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In this study, we provide a nonconforming spectral approach for the Stokes equations with nonstandard boundary conditions on a single domain. The discrete spaces are defined in such a way that the discrete approximations for the velocity are exactly divergence-free. We provide a novel discrete inf-sup condition from which pressure error estimates are derived. Several numerical experiments are provided to demonstrate the method's interest. [ABSTRACT FROM AUTHOR]

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms. [ABSTRACT FROM AUTHOR]

The problem of the numerical analysis of natural convection from a spinning cone with variable wall temperature, viscous dissipation and pressure work effect is studied. The numerical method used is based on the spectral analysis. The method used to solve the system of partial differential equations is the multi-domain bivariate spectral quasi-linearization method (MD-BSQLM). The numerical method is compared with other methods in the literature, and the results show that the MD-BSQLM is robust and accurate. The method is also stable for large parameters. The numerical errors do not deteriorate with increasing iterations for different values of all parameters. The numerical error size is of the order of 10−101{0}^{-10}. With the increase in the suction parameter ξ\xi , fluid velocity, spin velocity and temperature profiles decrease.

In this paper, we consider the time‐dependent Stokes problem in a three‐dimensional domain with mixed boundary conditions. The discretization relies on spectral methods with respect to the space variables and Euler's implicit scheme with respect to the time variable, then by the second order BDF method. A detailed numerical analysis leads to a priori error estimates for each numerical scheme. [ABSTRACT FROM AUTHOR]

We present a generalized model to describe the flow of three non‐Newtonian nanofluids, namely, Jeffrey, Maxwell, and Oldroyd‐B nanofluids. Using this model, we study entropy generation and heat transfer in laminar nanofluid boundary‐layer stagnation‐point flow. The flow is subject to an external magnetic field. The conventional energy equation is modified by the incorporation of nanoparticle Brownian motion and thermophoresis effects. A hydrodynamic slip velocity is used in the initial condition as a component of the stretching velocity. The system of nonlinear equations is solved numerically using three different methods, a spectral relaxation method, spectral quasilinearization method, and the spectral local linearization method, first to determine the most accurate of these methods, and second as a measure to validate the numerical simulations. The residual errors for each method are presented. The numerical results show that the spectral relaxation method is the most accurate of the three methods, and this method is used subsequently to solve the transport equations and thus to determine the empirical impact of the physical parameters on the fluid properties and entropy generation. [ABSTRACT FROM AUTHOR]

The current manuscript focuses on solving fractional logistic population models (FLPMs). The spectral tau method is developed for solving FLPMs with shifted Jacobi polynomials as basis functions. We express the nonlinear terms of such FLPMs as linear expansions in shifted Jacobi polynomials. The proposed method is applied on two real-life applications and compared with other numerical schemes. [ABSTRACT FROM AUTHOR]

It is of great concern to produce numerically efficient methods for moisture diffusion through porous media, capable of accurately calculate moisture distribution with a reduced computational effort. In this way, model reduction methods are promising approaches to bring a solution to this issue since they do not degrade the physical model and provide a significant reduction of computational cost. Therefore, this article explores in details the capabilities of two model reduction techniques—the Spectral reduced-order model and the proper generalized decomposition—to numerically solve moisture diffusive transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. The methodology of both approaches is explained extensively so that the article can be used as a numerical benchmark by anyone interested in building a reduced-order model for diffusion problems in porous materials. Linear and nonlinear unsteady behaviors of unidimensional moisture diffusion are investigated. The last case focuses on solving a parametric problem in which the solution depends on space, time and the diffusivity properties. Results have highlighted that both methods provide accurate solutions and enable to reduce significantly the order of the model around 10 times lower than the large original model. It also allows an efficient computation of the physical phenomena with an error lower than 10-2 when compared to a reference solution. [ABSTRACT FROM AUTHOR]

In this work, we develop a validated numeric method for the solution of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for numerically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are nonconstructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity analysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map. Specifically, given a polynomial approximation, obtained by some numerical algorithm and expressed on a Chebyshev basis, our algorithm efficiently computes an accurate and rigorous error bound. For this, we study theoretical properties like compactness, convergence, and invertibility of associated linear integral operators and their truncations in a suitable coefficient space of Chebyshev series. Then, we analyze the almost-banded matrix structure of these operators, which allows for very efficient numerical algorithms for both numerical solutions of LODEs and rigorous computation of the approximation error. Finally, several representative examples show the advantages of our algorithms as well as their theoretical and practical limits. [ABSTRACT FROM AUTHOR]

In this paper a novel numerical scheme for finding the sparse self-localized states of a nonlinear system of equations with missing spectral data is introduced. As in the Petviashivili's and the spectral renormalization method, the governing equation is transformed into Fourier domain, but the iterations are performed for far fewer number of spectral components (M) than classical versions of the these methods with higher number of spectral components (N). After the converge criteria is achieved for M components, N component signal is reconstructed from M components by using the l1 minimization technique of the compressive sampling. This method can be named as compressive spectral renormalization (CSRM) method. The main advantage of the CSRM is that, it is capable of finding the sparse self-localized states of the evolution equation(s) with many spectral data missing. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods. [ABSTRACT FROM AUTHOR]

