In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.
{"title":"A higher order multiscale method for the wave equation","authors":"Felix Krumbiegel, Roland Maier","doi":"10.1093/imanum/drae059","DOIUrl":"https://doi.org/10.1093/imanum/drae059","url":null,"abstract":"In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275655","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 existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.
{"title":"High-order energy stable discrete variational derivative schemes for gradient flows","authors":"Jizu Huang","doi":"10.1093/imanum/drae062","DOIUrl":"https://doi.org/10.1093/imanum/drae062","url":null,"abstract":"The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275656","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}
This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.
{"title":"A mesh-independent method for second-order potential mean field games","authors":"Kang Liu, Laurent Pfeiffer","doi":"10.1093/imanum/drae061","DOIUrl":"https://doi.org/10.1093/imanum/drae061","url":null,"abstract":"This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245188","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}
Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $tau ^{s/2}$ in $L^{2}(mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.
{"title":"Low regularity error estimates for the time integration of 2D NLS","authors":"Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz","doi":"10.1093/imanum/drae054","DOIUrl":"https://doi.org/10.1093/imanum/drae054","url":null,"abstract":"A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $tau ^{s/2}$ in $L^{2}(mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142231539","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}
This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.
{"title":"A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes","authors":"Haifeng Ji, Dong Liang, Qian Zhang","doi":"10.1093/imanum/drae053","DOIUrl":"https://doi.org/10.1093/imanum/drae053","url":null,"abstract":"This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160426","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}
Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier
In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.
{"title":"New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations","authors":"Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier","doi":"10.1093/imanum/drae052","DOIUrl":"https://doi.org/10.1093/imanum/drae052","url":null,"abstract":"In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142448","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}
It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.
{"title":"Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes","authors":"Sören Bartels, Philipp Tscherner","doi":"10.1093/imanum/drae050","DOIUrl":"https://doi.org/10.1093/imanum/drae050","url":null,"abstract":"It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084952","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 study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.
{"title":"Computing Klein-Gordon Spectra","authors":"Frank Rösler, Christiane Tretter","doi":"10.1093/imanum/drae032","DOIUrl":"https://doi.org/10.1093/imanum/drae032","url":null,"abstract":"We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084953","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}
Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim
We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.
{"title":"Mixed finite elements for the Gross–Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound","authors":"Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim","doi":"10.1093/imanum/drae048","DOIUrl":"https://doi.org/10.1093/imanum/drae048","url":null,"abstract":"We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142042431","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}
Valentin Carlier, Martin Campos Pinto, Francesco Fambri
In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.
在这篇文章中，我们针对纳维-斯托克斯方程提出了两种有限元方案，这两种方案都是基于德拉姆序列微分算子和具有显式弱形式倾斜对称性的平流算子的重新表述。我们的第一个方案是通过符合 FEEC（有限元外部计算）的空间对这一公式进行离散化而获得的：它保留了速度、总动量和能量的无发散点约束，此外还具有压力稳健性。我们的第二种方案沿用了断裂 FEEC 方法，使用完全不连续空间和局部保角投影来定义离散微分算子。它保留了相同的不变性，直到能量耗散，以稳定数值不连续性。对于这两种方案，我们都采用了中间点时间离散化，在完全离散的水平上保留了这些不变式，并根据 CFL 条件分析了其良好拟合性。虽然我们的理论结果适用于保留 de Rham 结构的一般有限元，但我们考虑了张量乘积样条空间的一个应用。具体地说，我们进行了几个数值测试案例，以验证所产生的数值方法的高阶精度及其处理一般边界条件的能力。
{"title":"Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations","authors":"Valentin Carlier, Martin Campos Pinto, Francesco Fambri","doi":"10.1093/imanum/drae047","DOIUrl":"https://doi.org/10.1093/imanum/drae047","url":null,"abstract":"In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141991902","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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