Search

Generalized isogeometric analysis (GIGA) enables accurate numerical solutions for interface problem with optimal convergence rates. However, the approach is not stable in terms of the spline functions' degree and is not robust with the position of the interface. In this paper, we introduce a stable GIGA (SGIGA) through a specific modification of the enrichment space. We prove that SGIGA can retain optimal convergence rates, and provide necessary well-conditioning for arbitrary degree and arbitrary location of the interface. We validate the theoretical results in various illustrations and compare the method with the stable generalized finite element method and GIGA. [ABSTRACT FROM AUTHOR]

• The improved HOMM is introduce dintoreliability analysis by the combination of Latin hypercube sampling and the Maximum Entropy Model. The error in failure probability is controlled by iteratively increasing the number of samples. • The time-dependent seismic reliability analysis frame work of structural level is recommended based on the improved HOMM. It directly targets the objective reliability index of the global limit states of the structure in the life cycle. • For the cold-formed steel structures subjected to prolonged exposure to the corrosive environment, the investigation indicates that the differences between time-dependent and time-independent seismic reliability indexes are substantial. Cold-formed steel structures are particularly sensitive to seismic loads due to their relatively low ductility and limited energy dissipation capacity. Moreover, the small thickness of cold-formed steel structures makes them more susceptible to environmental influences, leading to corrosion and material degradation. Therefore, it is essential to quantitatively assess the seismic reliability of cold-formed steel structures from a lifecycle perspective. Current structural reliability design standards often focus on the ultimate strength at the component level while neglecting the global safety of structures. Additionally, it tends to neglect the time-variant characteristic of structural resistance due to the corrosion and degradation of materials. Therefore, the time-dependent seismic reliability analysis framework was proposed at the global structural level based on an improved high-order moment method (HOMM). It directly addresses the objective failure probabilities at the global structural level and takes into account the resistance degradation of structures in the design reference period. A comparative analysis of the time-dependent seismic reliability between rocking cold-formed steel frames and rigid cold-formed steel frames is conducted, focusing on load-bearing capacity limit states and deformation capacity limit states. The investigation indicates that the differences between time-dependent and time-independent seismic reliability indexes are substantial. For cold-formed steel structures subjected to prolonged exposure to the corrosive environment, it is recommended to consider the time-dependent degradation of structures. [Display omitted] [ABSTRACT FROM AUTHOR]

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important. [ABSTRACT FROM AUTHOR]

The Double Absorbing Boundary (DAB) is a recently proposed absorbing layer used to truncate an unbounded domain with high-order accuracy. While it was originally designed for time-dependent acoustics and elastodynamics, here the DAB construction is adapted and applied to the 2D Helmholtz equation. Both wave-guide and corner configurations are considered. A high-order spectral finite element scheme is used in order to match the discretization accuracy to the accuracy of the DAB. The DAB scheme is analyzed, and numerical experiments demonstrate its performance. [ABSTRACT FROM AUTHOR]

In this paper, we develop second and third order accurate positivity-preserving (PP) nodal discontinuous Galerkin (DG) methods for one and two dimensional anisotropic diffusion problems. The key idea is to first represent the cell average of its numerical approximation as a weighted summation of Gaussian quadrature point values used in the updating of nodal DG methods, and then transform these Gaussian quadrature point values to some other special chosen point values. We prove that by taking parameters in the definition of nodal DG methods appropriately, together with a suitable time stability condition, the cell averages can be kept positive. A polynomial scaling limiter is then applied to obtain positive numerical approximations on the whole cell without sacrificing accuracy. Stability analysis without the PP limiter is also rigorously established. Numerical experiments are performed to demonstrate desired orders of accuracy, PP and good performances of our proposed approach. [ABSTRACT FROM AUTHOR]

We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementation on Graphics Processing Units (GPUs) and provide comparisons to existing GPU-accelerated implementations of high-order DG methods on quadrilateral meshes. [ABSTRACT FROM AUTHOR]

