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.

Based on the study of two commonly used stochastic elliptic models: I: − ∇· (a(x,w)·∇u(x,w)) = f (x) and II: − ∇·(a(x,w)◊∇u(x,w)) = f (x), we constructed a new stochastic elliptic model III: −∇ ◊ ((a−1)·(−1)◊∇u(x,w)) = f (x), in. The difference between models I and II is twofold: a scaling factor induced by the way of applying the Wick product and the regularization induced by the Wick product itself. In, we showed that model III has the same scaling factor as model I. In this paper we present a detailed discussion about the difference between models I and III with respect to the two characteristic parameters of the random coefficient, i.e., the standard deviation σ and the correlation length lc. Numerical results are presented for both one- and two-dimensional cases.

We propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

A biphase mixture continuum mechanics model is derived for neutral heat-shrinkable thermo-sensitive hydrogels in this paper. The mixing free energy of the special mixture is recalculated based on the partition function of Bose system, and it evaluates the contribution of the hydrophilic, hydrophobic interaction and hydrogen bonding to the volume phase transition behaviors. The ideas of the Flory lattice theory and the UNIFAC group contribution method are employed to get the expression of the mixing free energy. Then we deduce a particular model by combining this mixing free energy with the conservation laws equations and constitutive relations of both phases to predict the volume transition behaviors of these special hydrogels.

An algebraic multilevel method is proposed for efficiently simulating linear wave propagation using higher-order numerical schemes. This method is used in conjunction with the Finite Volume Time Domain (FVTD) technique for the numerical solution of the time-domain Maxwell’s equations in electromagnetic scattering problems. In the multilevel method the solution is cycled through spatial operators of varying orders of accuracy, while maintaining highest-order accuracy at coarser approximation levels through the use of the relative truncation error as a forcing function. Higher-order spatial accuracy can be enforced using the multilevel method at a fraction of the computational cost incurred in a conventional higher-order implementation. The multilevel method is targeted towards electromagnetic scattering problems at large electrical sizes which usually require long simulation times due to the use of very fine meshes dictated by point-per-wavelength requirements to accurately model wave propagation over long distances.

We introduce and study a parallel domain decomposition algorithm for the simulation of blood flow in compliant arteries using a fully-coupled system of nonlinear partial differential equations consisting of a linear elasticity equation and the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition plays an interesting role in the accuracy of the blood flow simulation and we provide a numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition method for solving the fully coupled nonlinear system on a supercomputer with a few hundred processors.

In this paper we consider an anisotropic convection-diffusion (ACD) filter for image denoising and compression simultaneously. The ACD filter is discretized by a tailored finite point method (TFPM), which can tailor some particular properties of the image in an irregular grid structure. A quadtree structure is implemented for the storage in multi-levels for the compression. We compare the performance of the proposed scheme with several well-known filters. The numerical results show that the proposed method is effective for removing a mixture of white Gaussian and salt-and-pepper noises.

The “analogue gravity formalism”, an interdisciplinary theoretical scheme developed in the past for studying several non relativistic classical and quantum systems through effective relativistic curved space-times, is here applied to largely de-formable elastic bodies described by the nonlinear theory of solid mechanics. Assuming the simplest nonlinear constitutive relation for the elastic material given by a Kirchhoff-St Venant strain-energy density function, it is possible to write for the perturbations an effective space-time metric if the deformation is purely longitudinal and depends on one spatial coordinate only. Theoretical and numerical studies of the corresponding dynamics are performed in selected cases and physical implications of the results obtained are finally discussed.

Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate themembrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

First introduced in, the lumped particle framework is a flexible and numerically efficient framework for the modelling of particle transport in fluid flow. In this paper, the framework is expanded to simulate multicomponent particle-laden fluid flow. This is accomplished by introducing simulation protocols to model particles over a wide range of length and time scales. Consequently, we present a time ordering scheme and an approximate approach for accelerating the computation of evolution of different particle constituents with large differences in physical scales. We apply the extended framework on the temporal evolution of three particle constituents in sandladen flow, and horizontal release of spherical particles. Furthermore, we evaluate the numerical error of the lumped particle model. In this context, we discuss the Velocity-Verlet numerical scheme, and show how to apply this to solving Newton’s equations within the framework. We show that the increased accuracy of the Velocity-Verlet scheme is not lost when applied to the lumped particle framework.

