arXiv subjects

Yaolin Jiang

Publications and source records attributed to Yaolin Jiang.

10 recordsLinked to original sources

arXiv2025

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more efficient than the implicit integrators of the same order of accuracy. Up to now, all responses to this problem are negative. That is, there exist explicit symplectic integrators only for some special nonseparable Hamiltonian systems, whereas the universal design involving explicit symplectic integrators for general nonseparable Hamiltonian systems has not yet been studied sufficiently. In this paper, we present a constructive proof for the existence of explicit symplectic integrators for general nonseparable Hamiltonian systems via finding explicit symplectic mappings under which the special submanifold of the extended phase space is invariant. It turns out that the proposed explicit integrators are symplectic in both the extended phase space and the original phase space. Moreover, on the basis of the global modified Hamiltonians of the proposed integrators, the backward error analysis is made via a parameter relaxation and restriction technique to show the linear growth of global errors and the near-preservation of first integrals. In particular, the effective estimated time interval is nearly the same as classical implicit symplectic integrators when applied to (near-) integrable Hamiltonian systems. Numerical experiments with a completely integrable nonseparable Hamiltonian and a nonintegrable nonseparable Hamiltonian illustrate the good long-term behavior and high efficiency of the explicit symplectic integrators proposed and analyzed in this paper.

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arXiv2024

A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation

In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the employment of a twisted function tailored for trigonometric integrals. Robust error analysis is conducted, demonstrating that the proposed scheme achieves third-order accuracy in the energy space under a weak regularity requirement in $H^{1+\max(\mu,1)}(\mathbb{T}^d)\times H^{\max(\mu,1)}(\mathbb{T}^d)$ with $\mu> \frac{d}{2}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than some well-known exponential integrators of order three for approximating the Klein-Gordon equation with non-smooth solutions.

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arXiv2023

Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator

In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures its good long-time energy, momentum and action conservations which are rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods.

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arXiv2023

Two-scale exponential integrators with uniform accuracy for three-dimensional charged-particle dynamics under strong magnetic field

The numerical simulation of three-dimensional charged-particle dynamics (CPD) under strong magnetic field is a basic and challenging algorithmic task in plasma physics. In this paper, we introduce a new methodology to design two-scale exponential integrators for three-dimensional CPD whose magnetic field's strength is inversely proportional to a dimensionless and small parameter $0<\varepsilon \ll 1$. By dealing with the transformed form of three-dimensional CPD, we linearize the magnetic field and put the residual component in a new nonlinear function which is shown to be uniformly bounded. Based on this foundation and the proposed two-scale exponential integrators, a class of novel integrators is formulated and studied. The corresponding uniform accuracy of the proposed $r$-th order integrator is shown to be $\mathcal{O}(h^r)$, where $r=1,2,3,4$ and the constant symbolized by $\mathcal{O}$, the time stepsize $h$ and the computation cost are all independent of $\varepsilon$. Moreover, in the case of maximal ordering strong magnetic field, improved error bound $\mathcal{O}(\varepsilon^r h^r)$ is obtained for the proposed $r$-th order integrator. A rigorous proof of these uniform and improved error bounds is presented, and a numerical test is performed to illustrate the error and efficiency behaviour of the proposed integrators.

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arXiv2022

Time exponential integrator Fourier pseudospectral methods with high accuracy and multiple conservation laws for three-dimensional Maxwell's equations

Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining time exponential integrator and Fourier pseudospectral methods. Fast computation is implemented in the scheme by using the Fast Fourier Transform algorithm which is well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and infinite-order accuracy in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

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arXiv2022

Improved Bound on Vertex Degree Version of Erdős Matching Conjecture

For a $k$-uniform hypergraph $H$, let $δ_1(H)$ denote the minimum vertex degree of $H$, and $ν(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $β>0$, there exists an integer $n_0(β,k)$ such that for positive integers $n\geq n_0$ and $m\leq (\frac{k}{2(k-1)}-β)\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with $δ_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $ν(H)\geq m$. This improves upon earlier results of Bollobás, Daykin and Erdős (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.

math.CO↗

arXiv2022

Two-scale integrators with high accuracy and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

In this paper, we are concerned with two-scale integrators for the non-relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation {in the form of $\partial_{tt}u -\frac{\Delta}{\eps^2} u +\frac{1}{\eps^4}u +\frac{\lambda}{\varepsilon^2} f(u)=0$} {has a factor $1/\varepsilon^2$ in front of the nonlinearity which means that this part becomes strong when $\varepsilon$ is small. These} two aspects bring significantly numerical burdens in designing numerical methods. {We propose a class of two-scale integrators which is constructed based on some reformulations to the system, Fourier pseudo-spectral method and exponential integrators.} Two practical integrators up to order three and four are constructed by using some symmetric conditions and the stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the uniform accuracy in time is $\mathcal{O}(h^3)$ and $\mathcal{O}(h^4)$ for the time stepsize $h$. The near energy conservation over long times is also established for the multi-stage integrators by using modulated Fourier expansions.

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arXiv2022

Semi-discretization and full-discretization with optimal accuracy for charged-particle dynamics in a strong nonuniform magnetic field

The aim of this paper is to formulate and analyze numerical discretizations of charged-particle dynamics (CPD) in a strong nonuniform magnetic field. A strategy is firstly performed for the two dimensional CPD to construct the semi-discretization and full-discretization which have optimal accuracy. This accuracy is improved in the position and in the velocity when the strength of the magnetic field becomes stronger. This is a better feature than the usual so called "uniformly accurate methods". To obtain this refined accuracy, some reformulations of the problem and two-scale exponential integrators are incorporated, and the optimal accuracy is derived from this new procedure. Then based on the strategy given for the two dimensional case, a new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case. All the theoretical results of the accuracy are numerically illustrated by some numerical tests.

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arXiv2021

Structure-preserving algorithms with uniform error bound and long-time energy conservation for highly oscillatory Hamiltonian systems

Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear Hamiltonian systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Moreover, the proposed algorithms are shown to have uniform error bound for the highly oscillatory structure. A numerical experiment is carried out to support the theoretical results established in this paper by showing the performance of the blended algorithms.

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arXiv2021

Optimal convergence and long-time conservation of exponential integration for Schrödinger equations in a normal or highly oscillatory regime

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schrödinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of actions, momentum and density will be formulated and analysed. To this end, we derive continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. It is shown that these integrators exhibit optimal convergence and have near conservations of actions, momentum and density over long times. A numerical experiment is carried out to support all the theoretical results presented in this paper. Some applications of the integrators to other kinds of ordinary/partial differential equations are also presented.

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