A second-order discretization of the nonlinear Poisson–Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids

In this paper we present a finite difference scheme for the discretization of the nonlinear Poisson–Boltzmann (PB) equation over irregular domains that is second-order accurate. The interface is represented by a zero level set of a signed distance function using Octree data structure, allowing a natural and systematic approach to generate non-graded adaptive grids. Such a method guaranties computational efficiency by ensuring that the finest level of grid is located near the interface. The nonlinear PB equation is discretized using finite difference method and several numerical experiments are carried which indicate the second-order accuracy of method. Finally the method is used to study the supercapacitor behaviour of porous electrodes.     

An efficient method for the incompressible Navier–Stokes equations on irregular domains with no-slip boundary conditions,high order up to the boundary

Common efficient schemes for the incompressible Navier–Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier–Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L^∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.     

The Riemann–Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann,Robin and Dirichlet boundary conditions

We revisit the Helmholtz equation in a quarter-plane in the framework of the Riemann–Hilbert approach to linear boundary value problems suggested in late 1990s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas’ scheme.     

Approximate solutions to Poisson–Boltzmann systems with Sobolev gradients

A weighted Sobolev gradient approach [1] is presented to a nonlinear PBE [2] with discontinuous coefficient functions. A comparison is given between the weighted and unweighted Sobolev gradient in the finite element setting in two and three dimensions. Behavior of the various Sobolev gradients is discussed for large jump size in the coefficient. A comparison with Newton’s method is given where the failure of Newton’s method is demonstrated for a test problem.     

Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

A Fourier spectral embedded boundary method, for solution of the Poisson’s equation with Dirichlet boundary conditions and arbitrary forcing functions (including zero forcing function), is presented in this paper. This iterative method begins by transformation of the Dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points (which are called the numerical boundaries), using a second order interpolation method. Then the transformed boundary conditions and the forcing function are extended to a square, smoothly and periodically, via multiplying them by some suitable error functions. Instead of direct solution of the resulting extended Poisson’s problem, it is suggested to define and solve an equivalent transient diffusion problem on the regular domain, until achievement of the steady solution (which is considered as the solution of the original problem). Without need of any numerical time integration method, time advancement of the solution is obtained directly, from the exact solution of the transient problem in the Fourier space. Consequently, timestep sizes can be chosen without stability limitations, which it means higher rates of convergence in comparison with the classical relaxation methods. The method is presented in details for one- and two-dimensional problems, and a new emerged phenomenon (which is called the saturation state) is illustrated both in the physical and spectral spaces. The numerical experiments have been performed on the one- and two-dimensional irregular domains to show the accuracy of the method and its superiority (from the rate of convergence viewpoint) to the other classical relaxation methods. Capability of the method, in dealing with complex geometries, and in presence of discontinuity at the boundaries, has been shown via some numerical experiments on a four-leaf shape geometry.     

A well-conditioned augmented system for solving Navier–Stokes equations in irregular domains

An augmented method based on a Cartesian grid is proposed for the incompressible Navier–Stokes equations in irregular domains. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be utilized in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results in an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improves the condition number of the system for the augmented variable significantly. Using the immersed interface method, we are able to achieve second order accuracy for the velocity. Numerical results and comparisons to benchmark tests are given to validate the new method. A lid-driven cavity flow with multiple obstacles and different geometries are also presented.     

Recovering doping profiles in semiconductor devices with the Boltzmann–Poisson model

We investigate numerically an inverse problem related to the Boltzmann–Poisson system of equations for transport of electrons in semiconductor devices. The objective of the (ill-posed) inverse problem is to recover the doping profile of a device, presented as a source function in the mathematical model, from its current–voltage characteristics. To reduce the degree of ill-posedness of the inverse problem, we proposed to parameterize the unknown doping profile function to limit the number of unknowns in the inverse problem. We showed by numerical examples that the reconstruction of a few low moments of the doping profile is possible when relatively accurate time-dependent or time-independent measurements are available, even though the later reconstruction is less accurate than the former. We also compare reconstructions from the Boltzmann–Poisson (BP) model to those from the classical drift–diffusion-Poisson (DDP) model, assuming that measurements are generated with the BP model. We show that the two type of reconstructions can be significantly different in regimes where drift–diffusion-Poisson equation fails to model the physics accurately. However, when noise presented in measured data is high, no difference in the reconstructions can be observed.     

Local Well-Posedness of Vlasov–Poisson–Boltzmann Equation with Generalized Diffuse Boundary Condition

Journal of Statistical Physics - The Vlasov–Poisson–Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We...     

An efficient multi-scale Poisson solver for the incompressible Navier–Stokes equations with immersed boundaries

The iterative-multi-scale-finite-volume (IMSFV) procedure is applied as an efficient solver for the pressure Poisson equation arising in numerical methods for the simulation of incompressible flows with the immersed-interface method (IIM). Motivated by the requirements of the specific IIM implementation, a modified version of the IMSFV algorithm is presented to allow the solution of problems, in which the varying coefficient of the elliptic equation (e.g. the permeability of the medium in the context of the simulation of flows in porous media) varies over several orders of magnitude or even becomes zero within the integration domain. Furthermore, a strategy is proposed to incorporate the iterative procedure needed by the IIM to converge out constraints at immersed boundaries into the iterative IMSFV cycle. No significant deterioration of performance of the IMSFV method is observed with respect to cases, in which no iterative improvement of the boundary conditions is considered.     