Electrophoresis of a cylinder suspended in a cylindrical tube is analytically studied in the limit of thin electric double layer approximation. The electric and fluid flow fields within the annulus, and the cylinder velocities are analytically obtained in bipolar coordinates. The results are analyzed with various values of dimensionless parameters: eccentricity, cylinder-to-tube radius ratio and tube-to-cylinder zeta potential ratio (i.e., tube-to-cylinder velocity scale ratio). The analysis shows that microvortices are generated within the annulus. By changing the parameters, different flow patterns can be created, which shows potential for mixing enhancement in micro/nanofluidics. Moreover, the cylinder not only translates but also rotates when the cylinder and tube are eccentric. The cylinder rotation might be utilized as a micromotor or an electric field detector. The cylinder trajectories show that the cylinder may approach the tube wall or rest within the tube depending on the zeta potential ratio.

The dynamic behaviors of continuous and discrete flows in superhydrophobic microchannels are investigated with a lattice Boltzmann model. Typical characters of the superhydrophobic phenomenon are well observed from our simulations, including air trapped in the surface microstructures, high contact angles, low contact angle hysteresis, and reduced friction to fluid motions. Increasing the roughness of a hydrophobic surface can produce a large flow rate through the channel due to the trapped air, implying less friction or large apparent slip. The apparent slip length appears to be independent to the channel width and could be considered as a surface property. For a moving droplet, its behavior is affected by the surface roughness from two aspects: the contact angle difference between its two ends and the surface-liquid interfacial friction. As a consequence, the resulting droplet velocity changes with the surface roughness as firstly decreasing and then increasing. Simulation results are also compared with experimental observations and better agreement has been obtained than that from other numerical method. The information from this study could be valuable for microfluidic systems.

Homogeneous isotropic turbulence has been playing a key role in the research of turbulence theory. And the pseudo-spectral method is the most popular numerical method to simulate this type of flow fields in a periodic box, where fast Fourier transform (FFT) is mostly effective. However, the bottle-neck in this method is the memory of computer, which motivates us to construct a memory-saving algorithm for spectral method in present paper. Inevitably, more times of FFT are needed as compensation. In the most memory-saving situation, only 6 three-dimension arrays are employed in the code. The cost of computation is increased by a factor of 4, and that 38 FFTs are needed per time step instead of the previous 9 FFTs. A simulation of isotropic turbulence on 20483 grid can be implemented on a 256G distributed memory clusters through this method.

In this contribution, we introduce numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrödinger equation. We extend previous work on the subject [1] to systems of coupled equations.

Bound and resonant states of the Schrödinger equation can be determined through the poles of the S-matrix, a quantity that can be derived from the asymptotic form of the wave function. We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties, thus making it amenable to numerical continuation. This allows us to automate the process of tracking bound and resonant states when parameters in the Schrödinger equation are varied. We have applied this approach to a number of model problems with satisfying results.

In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in ℝ 3 (spheres, cubes, and cylinders). The quadrature nodes are made of the tensor-product of 1-D Gauss nodes on [–1,1] for non-periodic variables or uniform nodes on [0,2π] or [0,π] for periodic ones. The quadrature weights are converted from a brute-force integration of the hypersingular function through interpolating the smooth component of the integrand. Numerical results are presented to validate the accuracy and efficiency of computing hypersingular integrals, as in the computations of Cauchy principal values, with a minimum number of quadrature nodes. The pre-calculated quadrature tables can be then readily used to implement Nyström collocation methods of hypersingular volume integral equations such as the one for Maxwell equations.

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.

We study a novel class of numerical integrators, the adapted nested force-gradient schemes, used within the molecular dynamics step of the Hybrid Monte Carlo (HMC) algorithm. We test these methods in the Schwinger model on the lattice, a well known benchmark problem. We derive the analytical basis of nested force-gradient type methods and demonstrate the advantage of the proposed approach, namely reduced computational costs compared with other numerical integration schemes in HMC.

The present work demonstrates the extension of a thermal-hydraulic model, THRUST, with an objective to simulate the fast transient flow dynamics in a supercritical water channel of circular cross section. THRUST is a 1-D model which solves the nonlinearly coupled mass, axial momentum and energy conservation equations in time domain based on a characteristics-dependent fully implicit finite difference scheme using an Eulerian approach. The model developed accounts for the compressibility of the supercritical flow by considering the finite value of acoustic speed in the solution algorithm and treats the boundary conditions naturally. A supercritical water channel of circular cross section, for which the experimental data is available at steady state operating conditions, is chosen for the transient simulations to start with. Two different case studies are undertaken with a purpose to assess the capability of the model to analyze the fast transient processes caused by the large reduction in system pressure. The first transient case study is where the initial exit pressure is reduced by 1 MPa exponentially in a time span of 5 s. In the second case study, the transient is initiated with a sudden step decrease in the exit pressure by the same amount. Results obtained for both the case studies show the desired performance from the model developed.