Search

In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O (h 2) in L 2 norm and O (h) in the H 1 semi-norm. Further the condition number increases with the mesh size similar to the FEM. [ABSTRACT FROM AUTHOR]

For transformer asset management, the dynamic state estimation can track transient phenomena of transformers or extract an unmeasurable quantity such as magnetic flux linkage. Even the extended Kalman filter or unscented Kalman filter, which are widely-used dynamic state estimations, might result in significant estimation errors when dealing with highly nonlinear dynamics such as transformer core saturation. In this sense, this paper proposes a dynamic state estimation based on the weighted least squares, using real-time measurements and the dynamic model induced by numerically integrating differential equations. The proposed method can directly use implicit functions including differential equations while Kalman filters must rearrange the implicit functions into the explicit ones to obtain the prediction process. Further, because the weighted least squares approach is implemented via the Newton iterative method, the estimation accuracy and robustness to various conditions can be improved despite strong nonlinearity. For the numerical integration of dynamics, the computationally efficient trapezoidal rule or the numerically stable quadratic rule is used. This paper presents numerical simulation results, with a comparison with other dynamic state estimators. [ABSTRACT FROM AUTHOR]

In many memristor models the internal variable is related to the width of a conductive layer and is therefore limited by the physical dimensions of the device. The necessity of keeping the internal variable within allowable limits may introduce discontinuities in right hand sides of differential equations describing dynamics of circuits with memristors. Such discontinuities are difficult to handle using standard numerical integration methods. An approach based on the concept of a Poincaré map is proposed to solve these difficulties. Two examples are discussed to show the usefulness of the proposed technique. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we generalize the recently developed Finite Integration Method (FIM) for the solutions of high-dimensional partial differential equations. Formulation of this Generalized Finite Integration Method (GFIM) can be derived due to the use of piecewise polynomials in the numerical integrations. The GFIM does not require the strict requirement for uniformly distributed nodal points in the original FIM. This robustness advantage extends the applicability of FIM to solve partial differential equations by using direct Kronecker product. Due to the unconditional stability of numerical integrations, the GFIM is effective and efficient to solve higher dimensional partial differential equations with stiffness. For numerical verification, we construct several 1D to 4D problems with different types of stiffness and make comparisons among existing numerical methods. [ABSTRACT FROM AUTHOR]

Delays are important processes represented by continuous simulation models; however, representing queueing delays efficiently within continuous simulations merits the development of new methodology. Rothkopf and Oren introduced the concept of using a surrogate distribution, viz., the negative-binomial, as a closure approximation to the infinite set of Chapman-Kolmogorov equations representing a nonstationary M/M/s queue. The method presented in this paper uses the Polya-Eggenberger distribution as a surrogate for the true distribution of the number in the queueing system at a particular time and only requires the numerical integration of five differential equations. The paper presents numerical results comparing the Polya surrogate and Rothkopf and Oren's approximation for a number of diverse cases, and these results indicate that the Polya surrogate is, in general, more accurate, although exceptions were encountered. Moreover, queueing delays represented by a closure approximation involving a surrogate distribution, in particular, the Polya, are suitable for use within a larger continuous simulation. [ABSTRACT FROM AUTHOR]

A new integration algorithm is found, and an implementation is compared with other programmed algorithms. The new algorithm is a step-by-step procedure for solving the initial value problem in ordinary differential equations. It is designed to approximate poles of small integer order in the solutions of the differential equations by continued fractions obtained by manipulating the sums of truncated Taylor series expansions. The new method is compared with the Gragg-Bulirsch-Stoer, and the Taylor series method. The Taylor series method and the new method are shown to be superior in speed and accuracy, while the new method is shown to be most superior when the solution is required near a singularity. The new method can finally be seen to pass automatically through singularities where all the other methods which are discussed will have failed. [ABSTRACT FROM AUTHOR]

