The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

In this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

This paper is a continuation of our earlier work [SIAM J. Sci. Comput., 32(2010), pp. 2875-2907] in which a numerical moment method with arbitrary order of moments was presented. However, the computation may break down during the calculation of the structure of a shock wave with Mach number M0 ≥ 3. In this paper, we concentrate on the regularization of the moment systems. First, we apply the Maxwell iteration to the infinite moment system and determine the magnitude of each moment with respect to the Knudsen number. After that, we obtain the approximation of high order moments and close the moment systems by dropping some high-order terms. Linearization is then performed to obtain a very simple regularization term, thus it is very convenient for numerical implementation. To validate the new regularization, the shock structures of low order systems are computed with different shock Mach numbers.

Lattice Boltzmann simulations based on the Cahn-Hilliard diffuse interface approach are performed for droplet dynamics in viscous fluid under shear flow, where the degree of confinement between two parallel walls can play an important role. The effects of viscosity ratio, capillary number, Reynolds number, and confinement ratio on droplet deformation and break-up in moderately and highly confined shear flows are investigated.

As an exploratory study for structural deformation and thermodynamic response induced by spacecraft reentry aerodynamic force and thermal environment, a finite element algorithm is presented on the basis of the classic Fourier heat conductive law to simulate the dynamic thermoelasticity coupling performance of the material. The Newmark method and Crank-Nicolson scheme are utilized to discretize the dynamic thermoelasticity equation and heat conductive equation in the time domain, respectively, and the unconditionally stable implicit algorithm is constructed. Four types of finite-element computing schemes are devised and discussed to solve the thermodynamic coupling equation, all of which are implemented and compared in the computational examples including the one-dimensional transient heat conduction in considering and not considering the vibration, the transient heat flow for the infinite cylinder, and the dynamic coupling thermoelasticity around re-entry flat plate from hypersonic aerothermodynamic environment. The computational results show that the transient responses of temperature and displacement field generate lag phenomenon in case of considering the deformation effect on temperature field. Propagation, rebounding, attenuation and stabilized phenomena of elastic wave are also observed by the finite-element calculation of thermodynamic coupling problem considering vibration and damping, and the oscillation of the temperature field is simultaneously induced. As a result, the computational method and its application research platform have been founded to solve the transient thermodynamic coupling response problem of the structure in strong aerodynamic heating and force environment. By comparing various coupling calculations, it is demonstrated that the present algorithm could give a correct and reliable description of transient thermodynamic responses of structure, the rationality of the sequentially coupling method in engineering calculation is discussed, and the bending deformation mechanism produced by the thermodynamic coupling response from windward and leeward sides of flying body is revealed, which lays the foundation in developing the numerical method to solve material internal temperature distribution, structural deformation, and thermal damage induced by spacecraft dynamic thermoelasticity coupling response under uncontrolled reentry aerothermodynamic condition.

In this paper a second-order two-scale (SOTS) analysis method is developed for a static heat conductive problem in a periodical porous domain with radiation boundary condition on the surfaces of cavities. By using asymptotic expansion for the temperature field and a proper regularity assumption on the macroscopic scale, the cell problem, effective material coefficients, homogenization problem, first-order correctors and second-order correctors are obtained successively. The characteristics of the asymptotic model is the coupling of the cell problems with the homogenization temperature field due to the nonlinearity and nonlocality of the radiation boundary condition. The error estimation is also obtained for the original solution and the SOTS’s approximation solution. Finally the corresponding finite element algorithms are developed and a simple numerical example is presented.