High order (HO) schemes are attractive candidates for the numerical solution of multiscale problems occurring in fluid dynamics and related disciplines. Among the HO discretization variants, discontinuous Galerkin schemes offer a collection of advantageous features which have lead to a strong increase in interest in them and related formulations in the last decade. The methods have matured sufficiently to be of practical use for a range of problems, for example in direct numerical and large eddy simulation of turbulence. However, in order to take full advantage of the potential benefits of these methods, all steps in the simulation chain must be designed and executed with HO in mind. Especially in this area, many commercially available closed-source solutions fall short. In this work, we therefore present the FLEXI framework, a HO consistent, open-source simulation tool chain for solving the compressible Navier–Stokes equations on CPU clusters. We describe the numerical algorithms and implementation details and give an overview of the features and capabilities of all parts of the framework. Beyond these technical details, we also discuss the important but often overlooked issues of code stability, reproducibility and user-friendliness. The benefits gained by developing an open-source framework are discussed, with a particular focus on usability for the open-source community. We close with sample applications that demonstrate the wide range of use cases and the expandability of FLEXI and an overview of current and future developments. [ABSTRACT FROM AUTHOR]

We show that applying any deterministic B-series method of order p d with a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order ⌊ p d / 2 ⌋. As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips. Existing numerical methods for such PDEs suffer from the constraint of this low regularity and fail to deliver an optimal high-order rate of convergence. We approach this problem by (i) choosing an artificial interface to enclose the center of low regularity and (ii) representing the solution in the interior of artificial interface as unknown linear combination of known low-regularity modes of the solution. This results in an interface formulation of the original PDE, and the linear combination is represented in the interface conditions. By enforcing the smooth component of the numerical solution in the interior domain to be approximately zero, a least squares problem is obtained for the unknown coefficients. The solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the PDE in the exterior domain. The potential of our interface formulation is favorably demonstrated by numerical experiments on one- and two-dimensional Poisson equations with low-regularity solutions. High-order numerical solutions of unknown coefficients and PDEs are obtained. This proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems. We address the relations between our interface formulation and domain decomposition methods, as well as a regularization strategy for the Poisson–Boltzmann equation with singular charge density. [ABSTRACT FROM AUTHOR]

Abstract Traditional clustering methods, such as the well-known fuzzy c-means algorithm, usually need sufficient data to find a good clustering partition, and these algorithms are ineffective in dealing with the cases with insufficient datasets. Insufficient datasets and noise data can easily interfere with the clustering results, which is difficult to solve the non-commensurability problem for the multi-type target clustering quality in the spatial region, and it is complex to solve the high-dimensional heterogeneous data of swarm intelligence sets by using the existing algorithms. In order to overcome the above problems in this paper, the swarm similarity is proposed to extend the multi-dimensional similar spatial region and use the optimal value of iterative clustering function as the clustering quality metric standard firstly. Furthermore, a fuzzy high-order hybrid clustering (F-HOHC-SIS) algorithm is proposed, which can effectively control the convergence rate and decrease the calculating time while improving the anti-interference ability. The proposed algorithm makes the high-dimensional heterogeneous data clustering objective function achieve the optimal value within finite iteration numbers. Some real datasets are used to analyze the quality of clustering. Simulation experimental results and comparisons actually demonstrate these good characteristics of F-HOHC-SIS with its effectiveness in practice. [ABSTRACT FROM AUTHOR]

We study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis. Such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh, which allows for large computational savings during the solution of the resulting linear system. The continuity reduction results in extra degrees of freedom that modify the approximation properties of the method. The convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis. We show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra, such as stopping bands and outliers, and present a unified description of these effects in finite element method, isogeometric analysis, and refined isogeometric analysis. Accuracy of the refined isogeometric analysis approximations can be improved by using non-standard quadrature rules. In particular, optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence, as it occurs in standard isogeometric analysis. [ABSTRACT FROM AUTHOR]

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces results in less stringent CFL restrictions than equivalent C 0 or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L 2 n -widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

