In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.
{"title":"Spatio‐temporal scalar auxiliary variable approach for the nonlinear convection–diffusion equation with discontinuous Galerkin method","authors":"Yaping Li, W. Zhao, Wenju Zhao","doi":"10.1002/num.23061","DOIUrl":"https://doi.org/10.1002/num.23061","url":null,"abstract":"In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48751612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in H1$$ {H}^1 $$ ‐norm and pressure in L2$$ {L}^2 $$ ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.
{"title":"Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations","authors":"Huaijun Yang, Xu Jia","doi":"10.1002/num.23060","DOIUrl":"https://doi.org/10.1002/num.23060","url":null,"abstract":"In this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in H1$$ {H}^1 $$ ‐norm and pressure in L2$$ {L}^2 $$ ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43185276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters.
{"title":"Two‐sided Krylov enhanced proper orthogonal decomposition methods for partial differential equations with variable coefficients","authors":"Li Wang, Zhen Miao, Yaolin Jiang","doi":"10.1002/num.23058","DOIUrl":"https://doi.org/10.1002/num.23058","url":null,"abstract":"In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47673516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.
{"title":"A priori error estimates of two monolithic schemes for Biot's consolidation model","authors":"H. Gu, M. Cai, Jingzhi Li, Guoliang Ju","doi":"10.1002/num.23059","DOIUrl":"https://doi.org/10.1002/num.23059","url":null,"abstract":"This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44862642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Neilan Pk$$ {P}_k $$ ‐ Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory.
{"title":"Neilan's divergence‐free finite elements for Stokes equations on tetrahedral grids","authors":"Shangyou Zhang","doi":"10.1002/num.23055","DOIUrl":"https://doi.org/10.1002/num.23055","url":null,"abstract":"The Neilan Pk$$ {P}_k $$ ‐ Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49393949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 consider leap‐frog finite element methods with EQ1rot$$ {mathrm{EQ}}_1^{mathrm{rot}} $$ element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived τ$$ tau $$ independently with order O(h2+hτ)$$ Oleft({h}^2+ htau right) $$ in H1$$ {H}^1 $$ ‐norm, where h$$ h $$ and τ$$ tau $$ denote the space and time step size. Then the unconditional optimal L2$$ {L}^2 $$ error and superclose result with order O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ are deduced, and the unconditional optimal H1$$ {H}^1 $$ error is obtained with order O(h+τ2)$$ Oleft(h+{tau}^2right) $$ by using interpolation theory. The final unconditional superconvergence result with order O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter 0
{"title":"Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator","authors":"Lingli Wang, Mike Meng-Yen Li, S. Peng","doi":"10.1002/num.23057","DOIUrl":"https://doi.org/10.1002/num.23057","url":null,"abstract":"In this paper, we consider leap‐frog finite element methods with EQ1rot$$ {mathrm{EQ}}_1^{mathrm{rot}} $$ element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived τ$$ tau $$ independently with order O(h2+hτ)$$ Oleft({h}^2+ htau right) $$ in H1$$ {H}^1 $$ ‐norm, where h$$ h $$ and τ$$ tau $$ denote the space and time step size. Then the unconditional optimal L2$$ {L}^2 $$ error and superclose result with order O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ are deduced, and the unconditional optimal H1$$ {H}^1 $$ error is obtained with order O(h+τ2)$$ Oleft(h+{tau}^2right) $$ by using interpolation theory. The final unconditional superconvergence result with order O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter 0","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42545899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 investigate the numerical approximation of an optimal control problem with fractional Laplacian and state constraint in integral form based on the Caffarelli–Silvestre expansion. The first order optimality conditions of the extended optimal control problem is obtained. An enriched spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is proposed. A priori error estimate for the enriched spectral discrete scheme is proved. Numerical experiments demonstrate the effectiveness of our method and validate the theoretical results.
