Pseudospectral Approximation Research Articles

Solitary-Wave Solutions of the Fractional Nonlinear Schrödinger Equation: I—Existence and Numerical Generation

The present paper is the first part of a project devoted to the fractional nonlinear Schrödinger (fNLS) equation. It is concerned with the existence and numerical generation of the solitary-wave solutions. For the first point, some conserved quantities of the problem are used to search for solitary-wave solutions from a constrained critical point problem and the application of the concentration-compactness theory. Several properties of the waves, such as the regularity and the asymptotic decay in some cases, are derived from the existence result. Some other properties, such as the monotone behavior and the speed-amplitude relation, will be explored computationally. To this end, a numerical procedure for the generation of the profiles is proposed. The method is based on a Fourier pseudospectral approximation of the differential system for the profiles and the use of Petviashvili’s iteration with extrapolation.

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Numerical analysis of a multistable capsule system under the delayed feedback control with a constant delay

The vibro-impact capsule system is a self-propelled mechanism that has abundant coexisting attractors and moves rectilinearly under periodic excitation when overcoming environmental resistance. In this paper, we study the control of coexisting attractors in this system by using a delayed feedback controller (DFC) with a constant delay. The aim of our control is to steer this complex system toward an attractor with preferable performance characteristics among multiple coexisting attractors, e.g., a periodically fast forward progression. For this purpose, we give an example of a feedback-controlled transition from a period-3 motion with low progression speed to a period-1 motion with high progression speed at the system parameters where both responses coexist. The effectiveness of this controller is investigated numerically by considering its convergence time and the required control energy input to achieve transition. We combine pseudo-spectral approximation of the delay, event detection for the discontinuities and path-following (continuation) techniques for non-smooth delay dynamical systems to carry out bifurcation analysis. We systematically study the dynamical performance of the controlled system when varying its control gain and delay time. Our numerical simulations show the effectiveness of DFC under a wide range of system parameters. We find that the desired period-1 motion is achievable in a range of control delays between a period-doubling and a grazing bifurcation. Therefore, two-parameter continuation of these two bifurcations with respect to the control delay and control gain is conducted to identify the delay-gain parameter region where the period-1 motion is stable. The findings of this work can be used for tuning control parameters in experiments, and similar analysis can be carried out for other non-smooth dynamical systems with a constant delay term.

Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems

<abstract><p>A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices.</p></abstract>

Preconditioned Pseudo-Spectral Gradient Flow for Computing the Steady-State of Space Fractional Cahn-Allen Equations With Variable Coefficients

The aim of this paper is to propose some efficient and accurate numerical methods to compute the steady-state of variable coefficients space fractional Cahn-Allen equations. The approach combines an adaptive time stepping semi-implicit gradient flow method to minimize the fractional energy functional and pseudo-spectral approximation schemes. Based on the use of a preconditioned GMRES, the space fractional Cahn-Allen equation is then solved efficiently. The full methodology is supported by the numerical solution of a one-dimensional problem.

An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus

We analyze the behavior of an ensemble of time integrators applied to the semi-discrete problem resulting from the spectral discretization of the equations describing Boussinesq thermal convection in a cylindrical annulus. The equations are cast in their vorticity-streamfunction formulation that yields a differential algebraic equation (DAE). The ensemble comprises 28 members: 4 implicit-explicit multistep schemes, 22 implicit-explicit Runge-Kutta (IMEX-RK) schemes, and 2 fully explicit schemes used for reference. The schemes whose theoretical order varies from 2 to 5 are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEX-RK methods exhibit their expected order of convergence under all circumstances. IMEX-RK methods of higher-order show occasional order reduction that impacts both algebraic and differential field variables. We ascribe the order reduction to the stiffness of the problem at hand and, to a larger extent, the presence of the DAE. Using the popular Crank-Nicolson Adams-Bashforth of order 2 (CNAB2) integrator as reference, performance is defined by the ratio of maximum admissible time step to the cost of performing one iteration; the maximum admissible time step is determined by inspection of the time series of viscous dissipation within the system, which guarantees a physically acceptable solution. Relative performance is bounded between 0.5 and 1.5 across all studied configurations. Considering accuracy jointly with performance, we find that 6 schemes consistently outperform CNAB2, meaning that in addition to allowing for a more efficient calculation, the accuracy that they achieve at their operational, dissipation-based limit of stability yields a lower error. In our most turbulent setup, where the behavior of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.

An improved pseudospectral approximation of coupled nonlinear partial differential equations

An improved pseudospectral approximation of coupled nonlinear partial differential equations

Convergence analysis of the energy-stable numerical schemes for the Cahn-Hilliard equation

In this paper we present a second order numerical scheme for the Cahn-Hilliard equation, with a Fourier pseudo-spectral approximation in space. An additional Douglas-Dupont regularization term is introduced, which ensures the energy stability. The bound of numerical solution in H2h and l? norms are obtained at a theoretical level. Moreover, for the global nature of the pseudo-spectral method, we propose a linear iteration algorithm to solve the non-linear system, due to the implicit treatment for the non-linear term. Some numerical simulations verify the efficiency of iteration algorithm.

