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In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters. [ABSTRACT FROM AUTHOR]

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method. [ABSTRACT FROM AUTHOR]

The aim of this work is to design an explicit finite volume scheme with high-order MOOD reconstruction for specific systems of conservation laws with stiff source terms which degenerate into diffusion equations. We propose a general framework to design an asymptotic preserving scheme that is stable and consistent under a classical hyperbolic CFL condition in both hyperbolic and diffusive regimes for any 2D unstructured mesh. Moreover, the developed scheme also preserves the set of admissible states, which is mandatory to conserve physical solutions in stiff configurations. This construction is achieved by using a non-linear scheme as a target scheme for the limit diffusion equation, which gives the form of the global scheme for the full system. The high-order polynomial reconstructions allow to improve the accuracy of the scheme without getting a full high-order scheme. Numerical results are provided to validate the scheme in every regime. [ABSTRACT FROM AUTHOR]

A highly accurate numerical shock and material interface fitting scheme composed of fifth-order spatial and third- or fifth-order temporal discretizations is applied to the two-dimensional reactive Euler equations in both slab and axisymmetric geometries. High rates of convergence are not typically possible with shock-capturing methods as the Taylor series analysis breaks down in the vicinity of discontinuities. Furthermore, for typical high explosive (HE) simulations, the effects of material interfaces at the charge boundary can also cause significant computational errors. Fitting a computational boundary to both the shock front and material interface ( i.e. streamline) alleviates the computational errors associated with captured shocks and thus opens up the possibility of high rates of convergence for multi-dimensional shock and detonation flows. Several verification tests, including a Sedov blast wave, a Zel'dovich–von Neumann–Döring (ZND) detonation wave, and Taylor–Maccoll supersonic flow over a cone, are utilized to demonstrate high rates of convergence to nontrivial shock and reaction flows. Comparisons to previously published shock-capturing multi-dimensional detonations in a polytropic fluid with a constant adiabatic exponent (PF-CAE) are made, demonstrating significantly lower computational error for the present shock and material interface fitting method. For an error on the order of 10 m / s , which is similar to that observed in experiments, shock-fitting offers a computational savings on the order of 1000. In addition, the behavior of the detonation phase speed is examined for several slab widths to evaluate the detonation performance of PBX 9501 while utilizing the Wescott–Stewart–Davis (WSD) model, which is commonly used in HE modeling. It is found that the thickness effect curve resulting from this equation of state and reaction model using published values is dramatically more steep than observed in recent experiments. Utilizing the present fitting strategy, in conjunction with a nonlinear optimizer, a new set of reaction rate parameters improves the correlation of the model to experimental results. Finally, this new model is tested against two dimensional slabs as a validation test. [ABSTRACT FROM AUTHOR]

In this work, we have applied a new technique which is a union of multiplier and Ibragimov's nonlocal conservation method for constructing the local conservation laws of nonlinear evolution equations. One can conclude that the higher order solutions of adjoint equation can be obtained by the multiplier functions. The Lax equation and generalized Hirota-Satsuma coupled KdV system are chosen to illustrate the effectiveness of the method. Thus, we have obtained a plenty of local (some of them are the higher order) conservation laws. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In this paper, high-order numerical methods are investigated in a system analysis-like code. The classical Welander oscillatory natural circulation problem, which resembles a simplified example for many types of natural circulation loops widely seen in nuclear reactor systems, was chosen to illustrate the applicability of such methods in system analysis codes, and to demonstrate the advantages of such methods over the low-order methods widely used in existing system analysis codes. As originally studied by Welander, the fluid motion in a differentially heated fluid loop can exhibit stable, weakly unstable, and strongly unstable modes. A theoretical stability map has also been originally derived from the stability analysis. Numerical results obtained in this paper show very good agreement with Welander's theoretical derivations. For stable cases, numerical results from both the high-order and low-order numerical methods agree well with the non-dimensional flow rate that were analytically derived. The high-order numerical methods give much less numerical errors compared to those using low-order numerical methods. For stability analysis, the high-order numerical methods perfectly predicted the stability map even with coarse mesh and large time step, while the low-order numerical methods failed to do so unless very fine mesh and time step are used. The result obtained in this paper is a strong evidence for the benefits of using high-order numerical methods over the low-order ones, when they are applied to simulate natural circulation phenomenon that has already gained increasing interests in many existing and advanced nuclear reactor designs. [ABSTRACT FROM AUTHOR]

