Adaptive meshless refinement schemes for RBF-PUM collocation

In this paper we present an adaptive discretization technique for solving elliptic partial differential equations via a collocation radial basis function partition of unity method. In particular, we propose a new adaptive scheme based on the construction of an error indicator and a refinement algorithm, which used together turn out to be ad-hoc strategies within this framework. The performance of the adaptive meshless refinement scheme is assessed by numerical tests.     

Parallel algorithm for the solutions of PDEs in linux clustered workstations

In this paper we propose parallel algorithm for the solution of partial differential equations over a rectangular domain using the Crank–Nicholson method by cooperation with the DuFort–Frankel method and apply it on a model problem, namely, the heat conduction equation. One of the well known parallel techniques in solving partial differential equations in cluster computing environment is the domain decomposition technique. Using this technique, the whole domain is decomposed into subdomains, each of them has its own boundaries that are called the interface points. Parallelization is realized by approximating interface values using the unconditionally stable DuFort–Frankel explicit scheme, and these values serve as Neumann boundary conditions for the Crank–Nicholson implicit scheme in the subdomains. The numerical results show that our algorithm is more accurate than the algorithm based on the forward explicit method to approximate the values of the interface points, especially, when we use a small number of time steps. Moreover, these numerical results show that increasing the number of processors which are used in the cluster, yields an increase in the algorithm speedup.     

Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical chaotic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in finite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the effectiveness of the proposed control scheme.     

An adaptive integration scheme for heat conduction in additive manufacturing

Solidification dynamics are important for determining final microstructure in additively manufactured parts. Recently, researchers have adopted semi-analytical approaches for predicting heat conduction effects at length and time scales not accessible to complex multi-physics numerical models. The present work focuses on improving a semi-analytical heat conduction model for additive manufacturing by designing an adaptive integration technique. The proposed scheme considers material properties, process conditions, and the inherent physical behavior of the transient heat conduction around both stationary and moving heat sources. Both the adaptive integration scheme and a technique for calculating only the points within the melt pools are described in detail. The full algorithm is then implemented and compared against a simple Riemann sum integration scheme for a variety of cases that highlight process and material variations relevant to additive manufacturing. The new scheme is shown to have significant improvements in computational efficiency, solution accuracy, and usability.     

奇异摄动反应扩散方程的后验误差估计及自适应算法

研究了一类奇异摄动半线性反应扩散方程的自适应网格方法.在任意非均匀网格上建立迎风有限差分离散格式,并推导出离散格式的后验误差界,然后用该误差界设计自适应网格移动算法.数值实验结果证明了所提出的自适应网格方法的有效性.     

Impulsive synchronization of time-varying dynamical network

Synchronization of time-varying dynamical network is investigated via impulsive control. Based on the Lyapunov function method and stability theory of impulsive differential equation, a synchronization criterion with respect to the system parameters and the impulsive gains and intervals is analytically derived. Further, an adaptive strategy is introduced for designing unified impulsive controllers, with a corresponding synchronization criterion derived. In this proposed adaptive control scheme, the impulsive instants adjust themselves to the needed values as time goes on, and an algorithm for determining the impulsive instants is provided and evaluated. The derived theoretical results are illustrated to be effective by several numerical examples.     

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.     

Fast computation of adaptive wavelet expansions

In this paper we describe and analyze an algorithm for the fast computation of sparse wavelet coefficient arrays typically arising in adaptive wavelet solvers. The scheme improves on an earlier version from Dahmen et al. (Numer. Math. 86, 49–101, 2000) in several respects motivated by recent developments of adaptive wavelet schemes. The new structure of the scheme is shown to enhance its performance while a completely different approach to the error analysis accommodates the needs put forward by the above mentioned context of adaptive solvers. The results are illustrated by numerical experiments for one and two dimensional examples.     

Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e., $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.     

AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

Optimization via Chebyshev polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices. In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

An adaptive refinement method for convection‐diffusion problems

An adaptive finite difference method for singularly perturbed convection‐diffusion problems is presented. The method is introduced using a first‐order upwind scheme and a suitable error estimator based on the first derivatives. To obtain the grid structure needed for the cross stencil a special refinement strategy is considered. To avoid the slave points we change the stencil at the interface points from a cross to a skew one. After the convergence of the refinement algorithm we use a combination of a first order upwind and a second order central schemes to achieve higher order of convergence. Several numerical examples show the efficiency of our treatment. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

多重网格法和自适应算法的组合及其对非线性Schrdinger方程的应用

在解非线性的进化型偏微分方程时,为了数值计算的稳定性常常采用无条件稳定的隐式差分格式.这样会引起两个问题:一是要解线性甚至非线性的代数方程组,这是费时间的;另一是在解代数方程组时,迭代法的收敛性依赖于时间步长,特别是非线性迭代的收敛性会对时间步长加以严格的限制.     

On the Properties of Conservative Finite Volume Scheme for the Two-Phase Stefan Problem

Differential Equations - We study the properties of a finite volume scheme for the two-phase Stefan problem. The numerical algorithm based on the explicit interface tracking is considered. The...     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation

In this paper, a Fourier spectral method with an adaptive time step strategy is proposed to solve the fractional nonlinear Schrödinger (FNLS) equation with periodic initial value problem. First, we prove the conservation law of the mass and the energy for the semi-discrete Fourier spectral scheme. Second, the error estimation of the semi-discrete scheme is given in the relevant fractional Sobolev space. Then, an adaptive time-step strategy is designed to reduce central processing unit (CPU) time. Finally, the numerical experiments for the one-, two- and three-dimensional FNLSs, show that the adaptive strategy, compared to the constant time step, can reduce the CPU-time by almost half.     

The truncated Stieltjes moment problem solved by using kernel density functions

In this work the problem of the approximate numerical determination of a semi-infinite supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The target space is carefully defined and an approximation theorem is proved, establishing that the set of all convex superpositions of appropriate Kernel Density Functions (KDFs) is dense in this space. A solution algorithm is provided, based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics. The solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution. A parsimonious formulation of the proposed solution procedure, based on a novel sequentially adaptive scheme is developed, enabling a very efficient moment data inversion. The whole methodology is fully illustrated by numerical examples.     

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.