In, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire in to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

Salt influences protein stability through electrostatic mechanisms as well as through nonpolar Hofmeister effects. In the present work, a continuum solvation based model is developed to explore the impact of salt on protein stability. This model relies on a traditional Poisson-Boltzmann (PB) term to describe the polar or electrostatic effects of salt, and a surface area dependent term containing a salt concentration dependent microscopic surface tension function to capture the non-polar Hofmeister effects. The model is first validated against a series of cold-shock protein variants whose salt-dependent protein fold stability profiles have been previously determined experimentally. The approach is then applied to HIV-1 protease in order to explain an experimentally observed enhancement in stability and activity at high (1M) NaCl concentration. The inclusion of the salt-dependent non-polar term brings the model into quantitative agreement with experiment, and provides the basis for further studies into the impact of ionic strength on protein structure, function, and evolution.

Based on biologically inspired algorithms, a thermodynamic model to describe the vapor-liquid equilibrium of binary complex mixtures containing supercritical fluids and ionic liquids, is presented. The Peng-Robinson equation of state with the Wong-Sandler mixing rules are used to evaluate the fugacity coefficient on the systems. Then, a hybrid particle swarm-ant colony optimization was used to minimize the difference between calculated and experimental bubble pressure, and calculate the binary interaction parameters for the excess Gibbs free energy of all systems used. Simulations are carried out in nine systems with imidazolium-based ionic liquids. The results show that the bubble pressures were correlated with low deviations between experimental and calculated values. These deviations show that the proposed hybrid algorithm is the preferable method to describe the phase equilibrium of these complex mixtures, and can be used for other similar systems.

A two-dimensional lattice Boltzmann model has been employed to simulate the impingement of a liquid drop on a dry surface. For a range of Weber number, Reynolds number and low density ratios, multiple phases leading to breakup have been obtained. An analytical solution for breakup as function of Reynolds and Weber number based on the conservation of energy is shown to match well with the simulations. At the moment breakup occurs, the spread diameter is maximum; it increases with Weber number and reaches an asymptotic value at a density ratio of 10. Droplet breakup is found to be more viable for the case when the wall is non-wetting or neutral as compared to a wetting surface. Upon breakup, the distance between the daughter droplets is much higher for the case with a non-wetting wall, which illustrates the role of the surface interactions in the outcome of the impact.

In an interdisciplinary field on mathematics and physics, we examine a physical problem, fluid flow in porous media, which is represented by a stochastic partial differential equation (SPDE). We first give a priori error estimates for the solutions to an optimization problem constrained by the physical model under lower regularity assumptions than the literature. We then use the concept of Galerkin finite element methods to establish a new numerical algorithm to give approximations for our stochastic optimal physical problem. Finally, we develop original computer programs based on the algorithm and use several numerical examples of various situations to see how well our solver works by comparing its outputs to the priori error estimates.

We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

The effective thermal conductivity of composite materials with thermal contact resistance at interfaces is studied by lattice Boltzmann modeling in this work. We modified the non-dimensional partial bounce-back scheme, proposed by Han et al. [Int. J. Thermal Sci., 2008. 47: 1276-1283], to introduce a real thermal contact resistance at interfaces into the thermal lattice Boltzmann framework by re-deriving the redistribution function of heat at the phase interfaces for a corrected dimensional formulation. The modified scheme was validated in several cases with good agreement between the simulation results and the corresponding theoretical solutions. Furthermore, we predicted the effective thermal conductivities of composite materials using this method where the contact thermal resistance was not negligible, and revealed the effects of particle volume fraction, thermal contact resistance and particle size. The results in this study may provide a useful support for materials design and structure optimization.

In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321-340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are optimally convergent in the L2-norm on quasi-uniform tetrahedral meshes if the solution is sufficiently smooth.

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

Abstract. The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick’s method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.

A two-grid method for solving the Cahn-Hilliard equation is proposed in this paper. This two-grid method consists of two steps. First, solve the Cahn-Hilliard equation with an implicit mixed finite element method on a coarse grid. Second, solve two Poisson equations using multigrid methods on a fine grid. This two-grid method can also be combined with local mesh refinement to further improve the efficiency. Numerical results including two and three dimensional cases with linear or quadratic elements show that this two-grid method can speed up the existing mixed finite method while keeping the same convergence rate.

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for defining potentials. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problemswhich involve conductive regionswith holes, cuts are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature.

The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.

The analysis of seismic wave propagation and amplification in complex geological structures requires efficient numerical methods. In this article, following up on recent studies devoted to the formulation, implementation and evaluation of 3-D single- and multi-region elastodynamic fast multipole boundary element methods (FM-BEMs), a simple preconditioning strategy is proposed. Its efficiency is demonstrated on both the single- and multi-region versions using benchmark examples (scattering of plane waves by canyons and basins). Finally, the preconditioned FM-BEM is applied to the scattering of plane seismic waves in an actual configuration (alpine basin of Grenoble, France), for which the high velocity contrast is seen to significantly affect the overall efficiency of the multi-region FM-BEM.