The high-order low-order (HOLO) method is a recently developed moment-based acceleration scheme for solving time-dependent thermal radiative transfer problems, and has been shown to exhibit orders of magnitude speedups over traditional time-stepping schemes. However, a linear stability analysis by Haut et al. (2015) revealed that the current formulation of the multigroup HOLO method was unstable in certain parameter regions. Since then, we have replaced the intensity-weighted opacity in the first angular moment equation of the low-order (LO) system with the Rosseland opacity. This results in a modified HOLO method (HOLO-R) that is significantly more stable. This paper has two primary components. First, we present results from a linear stability analysis of the HOLO-R method for a 2-group problem. These results show that HOLO-R is stable in all regions of the parameter space we considered, and strongly suggest that HOLO-R is unconditionally stable. The predicted decay factors from the stability analysis are verified with numerically computed decay factors from a spectral HOLO-R code. Second, we describe a preliminary (unoptimized) implementation of HOLO-R in Capsaicin – a deterministic, discrete-ordinates radiation transport code at Los Alamos National Laboratory. The results from this implementation further verify the stability of the HOLO-R method. Moreover, in the simulation of a 50-group, nonlinear Marshak wave problem, away from the 2-group, linearized stability analysis, we assess HOLO-R’s accuracy, efficiency, and robustness as compared to current methods in Capsaicin. To our knowledge, this is the first implementation of HOLO that uses a mass-lumped discontinuous Galerkin spatial discretization for both the high-order (HO) and LO systems, and details of this discretization are provided in the Appendix. [ABSTRACT FROM PUBLISHER]

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary can effectively reduce the dissipation in numerical flux. Differently from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, the BVD strategy presented in this paper switches between a high-order polynomial and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. This work provides a simple and accurate alternative of great practical significance to the current high-order Godunov paradigm which overly pursues the smoothness within mesh cells under the questionable premiss that discontinuities only appear at cell interfaces. [ABSTRACT FROM AUTHOR]

This manuscript is concerned with the detection of shock discontinuities in the solution of conservation laws for high-order discontinuous Galerkin methods. A shock detector based on the entropy residual is proposed to distinguish smooth and non-smooth parts of the solution. The numerical analysis shows that the proposed entropy residual converges if the true solution is smooth and sufficiently regularized in space and time. To precisely localize discontinuities of different natures, an approach is developed that dynamically sets the threshold on the detection function, such that the detection criterion retains its sensitivity to the characteristics of the local solution. The implementation is conducted in an entropy-bounded discontinuous Galerkin framework, and numerical tests confirm the convergence property of the entropy-residual formulation and the effectiveness of the thresholding procedure. This shock detector is combined with an artificial viscosity scheme for shock stabilization. Comparison with other detectors is performed to demonstrate the excellent performance of the entropy-residual based shock detector for a wide range of problems on regular and triangular grids. [ABSTRACT FROM AUTHOR]

The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes; however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric exibility, suitability for h-and p-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 8918-8934] and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of solving nonlinear Klein-Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate. [ABSTRACT FROM AUTHOR]

Theory of general linear methods (GLMs) for the numerical solution of autonomous system of ordinary differential equations of the form y ′ = f ( y ) is extended to include the second derivative y ″ = g ( y ) : = f ′ ( y ) f ( y ) . This extension of GLMs is called second derivative general linear methods (SGLMs). In this paper we will construct two-stage A - and L -stable SGLMs of order p and stage order q = p up to six with low error constants. We will show the efficiency of the proposed methods by implementing on some well-known stiff problems. [ABSTRACT FROM AUTHOR]