An improved pseudospectral approximation of coupled nonlinear partial differential equations

An improved pseudospectral approximation of coupled nonlinear partial differential equations

Exponential integrator preserving mass boundedness and energy conservation for nonlinear Schrödinger equation

In this paper, a novel linearly implicit conservative scheme is developed for the two-dimensional nonlinear Schrödinger equation. The scheme is based on a new linearly implicit exponential time differencing method for the temporal discretization and the Fourier pseudo-spectral approximation for the spatial derivative. By rigorous analysis, we prove that the proposed scheme can preserve the energy conservation. Furthermore, although in fact the proposed scheme does not preserve mass conservation, we can prove that it can preserve the discrete L2 norm boundedness of the numerical solution, which may be helpful for the analysis of unconditional convergence of the energy conservative scheme. Then an optimal error estimate of the proposed scheme is established without any restriction on the grid ration. Numerical experiments show that the proposed scheme is remarkable efficiency comparing with some other existing structure-preserving schemes.

Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

We develop fully discrete structure-preserving numerical algorithms of arbitrarily high order for the Allen-Cahn model with a nonlocal constraint subject to the Neumann boundary condition. Using the energy quadratization methodology, we reformulate the thermodynamically consistent model into an equivalent one with a quadratic free energy. For the reformulated model, we first apply a Cosine pseudo-spectral approximation in space to arrive at a semi-discrete system that inherits the volume conservation and energy dissipative property; then we use two distinct temporal discretization methods to derive fully discrete schemes of arbitrarily higher order. One is based on the symplectic Runge-Kutta (RK) method and the other is a linearized Runge-Kutta method by the prediction-correction strategy. The fully discrete schemes preserve both volume and the energy dissipative property. In addition, we show that the liner system resulting from the schemes warrants the unique solvability. A fast solver combined with the discrete Cosine transform (DCT) is exploited to implement the high-order scheme efficiently. Extensive numerical examples are presented to show the efficiency and accuracy of the newly proposed methods.

A New Adaptive Sparse Pseudospectral Approximation Method and its Application for Stochastic Power Flow

With the increasing integration of various renewable energy resources into electric power systems, the stochastic power flow calculation, which falls into the uncertainty propagation (UP) problems, has attracted renewed interest recently. The adaptive sparse pseudospectral approximation method (A-SPAM) is highly efficient to solve UP problems. However, there still exist two major drawbacks: 1) its global error estimate is not accurate enough to evaluate the actual approximation error; and 2) it is intended for single-output systems and there is no efficient method for its application in multi-output systems. In order to overcome these two defects, in this study a new adaptive sparse pseudospectral approximation method (NA-SPAM) is proposed based on two improvements on A-SPAM. First, a new global error estimate is presented which features much higher estimation accuracy. Second, a revised version of error estimates is given based on unification of the index set, which apparently improves the calculation efficiency for multi-output problems. At last, the effectiveness of the two improvements and NA-SPAM is validated by mathematical and practical cases.

Numerical bifurcation analysis of renewal equations via pseudospectral approximation

We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

Long time [formula omitted] stability of a classical scheme for Cahn-Hilliard equation with polynomial nonlinearity

In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in H−1 and Hαs (s=1,2,3) spaces independent of time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes. The uniform Hα3 bound is obtained on basis of the uniform H1 estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.

Pseudospectral Approximation of Hopf Bifurcation for Delay Differential Equations

Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.

Numerical simulation of advection–diffusion equation with caputo-fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method

The study of the unsteady fractional advection–diffusion equation (ADE) is carried out in cylindrical geometry along with time-exponential concentration on a cylindrical surface. We have used the Caputo-Fabrizio time-fractional derivative for the fractional model of the advection–diffusion. The analytical solutions for the solute concentration are determined by using integral transformations. For comparison, we also present a numerical scheme to get a numerical solution for different parameters' values. For time ordinary derivative approximation, we use the finite difference method. For space derivatives, we use a pseudo-spectral collocation method for higher-order accuracies. The advection-diffusion’ classical model is obtained by taking the fractional parameter to be unit. The influence of the memory, namely, the Caputo-Fabrizio time-fractional derivative on the solute concentration is studied and compared with the ordinary case. Also, the impact of the drift velocity is analyzed by employing the Peclet number. It has been found that the concentration is decreasing with the Peclet number and is increasing with the radial coordinate. The present investigation will be helpful in future research to use a higher-order approximation for ordinary derivatives. As the derivatives in space are ordinary derivatives, we use highly accurate pseudo-spectral collocation approximation for them.

Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.

An iterative algorithm for robust simulation of the Sylvester matrix differential equations

This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given.

Uncertainty Propagation Using Polynomial Chaos Expansions for Extreme Sea Level Hazard Assessment: The Case of the Eastern Adriatic Meteotsunamis

AbstractThis study quantifies the hazard associated with extreme sea levels due to eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances—and assesses the sensitivity of the ocean response to the disturbances responsible for those events. In this spirit, a surrogate model of meteotsunami maximum elevation based on generalized polynomial chaos expansion (gPCE) methods, is implemented. The approach relies on the definition of a synthetic pressure disturbance—depending on six different stochastic parameters known to be important for meteotsunami generation, which is used as forcing to produce series of meteotsunami simulations defined with sparse grid methods (up to 10 689 used in this study). The surrogate model and the sensitivity study are then obtained with a pseudo-spectral approximation (PSA) method based on the chosen meteotsunami simulations. This study mainly presents the developed methodology and discusses the feasibility of implementing such gPCE-based surrogate models to assess the hazard and to study the sensitivity of meteorologically driven extreme sea levels.