. A double population model D2Q9-D2Q4 is employed to calculate the plasma velocity and temperature fields. Along with the calculation process a conversion procedure is made between the LB and the physical unit systems, so that thermo-physical properties variation is fully accounted for and the convergence is checked in physical space. The configuration domain and the boundary condition treatment are selected based on the most cited studies in order to illustrate a realistic situation. The jet morphology analysis gives credible results by comparison with commonly published works. It was demonstrated also that accounting for the substrate as wall boundary condition modify greatly the flow and temperature structures with may affect absolutely the particles behavior during its in-flight in the hot gas.

In this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

After setting a mixed formulation for the propagation of linearized water waves problem, we define its spectral element approximation. Then, in order to take into account unbounded domains, we construct absorbing perfectly matched layer for the problem. We approximate these perfectly matched layer by mixed spectral elements and show their stability using the “frozen coefficient” technique. Finally, numerical results will prove the efficiency of the perfectly matched layer compared to classical absorbing boundary conditions.

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the L2 norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the L2 norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

Novel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation. The method developed can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion, and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method. The numerical results indicate that the high order moment system in [6] is a valid model for the Wigner equation, and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.

This paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi, SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.

We introduce a lattice-free hard sphere exclusion stochastic process. The resulting stochastic rates are distance based instead of cell based. The corresponding Markov chain build for this many particle system is updated using an adaptation of the kinetic Monte Carlo method. It becomes quickly apparent that due to the lattice-free environment, and because of that alone, the dynamics behave differently than those in the lattice-based environment. This difference becomes increasingly larger with respect to particle densities/temperatures. The well-known packing problem and its solution (Palasti conjecture) seem to validate the resulting lattice-free dynamics.

In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

A diesel particulate filter (DPF) is a key technology to meet future emission standards of particulate matters (PM), mainly soot. It is generally consists of a wall-flow type filter positioned in the exhaust stream of a diesel vehicle. It is difficult to simulate the thermal flow in DPF, because we need to consider the soot deposition and combustion in the complex geometry of filter wall. In our previous study, we proposed an approach for the conjugate simulation of gas-solid flow. That is, the gas phase was simulated by the lattice Boltzmann method (LBM), coupled with the equation of heat conduction inside the solid filter substrate. However, its numerical procedure was slightly complex. In this study, to reduce numerical costs, we have tested a new boundary condition with chemical equilibrium in soot combustion at the surface of filter substrate. Based on the soot oxidation rate with catalysts evaluated in experiments, the lattice Boltzmann simulation of soot combustion in the catalyzed DPF is firstly presented to consider the process in the after-treatment of diesel exhaust gas. The heat and mass transfer is shown to discuss the effect of catalysts.

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.