In 1917, Marian von Smoluchowski presented a simple mathematical description of diffusion-controlled reactions on the scale of individual molecules. His model postulated that a reaction would occur when two reactants were sufficiently close and, more specifically, presented a succinct relationship between the relative proximity of two reactants at the moment of reaction and the macroscopic reaction rate. Over the past century, the Smoluchowski reaction theory has been applied widely in the physical, chemical, environmental, and, more recently, biological sciences. Despite the widespread utility of the Smoluchowski theory, it only describes the rates of second order reactions and is inadequate for describing higher order reactions for which there is no equivalent method for theoretical investigation. In this paper, we derive a generalized Smoluchowski framework in which we define proximity in the context when more than two reactants are involved. We derive the relationship between the macroscopic reaction rate and the critical proximity at which a reaction occurs for higher order reactions. Using this theoretical framework and through numerical experiments, we explore various peculiar properties of multimolecular diffusion-controlled reactions which, due to there being no other numerical method of this nature, have not been previous reported. [ABSTRACT FROM AUTHOR]

We develop high-order Galerkin methods with graded meshes for solving the two- dimensional reaction-diFFusion problem on a rectangle. With the help of the comparison principle, we establish upper bounds for high order partial derivatives of an arbitrary order of its exact solution. According to prior information of the high order partial derivatives of the solution, we design both implicit and explicit graded meshes which lead to numerical solutions of the problem having an optimal convergence order. Numerical experiments are presented to confirm the theoretical estimate and to demonstrate the outperformance of the proposed meshes over the Shishkin mesh. [ABSTRACT FROM AUTHOR]

In this article, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for a Riemann-Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability, and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are O(t² +h6 and O(t² + h8), respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L 1 ( L ∞ ) norm with the convergence orders O ( τ 2 + h 2 + Δ α 2 ) and O ( τ 2 + h 4 + Δ α 4 ) , respectively, where τ , h and Δ α are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

A theoretical analysis of numerical dispersion in the high-order finite-difference time-domain (FDTD) method with weighted Laguerre polynomials (WLPs) is proposed in this paper. According to the numerical dispersion relation for the two-dimensional (2-D) case, the numerical phase velocities relevant to the direction of wave propagation, grid discretization and time-scale factor are obtained. For a fixed relative error of the numerical phase velocity, the suitable sampling point density and time-scale factor can be determined. Compared with the low-order WLP-FDTD, the high-order one shows its good dispersion characteristics while a low sampling density is used. Three numerical examples are included to validate the effectiveness of the high-order scheme. [ABSTRACT FROM AUTHOR]

Solutions to elliptic PDEs, in particular to the Helmholtz equation, become singular near the boundary if the boundary data do not possess sufficient regularity. In that case, the convergence of standard numerical approximations may slow down or cease altogether. We propose a method that maintains a high order of grid convergence even in the presence of singularities. This is accomplished by an asymptotic expansion that removes the singularities up to several leading orders, and the remaining regularized part of the solution can then be computed on the grid with the expected accuracy. The computation on the grid is rendered by a compact finite difference scheme combined with the method of difference potentials. The scheme enables fourth order accuracy on a narrow 3 × 3 stencil: it uses only one unknown variable per grid node and requires only as many boundary conditions as needed for the underlying differential equation itself. The method of difference potentials enables treatment of non-conforming boundaries on regular structured grids with no deterioration of accuracy, while the computational complexity remains comparable to that of a conventional finite difference scheme on the same grid. The method of difference potentials can be considered a generalization of the method of Calderon's operators in PDE theory. In the paper, we provide a theoretical analysis of our combined methodology and demonstrate its numerical performance on a series of tests that involve Dirichlet and Neumann boundary data with various degrees of “non-regularity”: an actual jump discontinuity, a discontinuity in the first derivative, a discontinuity in the second derivative, etc. All computations are performed on a Cartesian grid, whereas the boundary of the domain is a circle, chosen as a simple but non-conforming shape. In all cases, the proposed methodology restores the design rate of grid convergence, which is fourth order, in spite of the singularities and regardless of the fact that the boundary is not aligned with the discretization grid. Moreover, as long as the location of the singularities is known and remains fixed, a broad spectrum of problems involving different boundary conditions and/or data on “smooth” segments of the boundary can be solved economically since the discrete counterparts of Calderon's projections need to be calculated only once and then can be applied to each individual formulation at very little additional cost. [ABSTRACT FROM AUTHOR]

Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in [8] , and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with non-matching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach. [ABSTRACT FROM AUTHOR]

Numerical approximations and modeling of many physical, biological, and biomedical problems often deal with equations with highly varying coefficients, heterogeneous models (described by different types of partial differential equations (PDEs) in different domains), and/or have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient numerical method that can capture certain properties of analytical solutions in different domains/subdomains (such as positivity, different regularity/smoothness of the solutions, etc.), while handling the arbitrary geometries and complex structures of the domains. In this work, we employ one-dimensional elliptic type models as the starting point to develop and numerically test high-order accurate Difference Potentials Method (DPM) for variable coefficient elliptic problems in heterogeneous media. While the method and analysis are simple in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach. [ABSTRACT FROM AUTHOR]

A high-order finite difference method for the two-dimensional complex Ginzburg-Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876-899, 2015 [ABSTRACT FROM AUTHOR]

Abstract: High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in the Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split semigroups can easily be proved. A numerical example from interest rate theory, the CIR2 model, is considered. The numerical results are robust for drift-dominated problems and confirm our theoretical results. [Copyright &y& Elsevier]

In this paper we study the piecewise collocation method for a class of functional integral equations with state-dependent delays that is, where the delays depend on the solution. It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called 'singular points', which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. Here, we present a numerical algorithm based on the piecewise collocation method and an approach for tracking the singular points relying on the recent strategy for implicit delay differential equations which has been proposed by Guglielmi and Hairer in 2008. The convergence analysis of the method is investigated and some numerical experiments are presented for clarifying the robustness of the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new high accuracy numerical method of O(k 2 + k 2 h 2 + h 4) for the solution of three dimensional quasi-linear hyperbolic partial differential equations, where k >0 and h >0 are mesh sizes in time and space directions respectively. We mainly discretize the space derivative terms using fourth order approximation and time derivative term using second order approximation. We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. The proposed method behaves like a fourth order method for a fixed value of . Some examples and their numerical results are provided to justify the usefulness of the proposed method. [Copyright &y& Elsevier]

Abstract: High order spatial derivatives and stiff reactions often introduce severe temporal stability constraints on the time step in numerical methods. Implicit integration method (IIF) method, which treats diffusion exactly and reaction implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reactions and diffusions. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. Motivated by a compact representation for IIF (cIIF) for Laplacian operators in two and three dimensions, we introduce an array-representation technique for efficient handling of exponential matrices from a general linear differential operator that may include cross-derivatives and non-constant diffusion coefficients. In this approach, exponentials are only needed for matrices of small size that depend only on the order of derivatives and number of discretization points, independent of the size of spatial dimensions. This method is particularly advantageous for high-dimensional systems, and it can be easily incorporated with IIF to preserve the excellent stability of IIF. Implementation and direct simulations of the array-representation compact IIF (AcIIF) on systems, such as Fokker–Planck equations in three and four dimensions and chemical master equations, in addition to reaction–diffusion equations, show efficiency, accuracy, and robustness of the new method. Such array-presentation based on methods may have broad applications for simulating other complex systems involving high-dimensional data. [Copyright &y& Elsevier]

