We propose an efficient approach for time integration of Klein–Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.
{"title":"Effective highly accurate time integrators for linear Klein–Gordon equations across the scales","authors":"Karolina Kropielnicka, Karolina Lademann, Katharina Schratz","doi":"10.1515/jnma-2023-0070","DOIUrl":"https://doi.org/10.1515/jnma-2023-0070","url":null,"abstract":"We propose an efficient approach for time integration of Klein–Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"166 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192664","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}
In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝd (d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.
本文提出了一种高阶弱 Galerkin 有限元方法 (WG-FEM)，用于求解速度和压力在 ℝ d (d = 2, 3) 中不连续的斯托克斯静止界面问题。这种 WG 方法配备了稳定的有限元，其中速度由 k ≥ 1 阶的普通多项式组成，压力由 k - 1 阶的多项式组成，两者都是不连续的。在混合网格的 L 2 规范下，速度和压力的最佳收敛率分别为 k + 1 阶和 k 阶。数值实验验证了二维和三维实例的预期精度。此外，数值结果表明，所提出的 WG 算法能够适应具有尖锐边缘、尖角和尖端的复杂和非常不规则的几何界面。
{"title":"Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements","authors":"Raman Kumar, Bhupen Deka","doi":"10.1515/jnma-2023-0112","DOIUrl":"https://doi.org/10.1515/jnma-2023-0112","url":null,"abstract":"In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ<jats:sup> <jats:italic>d</jats:italic> </jats:sup> (<jats:italic>d</jats:italic> = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree <jats:italic>k</jats:italic> ≥ 1 for the velocity and polynomials of degree <jats:italic>k</jats:italic> – 1 for the pressure, both are discontinuous. Optimal convergence rates of order <jats:italic>k</jats:italic> + 1 for the velocity and order <jats:italic>k</jats:italic> for the pressure are established in <jats:italic>L</jats:italic> <jats:sup>2</jats:sup>-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"62 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575326","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}
We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant (or equi-variant) under permutations and rotations. This task arises in the evaluation linear models as well as equivariant neural network models of many-particle systems. The theoretical bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. The sparsity of this tensor makes it challenging to construct a highly efficient evaluation scheme. The references [10, 11] introduced a recursive evaluation strategy that relied on a number of heuristics, but performed well in tests. In the present work, we propose an explicit construction of such a recursive evaluation strategy and show that it is in fact optimal in the limit of infinite polynomial degree.
{"title":"Optimal evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors","authors":"Illia Kaliuzhnyi, Christoph Ortner","doi":"10.1515/jnma-2024-0025","DOIUrl":"https://doi.org/10.1515/jnma-2024-0025","url":null,"abstract":"We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant (or equi-variant) under permutations and rotations. This task arises in the evaluation linear models as well as equivariant neural network models of many-particle systems. The theoretical bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. The sparsity of this tensor makes it challenging to construct a highly efficient evaluation scheme. The references [10, 11] introduced a recursive evaluation strategy that relied on a number of heuristics, but performed well in tests. In the present work, we propose an explicit construction of such a recursive evaluation strategy and show that it is in fact optimal in the limit of infinite polynomial degree.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"38 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575164","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}
The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.
{"title":"A fully well-balanced hydrodynamic reconstruction","authors":"Christophe Berthon, Victor Michel-Dansac","doi":"10.1515/jnma-2023-0065","DOIUrl":"https://doi.org/10.1515/jnma-2023-0065","url":null,"abstract":"The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"91 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297416","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}
Abstract The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2 π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1 (Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2 π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4 π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.
{"title":"Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations","authors":"Jesús Bonilla, Juan Vicente Gutiérrez-Santacreu","doi":"10.1515/jnma-2023-0016","DOIUrl":"https://doi.org/10.1515/jnma-2023-0016","url":null,"abstract":"Abstract The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2 π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1 (Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2 π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4 π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"295 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136114648","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}
Philipp Bringmann, Carsten Carstensen, Julian Streitberger
Abstract The symmetric 0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. a detailed documentation of C 0 interior penalty method in.
{"title":"Local parameter selection in the C<sup>0</sup> interior penalty method for the biharmonic equation","authors":"Philipp Bringmann, Carsten Carstensen, Julian Streitberger","doi":"10.1515/jnma-2023-0028","DOIUrl":"https://doi.org/10.1515/jnma-2023-0028","url":null,"abstract":"Abstract The symmetric 0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. a detailed documentation of C 0 interior penalty method in.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135979905","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}
Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $mathbb{R}^{d} text { for } d in mathbb{N}^{*}, p in[1,+infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B Vleft(mathbb{R}^{d}right) text { into } mathrm{L}^{frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p > d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p > d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.
摘要本文证明了著名的Sobolev不等式[1]在rd中的离散形式，对于可能具有非凸单元的一般非正交网格，对于d∈N∗，p∈[1，+∞[$mathbb{R}^{d} text { for } d in mathbb{N}^{*}, p in[1,+infty[$。我们通过引入定向总变分的离散类比，密切关注基于bv R d嵌入到ld d−1 $B Vleft(mathbb{R}^{d}right) text { into } mathrm{L}^{frac{d}{d-1}}$[1，定理9.9]，[12，定理1.1]的连续Sobolev不等式的证明。在p b> d (Gagliardo-Nirenberg不等式)的情况下，我们采用连续情况([1，定理9.9]，[9，定理4.8])的证明，并使用[3,5]中的技术。在p b> d (Morrey’s不等式)的情况下，我们将[12，定理1.1]的证明简化并推广到更一般的网格。
{"title":"On the discrete Sobolev inequalities","authors":"Sédrick Kameni Ngwamou, Michael Ndjinga","doi":"10.1515/jnma-2023-0086","DOIUrl":"https://doi.org/10.1515/jnma-2023-0086","url":null,"abstract":"Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $mathbb{R}^{d} text { for } d in mathbb{N}^{*}, p in[1,+infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B Vleft(mathbb{R}^{d}right) text { into } mathrm{L}^{frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p > d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p > d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"66 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063195","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}
Abstract The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting and segmentation.
{"title":"Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source","authors":"K. F. Lam, Ru Wang","doi":"10.1515/jnma-2023-0021","DOIUrl":"https://doi.org/10.1515/jnma-2023-0021","url":null,"abstract":"Abstract The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting and segmentation.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"30 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83313876","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}
Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo
Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.
{"title":"POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution: Error bounds for the pressure","authors":"Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo","doi":"10.1515/jnma-2023-0039","DOIUrl":"https://doi.org/10.1515/jnma-2023-0039","url":null,"abstract":"Abstract Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181307","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}
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