For a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

The polynomial chaos expansion (PCE) is an efficient numerical method for performing a reliability analysis. It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimensional polynomial approximation (the so-called PCE). Numerically, such an approximation can be obtained by using a regression method with a suitable design of experiments. The cost of this approximation depends on the size of the design of experiments. If the design of experiments is large and the system is modeled with a computationally expensive FEA (Finite Element Analysis) model, the PCE approximation becomes unfeasible. The aim of this work is to propose an algorithm that generates efficiently a design of experiments of a size defined by the user, in order to make the PCE approximation computationally feasible. It is an optimization algorithm that seeks to find the best design of experiments in the D-optimal sense for the PCE. This algorithm is a coupling between genetic algorithms and the Fedorov exchange algorithm. The efficiency of our approach in terms of accuracy and computational time reduction is compared with other existing methods in the case of analytical functions and finite element based functions.

This paper is about the construction of a unified gas-kinetic scheme (UGKS) for a coupled system of radiative transport and material heat conduction with different diffusive limits. Different from the previous approach, instead of including absorption/emission only, the current method takes both scattering and absorption/emission mechanism into account in the radiative transport process. As a result, two asymptotic limiting solutions will appear in the diffusive regime. In the strong absorption/emission case, an equilibrium diffusion limit is obtained, where the system is mainly driven by a nonlinear diffusion equation for the equilibrium radiation and material temperature. However, in the strong scattering case, a non-equilibrium limit can be obtained, where coupled nonlinear diffusion system with different radiation and material temperature is obtained. In addition to including the scattering term in the transport equation, an implicit UGKS (IUGKS) will be developed in this paper as well. In the IUGKS, the numerical flux for the radiation intensity is constructed implicitly. Therefore, the conventional CFL constraint for the time step is released. With the use of a large time step for the radiative transport, it becomes possible to couple the IUGKS with the gas dynamic equations to develop an efficient numerical method for radiative hydrodynamics. The IUGKS is a valid method for all radiative transfer regimes. A few numerical examples will be presented to validate the current implicit method for both optical thin to optical thick cases.

Phase-field methods with a degenerate mobility have been widely used to simulate surface diffusion motion. However, apart from the motion induced by surface diffusion, adverse effects such as shrinkage, coarsening and false merging have been observed in the results obtained from the current phase-field methods, which largely affect the accuracy and numerical stability of these methods. In this paper, a flux-corrected phase-field method is proposed to improve the performance of phase-field methods for simulating surface diffusion. The three effects were numerically studied for the proposed method and compared with those observed in the two existing methods, the original phase-field method and the profile-corrected phase-field method. Results show that compared to the original phase-field method, the shrinkage effect in the profile-corrected phase-field method has been significantly reduced. However, coarsening and false merging effects still present and can be significant in some cases. The flux-corrected phase field performs the best in terms of eliminating the shrinkage and coarsening effects. The false merging effect still exists when the diffuse regions of different interfaces overlap with each other. But it has been much reduced as compared to that in the other two methods.

Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical structure of the manifold, it is difficult to get efficient numerical method to solve PDE on manifold. In the paper, we propose a method called point integral method (PIM) to solve the Poisson-type equations from point clouds. Among different kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM, the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function. This feature makes the integral equation easy to be discretized from point cloud. In the paper, we explain the derivation of the integral equations, describe the point integral method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.

A 3D lattice Boltzmann (LB) model with twenty-seven discrete velocities is presented and used for the simulation of three-dimensional porous media flows. Its accuracy in combination with the half-way bounce back boundary condition is assessed. Characteristic properties of the gas diffusion layers that are used in polymer electrolyte fuel cells can be determined with this model. Simulation in samples that have been obtained via X-ray tomographic microscopy, allows to estimate the values of permeability and relative effective diffusivity. Furthermore, the computational LB results are compared with the results of other numerical tools, as well as with experimental values.