The effects of leakage delay on the dynamics of neural networks with integer-order have lately been received considerable attention. It has been confirmed that fractional neural networks more appropriately uncover the dynamical properties of neural networks, but the results of fractional neural networks with leakage delay are relatively few. This paper primarily concentrates on the issue of bifurcation for high-order fractional bidirectional associative memory(BAM) neural networks involving leakage delay. The first attempt is made to tackle the stability and bifurcation of high-order fractional BAM neural networks with time delay in leakage terms in this paper. The conditions for the appearance of bifurcation for the proposed systems with leakage delay are firstly established by adopting time delay as a bifurcation parameter. Then, the bifurcation criteria of such system without leakage delay are successfully acquired. Comparative analysis wondrously detects that the stability performance of the proposed high-order fractional neural networks is critically weakened by leakage delay, they cannot be overlooked. Numerical examples are ultimately exhibited to attest the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this study, an adjoint-based hp-adaptation algorithm has been developed within a Petrov–Galerkin finite-element method. The developed mesh adaptation algorithm is able to perform non-conformal mesh adaptations. To account for hanging nodes in a consistent manner, the constrained approximation method has been utilized. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation. The methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier–Stokes (RANS) equations, equipped with negative variant of Spalart–Allmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. Fully implicit linearization has been used to advance the solution toward a steady-state. Dirichlet boundary conditions have been imposed weakly and the functional outputs have been modified according to the weak boundary conditions in order to provide a smooth adjoint solution near the boundaries. To accelerate the error reduction in presence of singularity points, an enhanced h-refinement, based on solution’s smoothness, has been used. Numerical results illustrate consistent accuracy improvement of the functional outputs for both h- and hp-adaptation, and also capability enhancement in capturing complex viscous effects such as shock-wave/turbulent boundary layer interaction. [ABSTRACT FROM AUTHOR]

In this paper, we combine high order local maximum-entropy schemes (HOLMES) with the integration framework developed in the NURBS-enhanced finite element method (NEFEM). We focus on the two-dimensional case where, given a domain described by some NURBS curves, a meshless formulation based on the HOLMES approximants is employed for the discretization and, at the same time, the geometric fidelity given by the NURBS boundary is preserved thanks to the NEFEM integration. Since HOLMES basis functions are not interpolatory on the boundary, different techniques are considered for the imposition of essential boundary conditions. The efficiency and the accuracy of the proposed methodology are confirmed with supportive numerical examples. [ABSTRACT FROM AUTHOR]

We present a high order time-domain nodal discontinuous Galerkin method for wave problems on hybrid meshes consisting of both wedge and tetrahedral elements. We allow for vertically mapped wedges which can be deformed along the extruded coordinate, and present a simple method for producing quasi-uniform wedge meshes for layered domains. We show that standard mass lumping techniques result in a loss of energy stability on meshes of vertically mapped wedges, and propose an alternative which is both energy stable and efficient. High order convergence is demonstrated, and comparisons are made with existing low-storage methods on wedges. Finally, the computational performance of the method on Graphics Processing Units is evaluated. [ABSTRACT FROM AUTHOR]

We study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P 1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H (d i v) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H 1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature. • A hybrid numerical approach for the Green–Naghdi equations on unstructured meshes. • Compatible data representation, in both phases, is critical for the consistency. • The numerical fluxes used are affecting the well-posedness of the solution. • A non-consistent reconstruction of the derivatives affects the accuracy of the model. [ABSTRACT FROM AUTHOR]

Summary CCCTC-binding factor (CTCF) is an organizer of higher-order chromatin structure and regulates gene expression. Genetic studies have implicated mutations in CTCF in intellectual disabilities. However, the role of CTCF-mediated chromatin structure in learning and memory is unclear. We show that depletion of CTCF in postmitotic neurons, or depletion in the hippocampus of adult mice through viral-mediated knockout, induces deficits in learning and memory. These deficits in learning and memory at the beginning of adulthood are correlated with impaired long-term potentiation and reduced spine density, with no changes in basal synaptic transmission and dendritic morphogenesis and arborization. Cognitive disabilities are associated with downregulation of cadherin and learning-related genes. In addition, CTCF knockdown attenuates fear-conditioning-induced hippocampal gene expression of key learning genes and loss of long-range interactions at the BDNF and Arc loci. This study thus suggests that CTCF-dependent gene expression regulation and genomic organization are regulators of learning and memory. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L 2 -gradient flow. We analyze two energy functionals: the area, which leads to the mean curvature flow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore flow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch tensor product NURBS and discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method. To approximate the high order geometric PDEs we use high order continuous NURBS basis functions. For the time discretization of the nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) with extrapolation of the geometric quantities involved in the weak formulation of the problem; in this manner, we solve a linear problem at each time step. We report numerical results concerning the mean curvature and Willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme. [ABSTRACT FROM AUTHOR]

