arXiv subjects

Fangyan Yao

Publications and source records attributed to Fangyan Yao.

6 recordsLinked to original sources

arXiv2025

Computing rough solutions of the KdV equation below ${\bf L^2}$

We establish a novel numerical and analytical framework for solving the Korteweg--de Vries (KdV) equation in the negative Sobolev spaces, where classical numerical methods fail due to their reliance on high regularity and inability to control nonlinear interactions at low regularities. Numerical analysis is established by combining a continuous reformulation of the numerical scheme, the Bourgain-space estimates for the continuous reformulation, and a rescaling strategy that reduces the reformulated problem to a small initial value problem, which allow us to bridge a critical gap between numerical analysis and theoretical well-posedness by designing the first numerical method capable of solving the KdV equation in the negative Sobolev spaces. The numerical scheme is proved to have nearly optimal-order convergence with respect to the spatial degrees of freedom in the $H^{-\frac{1}{2}}$ norm for initial data in $H^s$, with $-\frac{1}{2} < s \leq 0$, a result unattainable by existing numerical methods.

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arXiv2024

Numerical approximation of discontinuous solutions of the semilinear wave equation

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^γ\times H^{γ-1})$ for $γ\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $τ\sim N^{-1}$, where $τ$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.

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arXiv2024

Convergence of a moving window method for the Schrödinger equation with potential on $\mathbb{R}^d$

We propose a novel framework, called moving window method, for solving the linear Schrödinger equation with an external potential in $\mathbb{R}^d$. This method employs a smooth cut-off function to truncate the equation from Cauchy boundary conditions in the whole space to a bounded window of scaled torus, which is itself moving with the solution. This allows for the application of established schemes on this scaled torus to design algorithms for the whole-space problem. Rigorous analysis of the error in approximating the whole-space solution by numerical solutions on a bounded window is established. Additionally, analytical tools for periodic cases are used to rigorously estimate the error of these whole-space algorithms. By integrating the proposed framework with a classical first-order exponential integrator on the scaled torus, we demonstrate that the proposed scheme achieves first-order convergence in time and $γ/2$-order convergence in space for initial data in $H^γ(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2γ} dx)$ with $γ\geq 2$. In the case where $γ= 1$, the numerical scheme is shown to have half-order convergence under an additional CFL condition. In practice, we can dynamically adjust the window when waves reach its boundary, allowing for continued computation beyond the initial window. Extensive numerical examples are presented to support the theoretical analysis and demonstrate the effectiveness of the proposed method.

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arXiv2021

A second order low-regularity integrator for the nonlinear Schrödinger equation

In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the $d$ dimensional torus $\mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^γ(\mathbb T^d)$ for initial data in $H^{γ+2}(\mathbb T^d)$ for any $γ> d/2$. This improves the previous work in [Knöller, A. Ostermann, and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986]. The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.

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arXiv2021

A fully discrete low-regularity integrator for the nonlinear Schrödinger equation

For the solution of the cubic nonlinear Schrödinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform with a complexity of $\mathcal{O}(N\log N)$ operations per time step, where $N$ denotes the degrees of freedom in the spatial discretisation. We prove that the new scheme provides an $\mathcal{O}(τ^{\frac32γ-\frac12-\varepsilon}+N^{-γ})$ error bound in $L^2$ for any initial data belonging to $H^γ$, $\frac12<γ\leq 1$, where $τ$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.

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arXiv2020

A first-order Fourier integrator for the nonlinear Schrödinger equation on $\mathbb T$ without loss of regularity

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^γ$ for any initial data belonging to $H^γ$, for any $γ>\frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(\|u\|_{L^\infty([0,T]; H^γ)})>0$, such that $$ \|u^n-u(t_n)\|_{H^γ(\mathbb T)}\le C τ, $$ where $u^n$ denotes the numerical solution at $t_n=nτ$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$ \left|M(u^n)-M(u_0)\right|\le Cτ^5. $$ In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0\in H^1(\mathbb T)$, we rigorously prove that $$ \|u^n-u(t_n)\|_{H^1(\mathbb T)}\le Cτ^{\frac12-}, $$ where $C= C(\|u_0\|_{H^1(\mathbb T)})>0$.

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