Pub Date : 2024-01-11DOI: 10.1007/s10092-023-00562-0
Yanchen He, Christoph Schwab
In an open, bounded Lipschitz polygon (Omega subset mathbb {R}^2), we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in (Omega ). The boundary conditions on each edge of (partial Omega ) are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in (H^1(Omega )), quantized tensor formats and certain deep neural networks.
{"title":"Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon","authors":"Yanchen He, Christoph Schwab","doi":"10.1007/s10092-023-00562-0","DOIUrl":"https://doi.org/10.1007/s10092-023-00562-0","url":null,"abstract":"<p>In an open, bounded Lipschitz polygon <span>(Omega subset mathbb {R}^2)</span>, we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term <i>f</i> which is analytic in <span>(Omega )</span>. The boundary conditions on each edge of <span>(partial Omega )</span> are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: <i>hp</i>-finite elements, reduced order models via Kolmogorov <i>n</i>-widths of solution sets in <span>(H^1(Omega ))</span>, quantized tensor formats and certain deep neural networks.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"8 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139460559","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}
Pub Date : 2024-01-03DOI: 10.1007/s10092-023-00555-z
Jikun Zhao, Teng Chen, Bei Zhang, Xiaojing Dong
We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the (C^1) continuity. A (C^0)-continuous Hermite-type virtual element with local (H^2) regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.
{"title":"The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem","authors":"Jikun Zhao, Teng Chen, Bei Zhang, Xiaojing Dong","doi":"10.1007/s10092-023-00555-z","DOIUrl":"https://doi.org/10.1007/s10092-023-00555-z","url":null,"abstract":"<p>We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the <span>(C^1)</span> continuity. A <span>(C^0)</span>-continuous Hermite-type virtual element with local <span>(H^2)</span> regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093136","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}
Pub Date : 2024-01-02DOI: 10.1007/s10092-023-00558-w
Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang
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.
{"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":"<p>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 <span>(O(tau ^2 + h^2))</span> for the velocity in the <span>(L^infty )</span>-norm and the surface elevation in the <span>(L^2)</span>-norm, where <span>(tau )</span> denotes the temporal stepsize and <i>h</i> 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.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"43 1","pages":""},"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}
Pub Date : 2024-01-02DOI: 10.1007/s10092-023-00561-1
Abstract
In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples.
{"title":"An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors","authors":"","doi":"10.1007/s10092-023-00561-1","DOIUrl":"https://doi.org/10.1007/s10092-023-00561-1","url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"125 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079803","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}
Pub Date : 2023-12-29DOI: 10.1007/s10092-023-00560-2
Dmitriy Leykekhman, Boris Vexler
In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in (L^2(I;L^2(Omega )^d)) and (L^2(I;H^1(Omega )^d)) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type (L^2(I;H^1(Omega ))) error estimate are new even for scalar parabolic problems.
{"title":"$$L^2(I;H^1(Omega )^d)$$ and $$L^2(I;L^2(Omega )^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems","authors":"Dmitriy Leykekhman, Boris Vexler","doi":"10.1007/s10092-023-00560-2","DOIUrl":"https://doi.org/10.1007/s10092-023-00560-2","url":null,"abstract":"<p>In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in <span>(L^2(I;L^2(Omega )^d))</span> and <span>(L^2(I;H^1(Omega )^d))</span> norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type <span>(L^2(I;H^1(Omega )))</span> error estimate are new even for scalar parabolic problems.\u0000</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066021","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}
Pub Date : 2023-12-28DOI: 10.1007/s10092-023-00559-9
Abstract
A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.
{"title":"A convection–diffusion problem with a large shift on Durán meshes","authors":"","doi":"10.1007/s10092-023-00559-9","DOIUrl":"https://doi.org/10.1007/s10092-023-00559-9","url":null,"abstract":"<h3>Abstract</h3> <p>A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"41 6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139055805","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}
Pub Date : 2023-12-22DOI: 10.1007/s10092-023-00556-y
Andrew P. Miller
The aim of this paper is twofold. First, we prove (L^p) estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green’s function is of order (h^{-1}), which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.