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We introduce the generalized Jacobi-Gauss-Lobatto interpolation involving the values of functions and their derivatives at the endpoints, which play important roles in the Jacobi pseudospectral methods for high order problems. We establish some results on these interpolations in non-uniformly weighted Sobolev spaces, which serve as the basic tools in analysis of numerical quadratures and various numerical methods of differential and integral equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with a class of high-order neural networks (HONNs). Rigorous analysis shows that the state components exhibit different dynamical behaviors with respect to external inputs lying in different ranges. And by dividing the index set into four subsets , according to different external input ranges, we can conclude that the HONNs have exact equilibrium points, of them are locally stable and others are unstable, here represents the number of elements in the subset . The results obtained improve and extend some related works. A numerical example is presented to illustrate the effectiveness of our criteria. [Copyright &y& Elsevier]

A deferred correction method is utilized to increase the order of spatial accuracy of the Crank-Nikolson scheme for the numerical solution of the one-dimensional heat equation. Numerical examples are given for both Neumann and Dirichlet initial boundary value problems. The fourth-order methods proposed are compared with high-order compact schemes. The set of methods proposed demonstrate a better performance compared with high-order compact schemes in the case of the Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub-diffusion equation. The time fractional derivative is described in the Riemann–Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the solvability, stability and convergence of the proposed scheme using the Fourier method. The convergence order of method is . Numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme. [Copyright &y& Elsevier]

Abstract: This paper presents a general formulation to construct high order numerical schemes by using multi-moment constraint conditions on the flux function reconstruction. The new formulation, so called multi-moment constrained flux reconstruction (MMC-FR), distinguishes itself essentially from the flux reconstruction formulation (FR) of Huynh (2007) [3] by imposing not only the continuity constraint conditions on the flux function at the cell boundary, but also other types of constraints which may include those on the spatial derivatives or the point values. This formulation can be also interpreted as a blend of Lagrange interpolation and Hermite interpolation, which provides a numerical framework to accommodate a wider spectrum of high order schemes. Some representative schemes will be presented and evaluated through Fourier analysis and numerical tests. [Copyright &y& Elsevier]

Abstract: This letter develops high order finite difference weighted essentially non-oscillatory (WENO) schemes for fractional differential equations. First, the th, , Caputo fractional derivative is split into a classical second derivative and a weakly singular integral. Then the sixth-order finite difference WENO scheme is used to discretize the classical second derivative and the Gauss–Jacobi quadrature is applied to solve the weakly singular integral. The constructed scheme of approximation for the fractional derivative has high order accuracy in smooth regions and maintains a sharp discontinuity transition. Finally, numerical experiments are performed to demonstrate the effectiveness of the proposed schemes. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new high accuracy numerical method of O(k 2 + k 2 h 2 + h 4) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u tt = A(x,y,z,t)u xx + B(x,y,z,t)u yy + C(x,y,z,t)u zz + g(x,y,z,t,u,u x ,u y ,u z ,u t ), 0< x,y,z <1,t >0 subject to given appropriate initial and Dirichlet boundary conditions, where k >0 and h >0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in [3,4]. We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method. [Copyright &y& Elsevier]

Abstract: Edge elements are a popular method to solve Maxwell’s equations especially in time-harmonic domain. However, when non-affine elements are considered, elements of the Nédélec’s first family [19] are not providing an optimal rate of the convergence of the numerical solution toward the solution of the exact problem in H(curl)-norm. We propose new finite element spaces for pyramids, prisms, and hexahedra in order to recover the optimal convergence. In the particular case of pyramids, a comparison with other existing elements found in the literature is performed. Numerical results show the good behavior of these new finite elements. [Copyright &y& Elsevier]