A procedure is given for continuously computing and monitoring the step size to be used by a self-starting, pth-order numerical integration method to solve an initial value problem. The procedure uses an estimate of the truncation error to calculate the step size. [ABSTRACT FROM AUTHOR]

A study has been made to determine which methods of numerical integration require the least computation time for a given amount of truncation error when applied to a particular system of ordinary differential equations where function evaluations are relatively trivial. Recent methods due to Butcher and Gear are compared with classic Runge-Kutta Kutta-Nyström and Adams methods. Some of the newer one-step methods due to Butcher are found to be slightly superior, but no one method is found to have any great advantage over the others in the application to this particular problem. [ABSTRACT FROM AUTHOR]

We introduce NSC++, a header-only C++ library that simulates the evolution of the plasma and a decaying fluid in the early Universe. NSC++ can be used in C++ programs or called directly from python scripts without significant overhead. There is no special installation process or external dependencies. Furthermore, there are example programs that can be modified to handle several cases. Program title: NSC++ CPC Library link to program files: https://doi.org/10.17632/yn883k4wsr.1 Developer's repository link: https://github.com/dkaramit/NSCpp Programming language: C++ (C++17+) and python (3.7+). Licensing provisions: MIT license. Nature of problem: Solves equations that describe the evolution of the plasma along with a fluid that increases the entropy of the plasma during the early Universe. Solution method: Embedded Runge-Kutta for the numerical integration of the system of differential equations. The user can choose between explicit and Rosenbrock methods. There are several of Butcher tableaux already implemented, but the user can implement their own. The interpolations of the relativistic degrees of freedom are accomplished by cubic spline interpolation. Additional comments including restrictions and unusual features: The value of the pressure over the energy density and the energy loss parameter of the fluid are assumed to be constant. It is also assumed that there is no energy leaking from the plasma to the fluid. [ABSTRACT FROM AUTHOR]

Several questions and answers related to differential equations are presented including the algebraic equation of any tangent to a curve and integration of the differential equation.

A computational method for the numerical integration of the stochastic transport equation is proposed. We develop an approach to construct an explicit numerical scheme for the effective simulation of trajectories of solutions of this equation. Results on its convergence and details on its efficient computational implementation are presented. The performance of the proposed scheme is illustrated by means of computer simulations. • An easy-to-implement explicit method is proposed for stochastic transport equations. • The proposed approach is completely different from others available in the literature. • The computational burden is basically the same as solving systems of linear equations. • The approach can be used for transport equations driven by low regularity processes. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to apply a novel technique for the simulation of nonlinear systems subject to modulated chirp signals. Design/methodology/approach – The simulation technique is first described and its salient features are presented. Two examples are given to confirm the merits of the method. Findings – The results indicate that the method is appropriate for simulating nonlinear systems subject to modulated chirp signals. In particular, the efficiency and accuracy of the method is seen to improve as the chirp frequency increases. In addition, error bounds are given for the method. Originality/value – Chirp signals are employed in several important applications such as representing biological signals and in spread spectrum communications. Analysis of systems involving such signals requires accurate, appropriate and effective simulation techniques. [ABSTRACT FROM AUTHOR]

Exploring singular perturbation and stability region theory, the foundations for decomposing the stability analysis of power systems in time scales is developed and a general algorithm, for stability analysis of power systems, which has transient stability analysis and quasi-steady state analysis as particular cases, is proposed. This algorithm bridges the gap between short and mid-term stability analysis and clearly establishes the conditions to guarantee stability of the original electrical power system model in terms of stability of the fast and slow subsystems. Simple examples illustrate the application of the proposed algorithm and its ability in delineating the evolution of the power system dynamics and identifying unstable modes. [ABSTRACT FROM AUTHOR]