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we present a new spectral residual method for solving large-scale positive definite generalized eigenvalue problems. The proposed algorithm is equipped with a dwindling multidimensional nonmonotone filter, in which the envelope dwindles as the step length decreases. We have also employed a relaxed nonmonotone line search technique in the structure of the algorithm which allows it to enjoy the nonmonotonicity from scratch. Under some mild and standard assumptions, the global convergence property of the proposed algorithm is established. An implementation of the new algorithm on some test problems shows the efficiency and effectiveness of the proposed algorithm in practice. [ABSTRACT FROM AUTHOR]

In this paper, simulation of heat transfer in a heat sink with macroscopic and microscopic scales when one pin-fin is added to the system is investigated by the proposed spectral method. In the microscopic problems, heat transfer model uses dual-phase lag formulas in contrast with macroscopic problems when Fourier law is used to formulate the governing equation. In macroscopic problem, the results are compared with COMSOL multiphysics software results and a good agreement between the results are shown. In microscopic problems, 3D Gaussian heat source is used and boundary conditions obey the Newton law. Comparisons show the efficiency of the current method, while the results are compared with existed literature. It is shown that: by adding one pin fin to the heat sink problem, temperature of the system will decrease. Numerical results are given and graphs of the temperature for various kinds of pin-fin heat sink problems are depicted. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to present and analyze a numerical algorithm for the solution of eighth-order boundary value problems. The proposed solutions are spectral and they depend on a new operational matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the operational matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We solve the Dirichlet and mixed Dirichlet-Neumann boundary value problems for the variable coefficient Cauchy-Navier equations of elasticity in a square using a Legendre spectral Galerkin method. The resulting linear system is solved by the preconditioned conjugate gradient (PCG) method with a preconditioner which is shown to be spectrally equivalent to the matrix of the resulting linear system. Numerical tests demonstrating the convergence properties of the scheme and PCG are presented. [ABSTRACT FROM AUTHOR]

CFD Julia is a programming module developed for senior undergraduate or graduate-level coursework which teaches the foundations of computational fluid dynamics (CFD). The module comprises several programs written in general-purpose programming language Julia designed for high-performance numerical analysis and computational science. The paper explains various concepts related to spatial and temporal discretization, explicit and implicit numerical schemes, multi-step numerical schemes, higher-order shock-capturing numerical methods, and iterative solvers in CFD. These concepts are illustrated using the linear convection equation, the inviscid Burgers equation, and the two-dimensional Poisson equation. The paper covers finite difference implementation for equations in both conservative and non-conservative form. The paper also includes the development of one-dimensional solver for Euler equations and demonstrate it for the Sod shock tube problem. We show the application of finite difference schemes for developing two-dimensional incompressible Navier-Stokes solvers with different boundary conditions applied to the lid-driven cavity and vortex-merger problems. At the end of this paper, we develop hybrid Arakawa-spectral solver and pseudo-spectral solver for two-dimensional incompressible Navier-Stokes equations. Additionally, we compare the computational performance of these minimalist fashion Navier-Stokes solvers written in Julia and Python.

We consider a variational formulation of the three dimensional Navier–Stokes equations provided with mixed boundary conditions. We write this formulation with three independent unknowns: the vorticity, the velocity and the pressure. Next, we propose a discretization by spectral methods. A detailed numerical analysis leads to a priori error estimates for the three unknowns. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical method for solving fractional integro-differential equations with nonlocal boundary conditions using Bernstein polynomials. Some theoretical considerations regarding fractional order derivatives of Bernstein polynomials are discussed. The error analysis is carried out and supported with some numerical examples. It is shown that the method is simple and accurate for the given problem. [ABSTRACT FROM AUTHOR]

In this work we present a new numerical method for the solution of the distributed order time-fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed. [ABSTRACT FROM AUTHOR]

In this paper a novel operational matrix of derivatives of certain basis of Legendre polynomials is established. We show that this matrix is expressed in terms of the harmonic numbers. Moreover, it is utilized along with the collocation method for handling initial value problems of any order. The convergence and the error analysis of the proposed expansion are carefully investigated. Numerical examples are exhibited to confirm the reliability and the high efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Spectral analysis of the self-potential (SP) field for geometrically simple anomalous bodies is studied. In particular, three spectral techniques, i.e. Periodogram (PM), Multi Taper (MTM) and Maximum Entropy (MEM) methods, are proposed to derive the depth of the anomalous bodies. An extensive numerical analysis at varying the source parameters outlines that MEM is successful in determining the source depth with a percent error less than 5%. The application of the proposed spectral approach to the interpretation of field datasets has provided depth estimations of the SP anomaly sources in very good agreement with those obtained by other numerical methods. [ABSTRACT FROM AUTHOR]

