This paper presents two new non-classical Lagrange basis functions which are based on the new Jacobi-Müntz functions presented by the authors recently. These basis functions are, in fact, generalizations form of the newly generated Jacobi based functions. With respect to these non-classical Lagrange basis functions, two non-classical interpolants are introduced and their error bounds are proved in detail. The pseudo-spectral differentiation (and integration) matrices have been extracted in two different manners. Some numerical experiments are provided to show the efficiency and capability of these newly generated non-classical Lagrange basis functions.
In this paper, we introduce a new deep learning framework for discovering the phase field models from existing image data. The new framework embraces the approximation power of physics informed neural networks (PINN), and the computational efficiency of the pseudo-spectral methods, which we named pseudo-spectral PINN or SPINN. Unlike the baseline PINN, the pseudo-spectral PINN has several advantages. First of all, it requires less training data. A minimum of two snapshots with uniform spatial resolution would be adequate. Secondly, it is computationally efficient, as the pseudo-spectral method…
In this paper, we propose a linearized Fourier pseudo-spectral method, which preserves the total mass and energy conservation laws, for the damped nonlinear Schrödinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal $L^2$-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation methods based on Legendre and Chebyshev polynomials. The time integration is carried out suitably with robust schemes attending to qualitative features such as stiffness and preservation of strong stability to simulate nonregular problems more correctly. The corresponding semidiscrete and fully discrete schemes are described and the performance of the methods is …
In this paper, we design a novel linearized and momentum-preserving Fourier pseudo-spectral scheme to solve the Rosenau-Korteweg de Vries equation. With the aid of a new semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, a prior bound of the numerical solution in discrete $L^{\infty}$-norm is obtained from the discrete momentum conservation law. Subsequently, based on the energy method and the bound of the numerical solution, we show that, without any restriction on the mesh ratio, the scheme is convergent with order $O(N^{-s}+τ^2)$ in discrete $…
We devise a spectral divide-and-conquer scheme for matrices that are self-adjoint with respect to a given indefinite scalar product (i.e. pseudosymmetic matrices). The pseudosymmetric structure of the matrix is preserved in the spectral division, such that the method can be applied recursively to achieve full diagonalization. The method is well-suited for structured matrices that come up in computational quantum physics and chemistry. In this application context, additional definiteness properties guarantee a convergence of the matrix sign function iteration within two steps when Zolotarev fun…
The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov--Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang …
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the, necessarily finite, computational domain. To obtain an efficient numerical scheme we discretize space using a spectral method. This allows us to drastically reduce the number of grid points required for a given accuracy. Applying a fully implicit time integrator, however, would require us to invert full matrices. This is addressed by performing an operator spli…
Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation
Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；College of Science, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院）
In this paper, we consider the error analysis of a conservative Fourier pseudo-spectral method that conserves mass and energy for the space fractional nonlinear Schrödinger equation. We give a new fractional Sobolev norm that can construct the discrete fractional Sobolev space, and we also can prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo-spectral method is unconditionally convergent with order $O(τ^{2}+N^{α/2-r})$ in the disc…
Numerical simulation of viscoelastic flows is challenging because of the hyperbolic nature of viscoelastic constitutive equations. Despite their superior accuracy and efficiency, pseudo-spectral methods require the introduction of artificial diffusion (AD) for numerical stability in hyperbolic problems, which alters the physical nature of the system. This study presents a hybrid numerical procedure that integrates an upwind total variation diminishing (TVD) finite-difference scheme, which is known for its stability in hyperbolic problems, for the polymer stress convection term into an overall …
Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme …
Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge--Kutta pair of Bogacki \& Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Ra…
Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $ν>0,$ in the Hörmander class $Ψ^ν_{ρ,δ}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $ω\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral ineq…
Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations
University College Dublin, School of Mathematical Sciences（都柏林大学学院数学科学学院）；LAMA, UMR 5127 CNRS, Université de Savoie（拉马实验室，法国国家科学研究中心联合研究单位5127，萨瓦大学）；Laboratoire J.-A. Dieudonné, Université de Nice – Sophia Antipolis（J.-A.迪厄多内实验室，尼斯-索菲亚安蒂波利斯大学）；Deptartment of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；University of California, Merced（加州大学默塞德分校）