A mathematical model for anaerobic digestion in plug-flow reactors is proposed on the basis of mass balance considerations. The model consists of a system of parabolic partial differential equations for the variables representing the concentrations of the bio-components constituting the waste matrix and takes into account convective and diffusive phenomena. The plug-flow reactor is modelled as a one-dimensional domain; the waste matrix moves in the direction of the reactor axis and undergoes diffusive phenomena which reproduce the movement of the bio-components along the reactor axis due to a gradient in concentration. The velocity characterizing the convection of the waste matrix is not fixed a priori but it is considered as an additional unknown of the mathematical problem. The variation in the convective velocity allows to account the mass variation occurring along a plug-flow reactor due to the conversion of solids. The equation governing the convective velocity is derived by considering the density of the waste matrix within the reactor constant over time and the sum of the volume fractions of the bio-components constituting the waste matrix constrained to unity. The waste matrix undergoes biochemical transformations catalyzed​ by anaerobic microbial species which lead to the production of gaseous methane, the final product of the anaerobic digestion process. Biochemical processes are modelled using a simplified scheme and a differential equation is used to describe the dynamics of the produced gaseous methane. A finite difference scheme is used for the numerical integration. Model consistency is showed through numerical simulations which investigate the effect of the variation of some operating parameters on process performance. The model is then applied to a real case scenario of engineering interest. Simulations produce results in good agreement with experimental observations. • A model describing dry anaerobic digestion in plug-flow reactors is presented. • A system of convection-diffusion equations is derived on mass balance considerations. • The convective velocity is considered variable to account for the solids conversion. • Numerical simulations investigate the effect of operating parameters on the process. • Model outcomes are in agreement with experimental data from a real case scenario. [ABSTRACT FROM AUTHOR]

Abstract: XMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range from a single ordinary differential equation up to systems of coupled stochastic partial differential equations. The equations are described in a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the efficient solution of those equations. It combines the advantages of high-level simulations, namely fast and low-error development, with the speed, portability and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS package, and features support for a much wider problem space while also producing faster code. Program summary: Program title: XMDS2 Catalogue identifier: AENK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENK_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 2 No. of lines in distributed program, including test data, etc.: 872490 No. of bytes in distributed program, including test data, etc.: 45522370 Distribution format: tar.gz Programming language: Python and C++. Computer: Any computer with a Unix-like system, a C++ compiler and Python. Operating system: Any Unix-like system; developed under Mac OS X and GNU/Linux. RAM: Problem dependent (roughly 50 bytes per grid point) Classification: 4.3, 6.5. External routines: The external libraries required are problem-dependent. Uses FFTW3 Fourier transforms (used only for FFT-based spectral methods), dSFMT random number generation (used only for stochastic problems), MPI message-passing interface (used only for distributed problems), HDF5, GNU Scientific Library (used only for Bessel-based spectral methods) and a BLAS implementation (used only for non-FFT-based spectral methods). Nature of problem: General coupled initial-value stochastic partial differential equations. Solution method: Spectral method with method-of-lines integration Running time: Determined by the size of the problem [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. After changing general planar self-excited systems to the canonical forms by several linear transformations, the auxiliary linear operators and the initial guess of solutions are introduced. Hence, the homotopy analysis solving is set up. Importantly, in solving the higher-order deformation equations, the idea of a perturbation procedure of limit cycles’ approximation proposed in the setting of second-order self-excited equations is embedded. As an application, a Rosenzweig–MacArthur predator–prey model is studied in detail. By choosing the suitable convergence-control parameters, the accurately analytical approximations of the large amplitude limit cycles and their frequency of the model are obtained. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations. [Copyright &y& Elsevier]

Abstract: We describe a class of two-step continuous methods for the numerical integration of initial-value problems based on stiff ordinary differential equations (ODEs). These methods generalize the class of two-step Runge-Kutta methods. We restrict our attention to methods of order p = m, where m is the number of internal stages, and stage order q = p to avoid order reduction phenomenon for stiff equations, and determine some of the parameters to reduce the contribution of high order terms in the local discretization error. Moreover, we enforce the methods to be A-stable and L-stable. The results of some fixed and variable stepsize numerical experiments which indicate the effectiveness of two-step continuous methods and reliability of local error estimation will also be presented. [Copyright &y& Elsevier]

