International Journal of Computational Methods

We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.

Laminar three-fluid stratified flow which involves two different moving interfaces is numerically investigated in a two-dimensional domain in this paper. These interfaces are captured using the level-set method via two level-set functions. The effects of various parameters including Froude number Fr and Weber number We as well as the initial locations of the two interfaces on the evolution of the two interfaces are investigated. It is found that the decrease of We number increases the entry length. For a given volumetric flow rate ratio, the interfacial location at fully developed flow is identical irrespective of the Froude and Weber numbers as well as the initial interfacial location at the inlet. The interfacial locations for fully developed flow show distinct behaviors under different flow rate ratios and viscosity ratios. Increase of volumetric flow rate and viscosity for any one of the fluids increases the pressure drop in the channel. The study of pressure gradient reduction factor (PGRF) shows that it is possible to achieve pressure gradient reduction by introducing less viscous fluids in the transportation of a more viscous fluid.

Two-dimensional (2D) and axisymmetric minimum length nozzles (MLN) with constant and variable specific heat were strictly designed using the method of characteristics. Requirements for the exit Mach number and flow field uniformity were proposed for the nozzles design. In solutions to kernel zone flow field reported previously, violent vibrations of upper wall discrete points at the inlet were observed. Meanwhile, slight compressions could be observed in the flow field of axisymmetric nozzles designed by those methods. In this study, we proposed a novel technique in which the inlet grid is intensified to overcome the limitations mentioned above. Additionally, methods based on conservation of mass and eliminating wave theory were proposed to determine the contour of the nozzle’s upper wall. Inviscid numerical simulations by CFD revealed that the proposed nozzle could meet the requirements for exit Mach number and flow field uniformity in various situations, and axisymmetric nozzles designed from eliminating wave theory exhibited better flow field compared with those designed from conservation of mass.

In this study, the Lattice Boltzmann Method (LBM) is implemented through a finite-volume approach to perform 2D, incompressible, and time-accurate fluid flow analyses on structured grids. Compared to the standard LBM (the so-called stream and collide scheme), the finite-volume approach followed in this study necessitates more computational effort, but the major limitations of the former on grid uniformity and Courant–Friedrichs–Lewy (CFL) number that is to be one are removed. Even though these improvements pave the way for the possibility of solving more practical fluid flow problems with the LBM, time-accurate simulations are still restricted due to the stability criteria dictated by high-aspect ratio grid cells that are usually required for adequate resolution of boundary layers and the stiffness due to the nature of the equation that are being solved. To overcome this limitation, a Dual Time Stepping (DTS) scheme, which iterates the solution in pseudo time using an Implicit-Explicit (IMEX) Runge–Kutta method while advancing the solution in physical time with an explicit scheme (backward difference formula), is developed and implemented. The accuracy of the resulting flow solver is evaluated using benchmark flow problems and overall second-order accuracy is demonstrated.

This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to solve nonclassical parabolic initial-boundary value problems. Accordingly, the inverse problem is reduced into a system of ordinary differential equations (ODEs) and afterwards, such system can be solved numerically using implicit Runge–Kutta (IRK) method of order four. Four examples are introduced to demonstrate the applicability, validity, effectiveness and stable approximations of the present method. Numerical results show the exponential convergence property and error characteristics of presented method.

In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11 species.

This paper presents a search model that finds $n$-dimensional randomly moving target in which any information of the target position is not available to the searchers. There exist $n$-searchers, starting the searching process from the origin. Rather than finding the conditions that make the expected value of the first meeting time between one of the searchers and the target is finite, this work shows the existence of the fuzzy optimal search plan that minimizes the expected value of the first meeting time. The effectiveness of this method is illustrated by using an example with numerical results.

In this paper, we develop the operational approach to the Tau method to solve delay integro-differential equations (DIDEs). The differential and integral parts appearing in the equations are replaced by their operational Tau matrix representations. Some numerical results are given to demonstrate the superior performance of the method.

This paper presents an algorithm for structural topology optimization involving linear buckling. In this algorithm, finite element analysis (FEA) is conducted only in a domain with solid and gray elements, eliminating the contribution of low density elements; and the response function is constructed in the full design domain accounting the contribution of removed low density elements. The errors induced by removing void elements in FEA on eigenvalue and eigenvectors are analyzed. By introducing a dynamic low bound of the first eigenvalue and a load-path coefficient, the algorithm allows converged, nondisjointed and accurate solutions for topology optimization of structures involving buckling. Numerical results are presented for plate and column-beam structures against linear buckling to illustrate the efficiency and effectiveness of the present algorithm. Buckling experiments of the plates manufactured from the obtained topologies further verify the present algorithm.

This paper deals with a mathematical problem statement of unsteady fluid flow in heterogeneous porous media with various hydrogeological characteristics. In order to solve the problem the authors developed an approximate analytical method. The method allows multidimensional differential equation in partial derivatives describing the process of fluid filtration in layered porous media to become one-dimensional differential equation. There was obtained a calculation formula for determining the process of mass transfer between the layers, depending on the operating modes of production wells and hydrogeological parameters of filtration layers. The model and the algorithm can be used as a mathematical support for the processes of development of oil and gas fields.

The smoothed finite element method (S-FEM) proposed recently is more accurate and has higher convergence rate compared with standard four-node isoparametric finite element method (FEM). In this work, a modified S-FEM using four-node quadrilateral elements is proposed, which greatly reduces further the computation cost while maintaining the high accuracy and convergence rate. The key idea of the proposed modification is that the strain of the element is a weighted average value of the smoothed strains in the smoothing cells (SCs), which means that only one integration point is required to construct the stiffness matrix, similar to the single cell S-FEM. A stabilization item is proposed using the differences of the smoothed strains obtained in four SCs, which installs the stability of algorithm and increases the accuracy. To verify the efficiency, accuracy and stability of the present formulation, a number of numerical examples of static and free vibration problems, are studied in comparison with different existing numerical methods.

Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) are additional processor instruction sets available in contemporary personal computers, designed for vectorized floating point calculations. Unfortunately, in order to utilize the advantages of these instructions, one cannot rely on automatic options of high level language compilers. Instead, handwritten assembly language or intrinsic function call insertions are necessary. By using this idea an accelerated C$++$ code is devised, for solving (quasi-) block-tridiagonal linear algebraic equation systems by means of an extended Thomas algorithm. Speedups reaching 3.5 and 3 (relative to C$++$ without using SSE/AVX) are demonstrated for single and double precision calculations, respectively.