IF 1.7 2区数学

Calcolo Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00558-w

Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang

{"title":"Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa–Holm system with small energy","authors":"Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang","doi":"10.1007/s10092-023-00558-w","DOIUrl":"https://doi.org/10.1007/s10092-023-00558-w","url":null,"abstract":"A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are (O(tau ^2 + h^2)) for the velocity in the (L^infty )-norm and the surface elevation in the (L^2)-norm, where (tau ) denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors 弱本质不可还原非负张量谱半径的算法

IF 1.7 2区数学

Calcolo Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00561-1

引用次数: 0

$$L^2(I;H^1(Omega )^d)$$ and $$L^2(I;L^2(Omega )^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems 瞬态斯托克斯问题 Galerkin 解决方案的 $$L^2(I;H^1(Omega )^d)$$ 和 $$L^2(I;L^2(Omega )^d)$$ 最佳近似型误差估计值

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-29 DOI: 10.1007/s10092-023-00560-2

Dmitriy Leykekhman, Boris Vexler

引用次数: 0

A convection–diffusion problem with a large shift on Durán meshes 杜兰网格上的大位移对流扩散问题

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-28 DOI: 10.1007/s10092-023-00559-9

引用次数: 0

On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions 关于三维非结构化有限元离散的离散格林函数的实在性

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-22 DOI: 10.1007/s10092-023-00556-y

Andrew P. Miller

引用次数: 0

A class of two stage multistep methods in solutions of time dependent parabolic PDEs 求解随时间变化的抛物型 PDE 的一类两阶段多步骤方法

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-21 DOI: 10.1007/s10092-023-00557-x

Moosa Ebadi, Mohammad Shahriari

引用次数: 0

High-order schemes based on extrapolation for semilinear fractional differential equation 基于外推法的半线性分数微分方程高阶方案

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-11 DOI: 10.1007/s10092-023-00553-1

Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan

{"title":"High-order schemes based on extrapolation for semilinear fractional differential equation","authors":"Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan","doi":"10.1007/s10092-023-00553-1","DOIUrl":"https://doi.org/10.1007/s10092-023-00553-1","url":null,"abstract":"By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order (alpha in (1,2).) The error has the asymptotic expansion ( big ( d_{3} tau ^{3- alpha } + d_{4} tau ^{4-alpha } + d_{5} tau ^{5-alpha } + cdots big ) + big ( d_{2}^{*} tau ^{4} + d_{3}^{*} tau ^{6} + d_{4}^{*} tau ^{8} + cdots big ) ) at any fixed time (t_{N}= T, N in {mathbb {Z}}^{+}), where (d_{i}, i=3, 4,ldots ) and (d_{i}^{*}, i=2, 3,ldots ) denote some suitable constants and (tau = T/N) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order (alpha in (1,2)) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime 在同时无质量和非相对论条件下的狄拉克方程的显式和结构保持指数波积分器傅立叶伪谱方法

IF 1.7 2区数学

Calcolo Pub Date : 2023-12-11 DOI: 10.1007/s10092-023-00554-0

Jiyong Li

{"title":"Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime","authors":"Jiyong Li","doi":"10.1007/s10092-023-00554-0","DOIUrl":"https://doi.org/10.1007/s10092-023-00554-0","url":null,"abstract":"We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter (0 <varepsilon ll 1) which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition (tau lesssim 1) and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at (O(h^{m_0} + tau ^2/varepsilon ^2)) where h is mesh size, (tau ) is time step and the integer (m_0) is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the (varepsilon )-scalability as (h = O(1)) and (tau = O(varepsilon )) which is better than the (varepsilon )-scalability of the finite difference (FD) methods: (h =O(varepsilon ^{1/2})) and (tau = O(varepsilon ^{3/2})). Numerical experiments confirm that the theoretical results in this paper are correct.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source terms 具有不连续对流和源项的时滞奇摄动抛物型问题的鲁棒数值格式

IF 1.7 2区数学

Calcolo Pub Date : 2023-11-29 DOI: 10.1007/s10092-023-00552-2

S. Priyadarshana, J. Mohapatra, H. Ramos

引用次数: 0

Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential 具有狄拉克δ势的非线性Schrödinger方程的精确和有效的数值方法

IF 1.7 2区数学

Calcolo Pub Date : 2023-11-20 DOI: 10.1007/s10092-023-00551-3

Xuanxuan Zhou, Yongyong Cai, Xingdong Tang, Guixiang Xu

{"title":"Accurate and efficient numerical methods for the nonlinear Schrödinger equation with Dirac delta potential","authors":"Xuanxuan Zhou, Yongyong Cai, Xingdong Tang, Guixiang Xu","doi":"10.1007/s10092-023-00551-3","DOIUrl":"https://doi.org/10.1007/s10092-023-00551-3","url":null,"abstract":"In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal (H^1) error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

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