Invariant moments and transforms are a special kind of statistical measure designed to remain constant after some transformations, such as object rotation, scaling, translation or image illumination changes, in order to improve the reliability of a pattern recognition system. Polar harmonic transforms (PHTs) are one of such image descriptors based on a set of orthogonal projection bases, which are better rated in comparison with other moments and transforms because they possess less redundant information. PHTs suffer from geometric error and numerical integration error. Geometric error is caused when a square image is mapped into a unit circular disk, which cannot exactly match the square domain. Numerical integration error is caused when the integration is approximated by zeroth-order summation. We propose a computational framework based on numerical integration approach that reduces the geometric error and the numerical integration error simultaneously. The effect of the improved accuracy in the PHTs is analysed for rotation invariance, scale invariance and image noise. Various numerical experiments demonstrate that the effect of these errors is much more prominent in small images and, therefore the proposed method is suitable for applications involving small images, such as in optical character recognition and template-matching applications. [ABSTRACT FROM AUTHOR]

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourthorder ones, in the sense of Kung-Traub's conjecture. [ABSTRACT FROM AUTHOR]

The paper presents the modifications of the classic Takens-Ellner (TE) algorithm that are necessary to estimate the dimensional complexity (d) of the medium- and high-dimensional signals. The main idea is the fitting of the fourth-degree polynomial to the relation d = fn(W) (where W is window width) in order to find the point of minimum slope in this relation. This point corresponds to the plateau area being the feature of the low-dimensional signals and representing the range of the optimal W used in the calculations. The point of minimum slope is represented by the local minimum in the first derivative function (third-degree polynomial). The exclusion of the attractor pairs of points lying approximately closer than the autocorrelation time removes the tendency to down-estimate d observed in the classic TE algorithm. The procedure of the choice of the embedding parameters was modified: The lag (L) was calculated for the given embedding dimension (m) and W was calculated by using the formula L = W/(m - 1) in order to obtain the desired values of W for generating the precise relation d = fn(W) being the basis for the polynomial fitting. The cubic interpolation of the signal is proposed for the noninteger L's. Three signals possessing the dimensional complexities dA ≈ 4, dB ≈ 6, and dc ≈ 8 and being the sums of two, three, and four Lorenz signals, respectively, were analyzed. Their lengths were 65 536 points. The applied algorithm gives a precise estimation of d for every signal A, B, and C. Also presented are the results of (a) the method for the estimation of the cubic interpolation error, (b) the analysis for signals contaminated with the white noise, and (c) the comparison between the original signals and the surrogates. [ABSTRACT FROM AUTHOR]

This paper discusses quasi-interpolation and interpolation with Gaussians. Estimates are obtained showing a high-order approximation up to some saturation error negligible in numerical applications. The construction of local high-order quasi-interpolation formulas is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the discrete Maxwellian equilibrium distribution function for incompressible flow and force term into the two-stage third-order Discrete Unified Gas-Kinetic Scheme (DUGKS) for simulating low-speed turbulent flows. The Wall-Adapting Local Eddy-viscosity (WALE) and Vreman sub-grid models for Large-Eddy Simulations (LES) of turbulent flows are coupled within the present framework. Meanwhile, the implicit LES are also presented to verify the effect of LES models. A parallel implementation strategy for the present framework is developed, and three canonical wall-bounded turbulent flow cases are investigated, including the fully developed turbulent channel flow at a friction Reynolds number (Re) about 180, the turbulent plane Couette flow at a friction Re number about 93 and lid-driven cubical cavity flow at a Re number of 12000. The turbulence statistics, including mean velocity, the r.m.s. fluctuations velocity, Reynolds stress, etc. are computed by the present approach. Their predictions match precisely with each other, and they are both in reasonable agreement with the benchmark data of DNS. Especially, the predicted flow physics of three-dimensional lid-driven cavity flow are consistent with the description from abundant literature. The present numerical results verify that the present two-stage third-order DUGKS-based LES method is capable for simulating inhomogeneous wall-bounded turbulent flows and getting reliable results with relatively coarse grids. [ABSTRACT FROM AUTHOR]

We present a new reduced Monte Carlo simulation strategy for nonlinear high-order systems based on an extension of the proper orthogonal decomposition for transient excited structures. This novel approach enables the evaluation of the response statistics in a fractional amount of time compared to the full Monte Carlo simulation procedure. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]