Abstract: The GeodesicViewer realizes exocentric two- and three-dimensional illustrations of lightlike and timelike geodesics in the general theory of relativity. By means of an intuitive graphical user interface, all parameters of a spacetime as well as the initial conditions of the geodesics can be modified interactively. This makes the GeodesicViewer a useful instrument for the exploration of geodesics in four-dimensional Lorentzian spacetimes. Program summary: Program title: GeodesicViewer Catalogue identifier: AEFP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFP_v1_0.html Program obtainable from: CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 168 868 No. of bytes in distributed program, including test data, etc.: 6 076 202 Distribution format: tar.gz Programming language: C++, Qt, Qwt, OpenGL Computer: All platforms with a C++ compiler, Qt, Qwt, OpenGL Operating system: Linux, Mac OS X RAM: 24 Mbytes Classification: 1.5 External routines: [•] Gnu Scientific Library (GSL) (http://www.gnu.org/software/gsl/) [•] Motion4D (included in the package). The Motion4D library can also be downloaded from CPC. Catalogue identifier: AEEX [•] Qt (http://qt.nokia.com/downloads) [•] Qwt (http://qwt.sourceforge.net/) [•] OpenGL (http://www.opengl.org/) Nature of problem: Illustrate geodesics in four-dimensional Lorentzian spacetimes. Solution method: Integration of ordinary differential equations. 3D-Rendering via OpenGL. Running time: Interactive. The examples given take milliseconds. [Copyright &y& Elsevier]

Abstract: Simulations of biophysical systems inevitably include steps that correspond to time integrations of ordinary differential equations. These equations are often related to enzyme action in the synthesis and destruction of molecular species, and in the regulation of transport of molecules into and out of the cell or cellular compartments. Enzyme action is almost invariably modeled with the quasi-steady-state Michaelis–Menten formula or its close relative, the Hill formula: this description leads to systems of equations that may be stiff and hard to integrate, and poses unusual computational challenges in simulations where a smooth evolution is interrupted by the discrete events that mark the cells'' lives, like initiation and termination of DNA synthesis or mitosis. These discrete events lead to abrupt parameter changes and to variable system size. This is the case of a numerical model (Virtual Biophysics Lab – VBL) that we are developing to simulate the growth of three-dimensional tumor cell aggregates (spheroids). The underlying cellular events have characteristic timescales that span approximately 12 orders of magnitude, and thus the program must be robust and stable. Moreover the program must be able to manage a very large number of equations (of the order of 107–108), and finally it must be able to accept frequent modifications of the underlying theoretical model. Here we study the applicability of known integration methods to this context – quite unusual from the point of view of the standard theory of differential equations, but extremely relevant to biophysics – and we describe the results of numerical tests in situations similar to those found in actual simulations. [Copyright &y& Elsevier]

DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its associated package, is to provide a public implementation of these series expansion methods, which works for any family of integrals for which the user provides a set of differential equations and boundary conditions (and for which the program is not computationally constrained). The main functions of the DiffExp package are discussed, and its use is illustrated by applying it to the three loop equal-mass and unequal-mass banana graph families. Program title: DiffExp CPC Library link to program files: https://doi.org/10.17632/j43fkg8vjt.1 Developer's repository link: https://gitlab.com/hiddingm/diffexp Licensing provisions: GPLv3 Programming language: Mathematica (Wolfram Language) Nature of problem: Obtaining high-precision numerical results for families of Feynman integrals in dimensional regularization, at arbitrary points in phase-space and in an efficient manner, from the system of differential equations satisfied by the integrals and a set of suitable boundary conditions. Solution method: The program provides a public implementation of the integration strategy outlined in Ref. [1]. A line is taken from a given boundary point to a point in phase-space at which numerical results are desired. The line is subdivided into multiple segments, and along each line segment the differential equations are solved in terms of one-dimensional series expansions with a radius of convergence that covers the respective line segment, and up to high order in the line parameter. Boundary conditions are transported by matching results along neighboring line segments. Branch points are crossed by analytically continuing logarithms and square roots that appear in the series expansions. By evaluating the series solutions of the last line segment at the endpoint, the desired numerical results are obtained. Additional comments including restrictions and unusual features: Depending on the complexity of the differential equations of the integrals, the package is limited by CPU speed and/or memory. The prefactors of the Feynman integrals in the basis of master integrals for which the differential equations are given, should only contain rational functions and/or square roots. Fine-tuning of the configuration options may be required for families involving sectors with a large numbers of coupled integrals. [1] F. Moriello, J. High Energy Phys. 2020 (2020) 150. arXiv:1907.13234 , https://doi.org/10.1007/JHEP01(2020)150. [ABSTRACT FROM AUTHOR]