In this paper, we present calculations based on density functional theory using generalized gradient approximation (GGA) in double perovskite structure La2BB’O6 (B,B’ = 3d transition metal) out of combinational possibilities. Considering 4 types of magnetic states, namely, ferromagnetic (FM), ferrimagnetic (FiM), antiferromagnetics (AF), and nonmagnetic (NM) with full structure optimization, 13 possible surviving, stable FM/FiM-HM materials containing 6 FM-HM materials (La2ScNiO6, La2CrCoO6, La2CrNiO6, La2VScO6, La2VZnO6, and La2VNiO6) and 7 FiM-HM materials (La2VFeO6, La2ZnCoO6, La2TiCoO6, La2CrZnO6, La2CrMnO6, La2ScFeO6, and La2TiMnO6) are found. Considering the correlation effect (GGA+U), there are 6 possible half-metallic stable, surviving (HM) materials containing 3 FMHM materials (La2ScNiO6, La2CrCoO6, and La2CrNiO6) and 3 FiM-HM materials (La2VFeO6, La2ZnCoO6, and La2TiCoO6).

In this paper, the fluid flow and heat transfer around a circular cylinder are studied under various conditions (Reynolds number 10 < Re < 200; Prandtl number, 0.1 ≤ Pr ≤ 2). To solve the governing equations, we use the simplified thermal lattice Boltzmann model based on double-distribution function approach, and present a corresponding boundary treatment for both velocity and temperature fields. Extensive numerical results have been obtained to the flow and heat transfer behaviors. The vortices and temperature evolution processes indicate that the flow and temperature fields change synchronously, and the vortex shedding plays a determinant role in the heat transfer. Furthermore, the effects of Reynolds and Prandtl number on the flow and isothermal patterns and local and averaged Nusselt numbers are discussed in detail. Our simulations show that the local and averaged Nusselt numbers increase with the Reynolds and Prandtl numbers, irrespective of the flow regime. However, the minimum value of the local Nusselt number can shift from the rear point at the back of the cylinder with higher Prandtl number even in the steady flow regime, and the distribution of the local Nusselt number is almost monotonous from front stagnation point to rear stagnation point with lower Prandtl number in the unsteady flow regime.

We report an efficient phase field formalism to compute the stress distribution in polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using a spectral iterative perturbation method. We discuss its applications to computing residual stress distribution in systems containing arbitrarily shaped cavities and cracks (with zero elastic modulus) and to determining the effective elastic properties of polycrystals and multilayered composites.

Efficient algorithms for the simulation of strained heteroepitaxial growth with intermixing in 2+1 dimensions are presented. The first of these algorithms is an extension of the energy localization method [T. P. Schulze and P. Smereka, An energy localization principle and its application to fast kinetic Monte Carlo simulation of heteroepitaxial growth, J. Mech. Phys. Sol., 3 (2009), 521-538] from 1+1 to 2+1 dimensions. Two approximations of this basic algorithm are then introduced, one of which treats adatoms in a more efficient manner, while the other makes use of an approximation of the change in elastic energy in terms of local elastic energy density. In both cases, it is demonstrated that a reasonable level of fidelity is achieved. Results are presented showing how the film morphology is affected by misfit and deposition rate. In addition, simulations of stacked quantum dots are also presented.

We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation. The solver uses a level set framework to represent sharp, complex interfaces in a simple and robust manner. It also uses non-graded, adaptive octree grids which, in comparison to uniform grids, drastically decrease memory usage and runtime without sacrificing accuracy. The basic solver was introduced in earlier works [16,27], and here is extended to address biomolecular systems. First, a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained; this allows to accurately represent the location of the molecule’s surface. Next, a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface. Since the interface is implicitly represented by a level set function, imposing the jump boundary conditions is straightforward and efficient.

This study makes the first attempt to accelerate the singular boundary method (SBM) by the precorrected-FFT (PFFT) for large-scale three-dimensional potential problems. The SBM with the GMRES solver requires computational complexity, where N is the number of the unknowns. To speed up the SBM, the PFFT is employed to accelerate the SBM matrix-vector multiplication at each iteration step of the GMRES. Consequently, the computational complexity can be reduced to . Several numerical examples are presented to validate the developed PFFT accelerated SBM (PFFT-SBM) scheme, and the results are compared with those of the SBM without the PFFT and the analytical solutions. It is clearly found that the present PFFT-SBM is very efficient and suitable for 3D large-scale potential problems.