We are interested in a mixed reaction diffusion system describing the organic pollution in stream-waters. In this work, we propose a mixed-variational formulation and recall its well-posedness. Next, we consider a spectral discretization of this problem and establish nearly optimal error estimates. Numerical experiments confirm the interest of this approach. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the (3-α)-order convergence of the time scheme, where α is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the (3 - α)-order scheme for the time-fractional diffusion equation. [ABSTRACT FROM AUTHOR]

We propose a numerical solution for the solution of the Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein-Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher-KPP stochastic equation and a stochastic Burgers equation in dimension 1. [ABSTRACT FROM AUTHOR]

In this paper, we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping. Thanks to the efficiency and the accuracy of spectral method, we can check that discrete energy decreases to zero as time goes to infinity, uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration. We prove the convergence of the full Fourier–Galerkin discretization. Thus, we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions. [ABSTRACT FROM AUTHOR]

This paper proposes an efficient method for the computation of Galois field (GF) expressions for multiple-valued logic functions. The algorithm is based on the partitioning of the input function vector and uses both CPUs (central processing units) and GPUs (graphics processing units) for performing the computations in parallel. After the first step of the fast Fourier transform (FFT)-like algorithm is performed on the CPU, the function vector is divided into disjoint subvectors that are further processed in parallel on the CPU and GPU. The proposed computational method reduces the time needed for computing the coefficients in the GF-expressions and, in this way, might extend the possibilities for their practical application. The experimental comparison of the proposed solution and previously used methods for computing GFexpressions for ternary and quaternary functions, confirms the validity of the method. [ABSTRACT FROM AUTHOR]

In the last thirty years, the Chebyshev points of the first kind have not been given as much attention for numerical applications as the second-kind ones. This survey summarizes theorems and algorithms for first-kind Chebyshev points with references to the existing literature. Benefits from using the first-kind Chebyshev points in various contexts are discussed. [ABSTRACT FROM AUTHOR]

In this work we apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases. [ABSTRACT FROM AUTHOR]

We combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure-valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure-valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness of measure-valued solutions as a solution framework for the Euler equations. Furthermore, we report an extensive computational study of the two-dimensional vortex sheet, which indicates that the computed measure-valued solution is non-atomic and implies possible non-uniqueness of weak solutions constructed by Delort. [ABSTRACT FROM AUTHOR]

Approximation properties of divergence-free vector fields by global and local solenoidal bases are studied. A comparison between interpolants generated with radial kernels and multivariate polynomials is presented. Numerical results show higher rates of convergence for derivatives of the vector field being approximated in directions enforced by the divergence operator when a rectangular grid is used. We also compute the growth of Lebesgue constants for uniform and clustered nodes and study the flat limit of divergence-free interpolants based on radial kernels. Numerical results are presented for two- and three-dimensional vector fields. [ABSTRACT FROM AUTHOR]

The paper introduces a new concept for composing the input shapers by combining lumped and distributed delays. Particularly, an analytical method is proposed to parametrize zero vibration shapers with delays of various distribution. Involving these delays in the shaper structure, the accommodation part of the shaped response can be smoothened in a predefined manner, retarded spectral properties can be achieved and the response length can be optionally adjusted within the given range. Next, following the analytical developments, a numerical algorithm based on the constrained linear least squares optimization is proposed for direct design of robust distributed delay shapers. [ABSTRACT FROM AUTHOR]

In this paper we show numerically that the main characteristics of a dust devil-like vortex: vertical vorticity generation, eye formation, and tilting of the eye/axis of rotation, can be explained by thermoconvective mechanisms. By considering a cylinder non-homogeneously heated from below we prove that an intense localized heating on the ground generates a convective stationary axisymmetric flow that begins to spiral up around a central axis when perturbation vertical vorticity is permitted and a critical vertical temperature gradient is exceeded, thus forming an axisymmetric vortex. If the intense heating on the ground is not too localized and the temperature gradient continues increasing, central downdrafts appear in the vortex and an eye is formed. We show that the axisymmetric vortex loses stability towards a new state for which the axisymmetry is broken, the axis of rotation or proper eye displaces from the center and tilts. The vortical states found are comparable to dust devils. These findings establish the relevance of thermoconvection on the formation and evolution of these atmospheric phenomena. [ABSTRACT FROM AUTHOR]

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalised Klein–Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets. [ABSTRACT FROM AUTHOR]

This paper presents a novel constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with a constrained backpropogation training approach to obtain an artificial neural network representation of the PDE solution that approximately satisfies the boundary conditions at every integration step. The advantage of CINT over existing methods is that it is readily applicable to solving PDEs on irregular domains, and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. In this paper the CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems with a known analytical solutions that can be used for performance comparison. The numerical results show that, when compared to the most efficient finite element methods, the CINT method achieves significant improvements both in terms of computational time and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the time-fractional diffusion equation, JCP 2007] for the time-fractional diffusion equation. It is a method basing on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of (2 -α)-order convergence in time and spectral accuracy in space for smooth solutions, where αis the time-fractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests. [ABSTRACT FROM AUTHOR]

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of $$\mathbb {R}^n$$ . The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, FitzHugh-Nagumo and Gray-Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator. [ABSTRACT FROM AUTHOR]