Abstract: We describe an implementation technique for boundary element methods that greatly reduces the required number of evaluations of Green''s function. Assuming that evaluating Green''s function is a costly operation, our approach has a computational cost that is almost independent of the chosen discretization scheme. Nyström methods, collocation methods and Galerkin methods lead to similar computation times, although the latter requires the computation of a large number of double integrals. The result is obtained mainly by constructing a specific family of quadrature and cubature rules that maximizes the possibility of sharing function evaluations. We apply the technique to the problem of scattering of electromagnetic waves by diffraction gratings in two dimensions. In this case Green''s function is given by a slowly converging oscillatory series. We describe a novel approach for evaluating such series and we illustrate our implementation with numerical results. [Copyright &y& Elsevier]

Abstract: In this paper differential quadrature Trefftz method (DQTM), a new meshless method based on coupling the dual reciprocity method (DRM) with the differential quadrature method (DQM) and the Trefftz method, is used to analyze Poisson-type interior and exterior problems. In this method, the DRM is used to construct equivalent equations to the original differential equation. Then the DQM is employed to approximate the particular solutions, while Trefftz method leads to a boundary-only formulation for homogeneous solution. As a result, an inherently meshless, integration-free, boundary-only DQ Trefftz collocation technique is developed for solving Poisson-type problems. Due to the flexibility in choosing points on boundaries, the new method also works well on irregular domains. Numerical results show that the present method works efficiently with quite few points on both uniform and irregular domains. [Copyright &y& Elsevier]

We examine and discuss the general problem of how “denominator functions” are to be calculated when using nonstandard finite difference (NSFD) methods for constructing discrete models of differential equations. Using the example of the Fisher PDE with linear advection, we show that the denominator functions follow from a simple algorithm. An important class of equations for which this method can be applied are those evolution PDE’s describing nonlinear wave phenomena. [Copyright &y& Elsevier]

Abstract: New Runge–Kutta–Nyström methods especially designed for the numerical integration of perturbed oscillators are presented in this paper. They are capable of exactly integrating the harmonic or unperturbed oscillator. We construct an embedded 4(3) RKN pair that is based on the FSAL property. The new method is much more efficient than previously derived RKN methods for some subclasses of problems. [Copyright &y& Elsevier]

A nonstandard finite difference scheme is constructed for the Burgers partial differential equation having no diffusion and a nonlinear logistic reaction term. This scheme preserves the positivity and boundedness properties of the original differential equation and includes the a priori requirement of being semi-explicit. Several other nonstandard discretizations are constructed and their mathematical structures discussed. All of these schemes can be used to calculate numerical solutions for traveling waves problems involving phenomena modeled by the original differential equation. [Copyright &y& Elsevier]

Abstract: An exact 1-soliton solution of the Benjamin–Bona–Mahoney equation, with dual-power law nonlinearity, will be obtained in this paper. The solitary wave ansatz will be used to carry out this integration. This solution will also be used to evaluate a couple of integrals of motion of this equation. [Copyright &y& Elsevier]

The article presents information about an algorithm, utilized for rapid computation of weights of interpolatory quadrature rules. The algorithm related differential equation has been given.