Space-time Spectral Method Research Articles

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

A step-by-step Chebyshev space-time spectral method for force vibration of functionally graded structures

A step-by-step Chebyshev space-time spectral method for force vibration of functionally graded structures

Adjoint-based optimisation of time- and span-periodic flow fields with Space–Time Spectral Method: Application to non-linear instabilities in compressible boundary layer flows

We aim at computing time- and span-periodic flow fields in span-invariant configurations. The streamwise and cross-stream derivatives are discretised with finite volumes while time and the span-direction are handled with pseudo-spectral Fourier-collocation methods. Doing so, we extend the classical Time Spectral Method (TSM) to a Space–Time Spectral Method (S-TSM), by considering non-linear interactions of a finite number of time and span harmonics. For optimisation, we introduce an adjoint-based framework that allows efficient computation of the gradient of any cost functional with respect to a large-dimensional control parameter. Both theoretical and numerical aspects of the methodology are described: evaluation of matrix–vector products with S-TSM Jacobian (or its transpose) by algorithmic differentiation, solution of fixed-points with quasi-Newton method and de-aliasing in time and space, solution of direct and adjoint linear systems by iterative algorithms with a block-circulant preconditioner, performance assessment in CPU time and memory. We illustrate the methodology on the case of 3D instabilities (first Mack mode) triggered within a developing adiabatic boundary layer at M = 4.5. A gradient-ascent method allows to identify a finite-amplitude 3D forcing that triggers a non-linear response exhibiting the strongest time- and span-averaged drag on the flat-plate. In view of flow control, a gradient-descent method finally determines a finite amplitude 2D wall-heat flux that minimises the averaged drag of the plate in presence of the previously determined non-linear optimal forcing.

Determination of an unknown coefficient in the Korteweg–de Vries equation

Abstract In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

Error estimates of a space–time Legendre spectral method for solving the Korteweg–de Vries equation

In this paper, a space–time spectral method for solving the Korteweg–de Vries equation is considered. The discrete schemes of the method are based on the Legendre–Petrov–Galerkin method in spatial direction and the Legendre-tau method in temporal direction with nonperiodic boundary conditions. Stability analysis results and error estimates are obtained in L2-norm by introducing a cut-off function without Lipschitz condition. The method is also applicable to solve some (2m+1)th-order differential equations. Comparison of our numerical results with those of other spectral methods exhibits the accuracy of our methods for the Korteweg–de Vries equation.

Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.

A high-order space-time spectral method for the distributed-order time-fractional telegraph equation

A high-order space-time spectral method for the distributed-order time-fractional telegraph equation

A space-time spectral method for solving the nonlinear Klein-Gordon equation

A space-time spectral method for the nonlinear Klein-Gordon equation is proposed. We use the Legendre-Galerkin method in space and Legendre-Petrov-Galerkin method in time. The nonlinear term in the equation is dealt with the Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. The stability and the optimal error estimates in L2-norm for the single and multiple interval methods are given. Numerical experiments are shown confirming the theoretical results.

A Space-Time Legendre-Petrov-Galerkin Method for Third-Order Differential Equations

In this article, a space-time spectral method is considered to approximate third-order differential equations with non-periodic boundary conditions. The Legendre-Petrov-Galerkin discretization is employed in both space and time. In the theoretical analysis, rigorous proof of error estimates in the weighted space-time norms is obtained for the fully discrete scheme. We also formulate the matrix form of the fully discrete scheme by taking appropriate test and trial functions in both space and time. Finally, extensive numerical experiments are conducted for linear and nonlinear problems, and spectral accuracy is derived for both space and time. Moreover, the numerical results are compared with those computed by other numerical methods to confirm the efficiency of the proposed method.

Space-time spectral method for the Stokes problem

The Stokes equations are a linearized version of the Navier-Stokes equations and model incompressible viscous fluid flow with low Reynolds numbers. Several spectral methods, exhibiting exponential decay in error when the solution is analytic, are known to solve the steady-state Stokes problem numerically. A common strategy to solve such a problem in the time-dependent case involves extending the spectral scheme in spatial derivatives by implementing a low-order finite difference scheme for the time derivatives. Instead, we implement and analyze a space-time spectral method for the Stokes problem, which converges exponentially in both space and time. This numerical scheme imposes spectral collocation in time and PN−PN−2 spectral Galerkin scheme in space by using a recombined Legendre polynomial basis, resulting in a global spectral operator that is a saddle point matrix. The main objectives of the research are estimating the condition number of the global spectral operators and proving the spectral convergence of this scheme in space and time. The analysis is not quite complete because two of the estimates are based on numerical evidence. However, throughout the project, some intermediate results-such as the 2-norm of the pseudospectral derivative matrix for Chebyshev-Gauss-Lobatto nodes as well as the condition number of the mass matrix and discrete Laplacian for a recombined Legendre basis-were proved to obtain the aforementioned findings. Numerical experiments of this scheme verify the theoretical results. Also, the numerical result of applying this scheme to the Navier-Stokes equations is presented.