Applying Higher Order Campbelling methods in neutron flux monitoring with fission chambers is advantageous due to their capabilities to suppress the impact of unwanted noises and signal contributions (such as gamma radiation). This work aims to verify through experimental results that the basic assumptions behind the Higher Order Campelling methods are valid in critical reactors. The experiments, reported in this work, were performed at the MINERVE reactor in Cadarache. It is shown that the calibration of a fission chamber and the associated electronic system is possible in higher order mode. With the use of unbiased cumulant estimators and with digital processing, it is shown that over a wide count rate range, accurate count rate estimation can be achieved based on signal samples of a few ms , which is a significant progress compared to similar experimental results in the literature. The difference between the count rate estimated by pulse counting and by the Higher Order Campelling is less than 4%. The work also investigates the possibility of monitoring transient events. For this purpose, a control rod drop event was followed in Higher Order Campbelling mode. [ABSTRACT FROM AUTHOR]

This paper investigates, through numerical simulations, the performance of a signal analysis method by which a high temperature fission chamber can be used over a wide range of count rates. Results reported in a previous paper (Elter et al., 2015 [1] ) indicated that the traditional Campbell method and the pulse mode cannot provide a sufficient overlap at medium count rates. Hence the use of the so-called Higher Order Campbell (HOC) methods is proposed and their performance is investigated. It is shown that the HOC methods can guarantee the linearity (i.e. correctness) of the neutron flux estimation over a wide count rate, even during transient conditions. The capabilities of these methods for suppressing parasitic noise (originating from various sources) are verified. [ABSTRACT FROM AUTHOR]

In this paper, an effective method for substrate integrated waveguide (SIW) filter with improved stop band performance is proposed. By configuring the position and the direction of the coupling window, the transmission of the parasitic mode such as TE 102 and TE 201 can be effectively suppressed without adding any extra structure and element. Under the above theory, a Ka-band SIW filter centered at 35.71 GHz based on quartz glass substrate with compact size and simple structure is designed and tested. The stop rejection of the design is increased from 1.03 times to 1.95 times the center frequency within −35 dB of the filter with the insertion loss of −2.5 dB and the return loss of −20 dB. Finally, the measurement results validated the proposed design. [ABSTRACT FROM AUTHOR]

Legendre spectral finite elements (LSFEs) are examined in their application to Reissner–Mindlin composite plates for static and dynamic deformation on unstructured grids. LSFEs are high-order Lagrangian-interpolant finite elements whose nodes are located at the Gauss–Lobatto–Legendre quadrature points. Nodal quadrature is employed for mass-matrix calculations, which yields diagonal mass matrices. Full quadrature or mixed-reduced quadrature is used for stiffness-matrix calculations. Solution accuracy is examined in terms of model size, computation time, and memory storage for LSFEs and for quadratic serendipity elements calculated in a commercial finite-element code. Linear systems for both model types were solved with the same sparse-system direct solver. At their best, LSFEs provide many orders of magnitude more accuracy than the quadratic elements for a fixed measure (e.g., computation time). At their worst, LSFEs provide the same accuracy as the quadratic elements for a given measure. The LSFEs were insensitive to shear locking and were shown to be more robust in the thin-plate limit than their low-order counterparts. [ABSTRACT FROM AUTHOR]