{"title":"An efficient and accurate numerical method for the fractional optimal control problems with fractional Laplacian and state constraint","authors":"Jiaqi Zhang, Y. Yang","doi":"10.1002/num.23056","DOIUrl":"https://doi.org/10.1002/num.23056","url":null,"abstract":"In this paper, we investigate the numerical approximation of an optimal control problem with fractional Laplacian and state constraint in integral form based on the Caffarelli–Silvestre expansion. The first order optimality conditions of the extended optimal control problem is obtained. An enriched spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is proposed. A priori error estimate for the enriched spectral discrete scheme is proved. Numerical experiments demonstrate the effectiveness of our method and validate the theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4403 - 4420"},"PeriodicalIF":3.9,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44607471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.
{"title":"Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases","authors":"Qifeng Zhang, Jiyuan Zhang, Zhi‐zhong Sun","doi":"10.1002/num.23054","DOIUrl":"https://doi.org/10.1002/num.23054","url":null,"abstract":"Abstract This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"14 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135608808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article proposes an improved hybrid weighted essentially non‐oscillatory (WENO) scheme based on the third‐ and fifth‐order finite‐difference modified WENO (MWENO) schemes developed by Zhu et al. in (SIAM J. Sci. Comput. 39 (2017), A1089–A1113.) for solving hyperbolic conservation laws. The MWENO schemes give a guideline on whether to use the WENO scheme or the linear upwind scheme. Unfortunately, because there is no explicit formula for computing the roots of algebraic polynomials of order four or higher, it is difficult to generalize this criterion to higher order cases. Therefore, this article proposes a simple criterion for constructing a series of seventh‐, ninth‐, and higher‐order hybrid WENO schemes, and then designs a class of improved smooth indicator WENO (WENO‐MS) schemes. Compared with the classical WENO schemes, the main advantages of the WENO‐MS schemes are their robustness and efficiency. And these WENO‐MS schemes are more efficient, have better resolution, and can solve many extreme problems without any additional techniques. Furthermore, a simplification criterion is proposed to further improve the computational efficiency of the WENO‐MS schemes, and these simple WENO‐MS schemes are abbreviated as WENO‐SMS schemes in this article. Extensive numerical results demonstrate the good performance of the WENO‐MS schemes and the WENO‐SMS schemes.
{"title":"A new high order hybrid WENO scheme for hyperbolic conservation laws","authors":"Liang Li, Zhenming Wang, Zhonglong Zhao, Jun Zhu","doi":"10.1002/num.23052","DOIUrl":"https://doi.org/10.1002/num.23052","url":null,"abstract":"This article proposes an improved hybrid weighted essentially non‐oscillatory (WENO) scheme based on the third‐ and fifth‐order finite‐difference modified WENO (MWENO) schemes developed by Zhu et al. in (SIAM J. Sci. Comput. 39 (2017), A1089–A1113.) for solving hyperbolic conservation laws. The MWENO schemes give a guideline on whether to use the WENO scheme or the linear upwind scheme. Unfortunately, because there is no explicit formula for computing the roots of algebraic polynomials of order four or higher, it is difficult to generalize this criterion to higher order cases. Therefore, this article proposes a simple criterion for constructing a series of seventh‐, ninth‐, and higher‐order hybrid WENO schemes, and then designs a class of improved smooth indicator WENO (WENO‐MS) schemes. Compared with the classical WENO schemes, the main advantages of the WENO‐MS schemes are their robustness and efficiency. And these WENO‐MS schemes are more efficient, have better resolution, and can solve many extreme problems without any additional techniques. Furthermore, a simplification criterion is proposed to further improve the computational efficiency of the WENO‐MS schemes, and these simple WENO‐MS schemes are abbreviated as WENO‐SMS schemes in this article. Extensive numerical results demonstrate the good performance of the WENO‐MS schemes and the WENO‐SMS schemes.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4347 - 4376"},"PeriodicalIF":3.9,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48895129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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