A Kohn–Sham equation solver based on hexahedral finite elements

We design a Kohn–Sham equation solver based on hexahedral finite element discretizations. The solver integrates three schemes proposed in this paper. The first scheme arranges one a priori locally-refined hexahedral mesh with appropriate multiresolution. The second one is a modified mass-lumping procedure which accelerates the diagonalization in the self-consistent field iteration. The third one is a finite element recovery method which enhances the eigenpair approximations with small extra work. We carry out numerical tests on each scheme to investigate the validity and efficiency, and then apply them to calculate the ground state total energies of nanosystems C₆₀, C₁₂₀, and C₂₇₅H₁₇₂. It is shown that our solver appears to be computationally attractive for finite element applications in electronic structure study.     

A spectral FC solver for the compressible Navier–Stokes equations in general domains I: Explicit time-stepping

We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier–Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods—since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.     

A WENO-solver for the transients of Boltzmann–Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods

In this paper we develop a deterministic high order accurate finite-difference WENO solver to the solution of the 1-D Boltzmann–Poisson system for semiconductor devices. We follow the work in Fatemi and Odeh [9] and in Majorana and Pidatella [16] to formulate the Boltzmann–Poisson system in a spherical coordinate system using the energy as one of the coordinate variables, thus reducing the computational complexity to two dimensions in phase space and dramatically simplifying the evaluations of the collision terms. The solver is accurate in time hence potentially useful for time-dependent simulations, although in this paper we only test it for steady-state devices. The high order accuracy and nonoscillatory properties of the solver allow us to use very coarse meshes to get a satisfactory resolution, thus making it feasible to develop a 2-D solver (which will be five dimensional plus time when the phase space is discretized) on today’s computers. The computational results have been compared with those by a Monte Carlo simulation and excellent agreements have been found. The advantage of the current solver over a Monte Carlo solver includes its faster speed, noise-free resolution, and easiness for arbitrary moment evaluations. This solver is thus a useful benchmark to check on the physical validity of various hydrodynamic and energy transport models. Some comparisons have been included in this paper.     

On the effect of boundary conditions on I–V characteristics of HTSC tunnel junctions

The pairing potential distribution over the thickness of superconducting CuO₂ layers in cuprate HTSCs is determined within the Ginzburg–Landau (GL) theory using the microscopic justification of this theory by Gor’kov. It is found that the pairing potential in them is significantly suppressed due to the effect of non-superconducting interlayers, which results in a decrease in the critical temperature of these superconductors. The temperature dependences of the effective energy gap and current–voltage (I–V) characteristic of tunnel junctions of the “break junction” type made of these superconductors are calculated.     

PBE–DFT theoretical study of organic photovoltaic materials based on thiophene with 1D and 2D periodic boundary conditions

Conjugated organic systems such as thiophene are interesting topics in the field of organic solar cells. We theoretically investigate π-conjugated polymers constituted by n units (n = 1–11) based on the thiophene (Tn) molecule. The computations of the geometries and electronic structures of these compounds are performed using the density functional theory (DFT) at the 6–31 G(d, p) level of theory and the Perdew–Burke–Eenzerhof (PBE) formulation of the generalized gradient approximation with periodic boundary conditions (PBCs) in one (1D) and two (2D) dimensions. Moreover, the electronic properties (HOCO, LUCO, E _gap, V _oc, and V _bi) are determined from 1D and 2D PBC to understand the effect of the number of rings in polythiophene. The absorption properties—excitation energies (E _ex), the maximal absorption wavelength (λ_max), oscillator strengths, and light harvesting—efficiency are studied using the time-dependent DFT method. Our studies show that changing the number of thiophene units can effectively modulate the electronic and optical properties. On the other hand, our work demonstrates the efficiency of theoretical calculation in the PBCs.     

Absorbing boundary conditions for the Euler and Navier–Stokes equations with the spectral difference method

Two absorbing boundary conditions, the absorbing sponge zone and the perfectly matched layer, are developed and implemented for the spectral difference method discretizing the Euler and Navier–Stokes equations on unstructured grids. The performance of both boundary conditions is evaluated and compared with the characteristic boundary condition for a variety of benchmark problems including vortex and acoustic wave propagations. The applications of the perfectly matched layer technique in the numerical simulations of unsteady problems with complex geometries are also presented to demonstrate its capability.     

A lattice Boltzmann–cellular automaton study on dendrite growth with melt convection in solidification of ternary alloys

A lattice Boltzmann(LB)–cellular automaton(CA) model is employed to study the dendrite growth of Al-4.0 wt%Cu–1.0 wt%Mg alloy. The effects of melt convection, solute diffusion, interface curvature, and preferred growth orientation are incorporated into the coupled model by coupling the LB–CA model and the CALPHAD-based phase equilibrium solver,Pan Engine. The dendrite growth with single and multiple initial seeds was numerically studied under the conditions of pure diffusion and melt convection. Effects of initial seed number and melt convection strength were characterized by newdefined solidification and concentration entropies. The numerical result shows that the growth behavior of dendrites, the final microstructure, and the micro-segregation are significantly influenced by melt convection during solidification of the ternary alloys. The proposed solidification and concentration entropies are useful characteristics bridging the solidification behavior and the microstructure evolution of alloys.