In this note, a general scheme is proposed for the high order unstable delay process with one or more positive poles, using the mirror mapping technique. The Nyquist criteria is employed to establish a systematic methodology to tune the parameter. The stabilizing parameter region could guarantee the prespecified robustness specification. In the scheme, a control law is designed based on the all-pole Padé approximated model. The unstable process was first mapped into a minimum-phase system, and the actual control is obtained by the closed-loop gain shaping algorithm (CGSA). The advantages are that one has a concise design procedure and can achieve good performance such as disturbance rejection and robustness. Finally, three highly cited examples are used to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Linearly implicit Runge–Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction–diffusion systems. However, the use of approximate factorization usually leads to loss of accuracy, which makes it attractive only for low order time integration schemes. This paper discusses the application of approximate matrix factorization with high order methods; an inexpensive correction procedure applied to each stage allows to retain the high order of the underlying linearly implicit Runge–Kutta scheme. The accuracy and stability of the methods are studied. Numerical experiments on reaction–diffusion type problems of different sizes and with different degrees of stiffness illustrate the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method based on mixed finite element, discontinuous Galerkin methods in space and high order Runge–Kutta method in time for solving the miscible displacement problem. No slope limiters are needed. The proposed method exhibits high order of convergence in space and time when comparing with analytical solutions. The simulation shows robustness of the method for heterogeneous media with highly varying permeabilities. [ABSTRACT FROM AUTHOR]

A high order method is formulated for solving the compressible black-oil problem with capillary effects in one-dimensional domains. The sequential implicit formulation combined with the interior penalty discontinuous Galerkin method is employed. The contribution of this work is to show convergence and robustness of high order polynomial approximations (up to fourth order) for saturated and undersaturated reservoirs. [ABSTRACT FROM AUTHOR]

Polynomial trace inequalities typically involve an unknown constant, depending on the order N of the polynomial. The dependence of these constants on N was made more explicit in Warburton and Hesthaven (2003) for the general d -simplex, and they were determined for quadrilaterals, hexes, and wedges in Hillewaert (2013). In this note, we derive explicit expressions for the constant in the trace inequality for two sets of basis functions on the pyramid. [ABSTRACT FROM AUTHOR]

We present a minimum-residual finite element method (based on a dual Petrov–Galerkin formulation) for convection–diffusion problems in a higher order, adaptive, continuous Galerkin setting. The method borrows concepts from both the Discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (2011) and the method of variational stabilization by Cohen, Dahmen, and Welper (2012), and it can also be interpreted as a variational multiscale method in which the fine-scales are defined through a dual-orthogonality condition. A key ingredient in the method is the proper choice of dual norm used to measure the residual, and we present two choices which are observed to be robust in both convection and diffusion-dominated regimes, as well as a proof of stability for quasi-uniform meshes and a method for the weak imposition of boundary conditions. Numerically obtained convergence rates in 2D are reported, and benchmark numerical examples are given to illustrate the behavior of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation. [Copyright &y& Elsevier]

Highlights: [•] Highly ordered copper phthalocyanine (CuPc) thin films were prepared. [•] Dependence of p-6P thickness on the performance of CuPc transistors was studied. [•] High field-effect mobility of 0.18cm2/Vs was obtained by CuPc transistors. [ABSTRACT FROM AUTHOR]

Abstract: A spatially arbitrary high order, semi-implicit spectral discontinuous Galerkin (DG) scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed. The free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis, and as test functions. Each unknown is, however, defined on a different set of control volumes that are spatially staggered with respect to each other. A semi-implicit time integration yields a stable and efficient mass conservative algorithm. The use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block-penta-diagonal system must be solved for the new free surface location, which is a scalar quantity. Subsequently the new momentum components are directly obtained. The staggered semi-implicit approach makes the present DG scheme different from other published DG schemes. The sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix-free GMRES algorithm. Furthermore, for the shallow water equations the proposed scheme can be written as a quadrature-free method, where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors. For the special case the proposed method reduces to a classical semi-implicit finite difference scheme. The proposed semi-implicit scheme is particularly well suited for low Froude number flows. The method is validated on some typical academic benchmark problems, using polynomial degrees of up to N=20. [Copyright &y& Elsevier]

Abstract: We consider a coupled system of singularly perturbed first order ordinary differential equations. The equations have small positive parameters of different magnitudes associated with them. A decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method. We apply the HODIE (High Order Difference approximation with Identity Expansions) technique to construct a second order finite difference scheme and combine this with standard backward Euler difference scheme in a special way on a piecewise-uniform Shishkin mesh to solve the system numerically. It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters. Numerical experiments are conducted to support the theoretical results. [Copyright &y& Elsevier]