A Legendre-tau-Galerkin method in time for two-dimensional Sobolev equations

<abstract><p>This work is devoted to present the Legendre space-time spectral method for two-dimensional (2D) Sobolev equations. Considering the asymmetry of the first-order differential operator, the Legendre-tau-Galerkin method is employed in time discretization and its multi-interval form is also investigated. In the theoretical analysis, rigorous proof of the stability and $ L^2(\Sigma) $-error estimates is given for the fully discrete schemes in both single-interval and multi-interval forms. Being different from the general Legendre-Galerkin method, we specifically take the Fourier-like basis functions in space to save the computing time and memory in the algorithm of the proposed method. Numerical experiments were included to confirm that our method attains exponential convergence in both time and space and that the multi-interval form can achieve improved numerical results compared with the single interval form.</p></abstract>

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

<p style='text-indent:20px;'>In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.</p>

A Space-Time Spectral Method for the Inverse Source Problems of the Damped Wave Equation with Memory

A Space-Time Spectral Method for the Inverse Source Problems of the Damped Wave Equation with Memory

Error estimate of a Legendre-Galerkin Chebyshev collocation method for a class of parabolic inverse problem

A Legendre-Galerkin Chebyshev collocation method is presented for the parabolic inverse problem with control parameters. Optimal order of convergence of the semi-discrete method is obtained in L2-norm for the nonlinear term being not globally Lipschitz continuous. For time-discretization, a Legendre-tau method is applied. The method is implemented by the explicit-implicit iterative method. Suitable basis functions are constructed leading to sparse matrices, and the nonlinear term is collocated at the Chebyshev-Gauss-Lobatto points computed explicitly by the fast Legendre transform. Numerical results are given to show the efficiency and capability of this space-time spectral method.

A space-time spectral method for time-fractional Black-Scholes equation

The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence of the numerical scheme are analyzed. Two numerical examples are considered to validate the accuracy and illustrate the practicability of the proposed method. The results agree with the theoretical analysis and this approach can be applied in dealing with option pricing models with smooth payoff functions.

A space-time spectral method for multi-dimensional Sobolev equations

In this paper, a space-time spectral method is applied to approximate the linear multi-dimensional Sobolev equations. The Legendre-Galerkin method and a dual-Petrov-Galerkin discretization are employed in space and time, respectively. Being different from the general Legendre-Galerkin method, the technique adopted in this study lies in diagonalization of the mass matrix by using Fourier-like basis functions to save the computing time and memory. Moreover, the detailed stability analysis and error estimates are provided in the weighted space-time norms. We also formulate the matrix forms of the fully discrete scheme. Finally, extensive numerical tests are implemented to verify the theoretical results and demonstrate the efficiency of our method.

A space-time spectral method for the 1-D Maxwell equation

<abstract> A Legendre-tau space-time spectral method is established for the 1-D Maxwell equation. The polynomials of different degrees are used to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. Also, the time multi-interval Legendre-tau space-time spectral method is considered to keep the long-time computation stable. Error estimates for the method of single and multi-internal are given, respectively. Moreover, the space-time spectral method is applied to the numerical solutions of the 1-D nonlinear Maxwell equation and describes its implicit-explicit iteration scheme. Numerical examples are compared with some other methods, which verifies the effectiveness of the methods for the 1-D Maxwell equation. </abstract>

Space-time spectral method for the Cattaneo equation with time fractional derivative

This paper introduces a high-order accurate numerical method for solving the Cattaneo equation with time fractional derivative. It is based on the Galerkin–Legendre spectral method in space and the Chebyshev collocation method in time. Arbitrarily high-order accurate can be made in both space and time. Optimal priori error bound of the semi-discrete method and the stability and convergence of the full-discrete method are strictly given. Extensive experimental results confirm the theoretical claims of this method in both space and time.

New space–time spectral and structured spectral element methods for high order problems

We propose new space–time spectral and structured spectral element methods for high order problems. By matrix decomposition, we introduce a family of new basis functions which leads to a diagonal system for the fourth order problems with constant coefficients. Based on new basis functions in space and a dual-Petrov–Galerkin formulation in time, we propose a new space–time spectral method for a time-dependent problem, and proved their spectral accuracy both in space and time. Numerical results demonstrate their high efficiency and coincide well with theoretical analysis. Further, to ultimately promote the numerical performance and efficiency, we exploit the idea of locally simultaneous diagonalization to provide new structured spectral element methods for high-order problems. Besides, to increase the flexibility, a C 1 -conforming rectangular spectral method in analogy to Argyris or Bell triangular elements are proposed for fourth-order equations, which serves as a preparative work towards conforming quadrilateral spectral elements for high-order equations.

Error estimates of the space–time spectral method for parabolic control problems

This work is devoted to investigate the spectral approximation of optimal control of parabolic problems. The space–time method is used to boost high-order accuracy by applying dual Petrov–Galerkin spectral scheme in time and spectral method in space. The optimality conditions are derived, and the a priori error estimates indicate the convergence of the proposed method. Numerical tests confirm the theoretical results, and show the efficiency of the method.