{"title":"On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions","authors":"Andrew P. Miller","doi":"10.1007/s10092-023-00556-y","DOIUrl":"https://doi.org/10.1007/s10092-023-00556-y","url":null,"abstract":"<p>The aim of this paper is twofold. First, we prove <span>(L^p)</span> estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter <i>h</i>. Actually, we show that at the singularity the discrete Green’s function is of order <span>(h^{-1})</span>, which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139029511","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}
Pub Date : 2023-12-21DOI: 10.1007/s10092-023-00557-x
Moosa Ebadi, Mohammad Shahriari
In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, ((p=4,5,6,7)) and (A(alpha ))-stable for order p, ((p=8,9,ldots , 14).) Compared to the existing BDF based method, i.e. class (2+1,) hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.
本手稿以混合后向微分公式(BDF)为基础,阐述了一类新的高阶多步方法,用于数值求解由时间相关偏微分方程半离散化产生的常微分方程(ODE)系统。详细讨论了这些方法的阶次和稳定性分析。通过在解的一阶导数中使用离阶点和阶点,得到的新方法在阶数为p时是A稳定的((p=4,5,6,7)),在阶数为p时是(A(α))稳定的(((p=8,9,ldots , 14).)。与现有的基于BDF的方法(即类(2+1,)混合BDF方法(HBDF)、基于超未来点的方法(SFPBM)和MEBDF)相比,在绝对稳定区域和阶数方面都有很好的改进。为了检验这些方法与 SFPBM 和 MEBDF 方法相比在减少 CPU 时间、从而提高低阶和高阶新方法精度方面的优势,我们给出了一些数值示例。
{"title":"A class of two stage multistep methods in solutions of time dependent parabolic PDEs","authors":"Moosa Ebadi, Mohammad Shahriari","doi":"10.1007/s10092-023-00557-x","DOIUrl":"https://doi.org/10.1007/s10092-023-00557-x","url":null,"abstract":"<p>In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are <i>A</i>-stable for order <i>p</i>, <span>((p=4,5,6,7))</span> and <span>(A(alpha )</span>)-stable for order <i>p</i>, <span>((p=8,9,ldots , 14).)</span> Compared to the existing BDF based method, i.e. class <span>(2+1,)</span> hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"70 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139029565","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}
Pub Date : 2023-12-11DOI: 10.1007/s10092-023-00553-1
Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan
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.
通过将黎曼-利奥维尔分数导数重写为哈达玛德有限部分积分,并借助片断二次插值多项式近似,建立了一个用于近似阶数为 (α in (1,2).) 的黎曼-利奥维尔分数导数的数值方案。误差具有渐近展开( 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 )),其中 (d_{i}, i=3, 4,ldots ) 和 (d_{i}^{*}, i=2, 3,ldots ) 表示一些合适的常数,而 (tau = T/N) 表示步长。在此离散化的基础上,推导出了一种新的近似线性分数微分方程的方案,其误差也有类似的渐近展开。因此,通过外推法得到了逼近线性分数微分方程的高阶方案。此外,还引入并分析了逼近半线性分数微分方程的高阶方案。我们进行了几次数值实验,结果表明数值结果与我们的理论发现是一致的。
{"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":"<p>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 <span>(alpha in (1,2).)</span> The error has the asymptotic expansion <span>( 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 ) )</span> at any fixed time <span>(t_{N}= T, N in {mathbb {Z}}^{+})</span>, where <span>(d_{i}, i=3, 4,ldots )</span> and <span>(d_{i}^{*}, i=2, 3,ldots )</span> denote some suitable constants and <span>(tau = T/N)</span> denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order <span>(alpha in (1,2))</span> 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.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"1 1","pages":""},"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}
Pub Date : 2023-12-11DOI: 10.1007/s10092-023-00554-0
Jiyong Li
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.
{"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":"<p>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 <span>(0 <varepsilon ll 1)</span> which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition <span>(tau lesssim 1)</span> 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 <span>(O(h^{m_0} + tau ^2/varepsilon ^2))</span> where <i>h</i> is mesh size, <span>(tau )</span> is time step and the integer <span>(m_0)</span> 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 <span>(varepsilon )</span>-scalability as <span>(h = O(1))</span> and <span>(tau = O(varepsilon ))</span> which is better than the <span>(varepsilon )</span>-scalability of the finite difference (FD) methods: <span>(h =O(varepsilon ^{1/2}))</span> and <span>(tau = O(varepsilon ^{3/2}))</span>. Numerical experiments confirm that the theoretical results in this paper are correct.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":"43 1","pages":""},"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}