We propose a deterministic solver for the time-dependent multi-subband Boltzmann transport equation (MSBTE) for the two dimensional (2D) electron gas in double gate metal oxide semiconductor field effect transistors (MOSFETs) with flared out source/drain contacts. A realistic model with six-valleys of the conduction band of silicon and both intra-valley and inter-valley phonon-electron scattering is solved. We propose a second order finite volume method based on the positive and flux conservative (PFC) method to discretize the Boltzmann transport equations (BTEs). The transport part of the BTEs is split into two problems. One is a 1D transport problem in the position space, and the other is a 2D transport problem in the wavevector space. In order to reduce the splitting error, the 2D transport problem in the wavevector space is solved directly by using the PFC method instead of splitting into two 1D problems. The solver is applied to a nanoscale double gate MOSFET and the current-voltage characteristic is investigated. Comparison of the numerical results with ballistic solutions show that the scattering influence is not ignorable even when the size of a nanoscale semiconductor device goes to the scale of the electron mean free path.

This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.

Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approach to such problems, the elastic forces generated by the immersed structure are applied to the surrounding fluid, and the motion of the immersed structure is determined by the local motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom. For such models, force and torque must both be applied to the fluid. The positional degrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the local angular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid. To describe the elasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced by our method to classical analytic results from elastic rod theory.

Aiming at simulating micro gas flows with accurate boundary conditions, an efficient hybrid algorithm is developed by combining the molecular dynamics (MD) method with the direct simulation Monte Carlo (DSMC) method. The efficiency comes from the fact that the MD method is applied only within the gas-wall interaction layer, characterized by the cut-off distance of the gas-solid interaction potential, to resolve accurately the gas-wall interaction process, while the DSMC method is employed in the remaining portion of the flow field to efficiently simulate rarefied gas transport outside the gas-wall interaction layer. A unique feature about the present scheme is that the coupling between the two methods is realized by matching the molecular velocity distribution function at the DSMC/MD interface, hence there is no need for one-to-one mapping between a MD gas molecule and a DSMC simulation particle. Further improvement in efficiency is achieved by taking advantage of gas rarefaction inside the gas-wall interaction layer and by employing the “smart-wall model” proposed by Barisik et al. The developed hybrid algorithm is validated on two classical benchmarks namely 1-D Fourier thermal problem and Couette shear flow problem. Both the accuracy and efficiency of the hybrid algorithm are discussed. As an application, the hybrid algorithm is employed to simulate thermal transpiration coefficient in the free-molecule regime for a system with atomically smooth surface. Result is utilized to validate the coefficients calculated from the pure DSMC simulation with Maxwell and Cercignani-Lampis gas-wall interaction models.

We combine the Shan-Chen multicomponent lattice Boltzmann model and the link-based bounce-back particle suspension model to simulate particle motion in binary immiscible fluids. The impact of the slightly mixing nature of the Shan-Chen model and the fluid density variations near the solid surface caused by the fluid-solid interaction, on the particle motion in binary fluids is comprehensively studied. Our simulations show that existing models suffer significant fluid mass drift as the particle moves across nodes, and the obtained particle trajectories deviate away from the correct ones. A modified wetting model is then proposed to reduce the non-physical effects, and its effectiveness is validated by comparison with existing wetting models. Furthermore, the first-order refill method for the newly created lattice node combined with the new wetting model significantly improves mass conservation and accuracy.

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.

In this work, we are concerned with a time-splitting Fourier pseudospectral (TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless parameter ɛ ∊ (0,1]. In the nonrelativistic limit regime, the small ɛ produces high oscillations in exact solutions with wavelength of (ɛ2) in time. The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time, with both the nonlinear and linear subproblems exactly integrable in time and, respectively, Fourier frequency spaces. The method is fully explicit and time reversible. Moreover, we establish rigorously the optimal error bounds of a second-order TSFP for fixed ɛ = (1), thanks to an observation that the scheme coincides with a type of trigonometric integrator. As the second task, numerical studies are carried out, with special efforts made to applying the TSFP in the nonrelativistic limit regime, which are geared towards understanding its temporal resolution capacity and meshing strategy for (ɛ2)-oscillatory solutions when 0 < ɛ « 1. It suggests that the method has uniform spectral accuracy in space, and an asymptotic (ɛ−2Δt2) temporal discretization error bound (Δt refers to time step). On the other hand, the temporal error bounds for most trigonometric integrators, such as the well-established Gautschi-type integrator in, are (ɛ−4Δt2). Thus, our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous or numerical, are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.