Abstract: We study a general class of high order Newton type methods. The schemes consist of the application of several steps of Newton type methods with frozen derivatives. We are interested to improve the order of convergence in each sub-step. In particular, we should finish the computation after some stop criteria and before the full computation of the current approximation. We prove that only two sequences of parameters can be derived verifying these properties. One corresponds to a very well known family and the other is a little (but not natural) modification. Finally, we study some dynamical aspects of these families in order to find differences. Surprisingly, the less natural family seems to have a simpler dynamic. [Copyright &y& Elsevier]

Abstract: In this contribution, a strategy is proposed for uncoupling geometrical description and approximation with the X-FEM. The strategy is based on an uniform coarse mesh that defines a high order approximation of the mechanical fields and an adapted mesh that defines the geometrical features by means of levelsets. The connection between the geometry and the approximation is obtained by sharing the quadtree trees of the two meshes. Numerical examples involving level-set based parts, convergence studies, mechanical computations and numerical homogenization show good promise for this approach. [Copyright &y& Elsevier]

Abstract: Legendre spectral finite elements (LSFEs) are examined through numerical experiments for static and dynamic Reissner–Mindlin plate bending and a mixed-quadrature scheme is proposed. LSFEs are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss–Lobatto–Legendre quadrature points. Solutions on unstructured meshes are examined in terms of accuracy as a function of the number of model nodes and total operations. While nodal-quadrature LSFEs have been shown elsewhere to be free of shear locking on structured grids, locking is demonstrated here on unstructured grids. LSFEs with mixed quadrature are, however, locking free and are significantly more accurate than low-order finite-elements for a given model size or total computation time. [Copyright &y& Elsevier]

Abstract: In a lattice spring model (LSM), the material is discretised into particles linked by springs. However, LSMs always adopt linear springs, which results in a stiff approximation of the corresponding elastic solution. In this work, a high order LSM is proposed to overcome this limitation by introducing additional degrees of freedoms (DOFs) to the particles. Based on the energy minimisation principle and the local strain technique, equations for the stiffness matrices of high order LSM are derived. Relationships between micro spring parameters and macro material constants are derived from the Cauchy-born rules and the hyperelastic theory. Numerical examples show that the high order LSM can provide a better solution than that of the linear LSM and that the LSM is more suitable for modelling singularity and fracture problems. [Copyright &y& Elsevier]

Abstract: One of the most efficient techniques for considering the unbounded media in a mathematical model is the local non-reflecting boundary conditions (NRBCs). A relatively recent approach in this context is the one proposed by Hagstrom and Warburton (H–W boundary condition). This approach is available for scalar wave equation. This equation governs the hydrodynamic pressure distribution inside a reservoir and hence, the H–W boundary condition may be used to solve the corresponding problems. A well-known case in which one has to deal with the dynamic analysis of unbounded reservoir is the dam–reservoir interaction problem. In this study the H–W NRBC has been applied to this problem and its harmonic response is calculated. By comparing the results with the exact solution, the performance and accuracy of this NRBC is examined. The numerical results confirm the very good behavior of the NRBC in the frequencies above the fundamental frequency of the reservoir. However, below this frequency range, this boundary condition does not perform very well, especially when it is applied in close distances from the dam. [Copyright &y& Elsevier]

Abstract: This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction–diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper investigates the problem of global strong stabilization by state feedback, for a family of high order feedforward nonlinear time-delay systems. The uncertain nonlinearities are assumed to satisfy a polynomial growth assumption with an input or delayed input dependent rate. With the help of the appropriate Lyapunov–Krasovskii functionals, and a rescaling transformation with a gain to be tuned online by a dynamic equation, we propose a dynamic low gain state feedback control scheme. A simulation example is given to demonstrate the effectiveness of the proposed design procedure. [Copyright &y& Elsevier]

Abstract: We consider a system of singularly perturbed equations of reaction–diffusion type coupled through the reaction term. A high order Schwarz domain decomposition method is developed to solve the system numerically. The method splits the original domain into three overlapping subdomains. On two boundary layer subdomains we use a compact fourth order difference scheme on a uniform mesh while on the interior subdomain we use a hybrid scheme on a uniform mesh. We prove that the method is almost fourth order -uniformly convergent. Furthermore, we prove that when is small, one iteration is sufficient to get almost fourth order -uniform convergence. Numerical experiments are performed to support the theoretical results. [Copyright &y& Elsevier]