In this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.
{"title":"A flux‐based HDG method","authors":"Issei Oikawa","doi":"10.1002/num.23117","DOIUrl":"https://doi.org/10.1002/num.23117","url":null,"abstract":"In this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191692","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 space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied.
{"title":"Extensions and investigations of space‐time generalized Riemann problems numerical schemes for linear systems of conservation laws with source terms","authors":"Rodolphe Turpault","doi":"10.1002/num.23118","DOIUrl":"https://doi.org/10.1002/num.23118","url":null,"abstract":"The space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191684","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, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.
{"title":"A parareal exponential integrator finite element method for semilinear parabolic equations","authors":"Jianguo Huang, Lili Ju, Yuejin Xu","doi":"10.1002/num.23116","DOIUrl":"https://doi.org/10.1002/num.23116","url":null,"abstract":"In this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191824","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 the numerical approximation for a two‐dimensional multiterm reaction‐subdiffusion equation, where we adopt an alternating direction implicit (ADI) method combined with the L1 approximation for the multiterm time Caputo fractional derivatives of orders between 0 and 1. Stability and convergence of the full‐discrete L1‐ADI scheme are established. The final convergence in time direction is point‐wise, that is, at . Numerical results are given to confirm our theoretical results.
{"title":"Convergence analysis of a L1‐ADI scheme for two‐dimensional multiterm reaction‐subdiffusion equation","authors":"Yubing Jiang, Hu Chen","doi":"10.1002/num.23115","DOIUrl":"https://doi.org/10.1002/num.23115","url":null,"abstract":"In this paper, we consider the numerical approximation for a two‐dimensional multiterm reaction‐subdiffusion equation, where we adopt an alternating direction implicit (ADI) method combined with the L1 approximation for the multiterm time Caputo fractional derivatives of orders between 0 and 1. Stability and convergence of the full‐discrete L1‐ADI scheme are established. The final convergence in time direction is point‐wise, that is, at . Numerical results are given to confirm our theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141117940","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 focuses on a new and optimal error analysis of a decoupled finite element scheme for the inductionless magnetohydrodynamic (MHD) equations. The method uses the classical inf‐sup stable Mini/Taylor‐Hood (Mini/TH) finite element pairs to appropriate the velocity and pressure, and Raviart–Thomas (RT) face element to discretize the current density spatially, and the semi‐implicit Euler scheme with an additional stabilized term and some delicate implicit–explicit handling for the coupling terms temporally. The method enjoys some impressive features that it is linear, decoupled, unconditional energy stable and charge‐conservative. Due to the errors from the explicit handing of the coupling terms and the existence of the artificial stabilized term, and the contamination of the lower‐order RT face discretization in the error analysis, the existing theoretical results are not unconditional and optimal. By utilizing the anti‐symmetric structure of the coupling terms and the existence of the extra dissipative term, and the negative‐norm estimate for the mixed Poisson projection, we establish the unconditional and optimal error estimates for all the variables. Numerical tests are presented to illustrate our theoretical findings.
{"title":"Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations","authors":"Xiaodi Zhang, Shitian Dong","doi":"10.1002/num.23108","DOIUrl":"https://doi.org/10.1002/num.23108","url":null,"abstract":"This article focuses on a new and optimal error analysis of a decoupled finite element scheme for the inductionless magnetohydrodynamic (MHD) equations. The method uses the classical inf‐sup stable Mini/Taylor‐Hood (Mini/TH) finite element pairs to appropriate the velocity and pressure, and Raviart–Thomas (RT) face element to discretize the current density spatially, and the semi‐implicit Euler scheme with an additional stabilized term and some delicate implicit–explicit handling for the coupling terms temporally. The method enjoys some impressive features that it is linear, decoupled, unconditional energy stable and charge‐conservative. Due to the errors from the explicit handing of the coupling terms and the existence of the artificial stabilized term, and the contamination of the lower‐order RT face discretization in the error analysis, the existing theoretical results are not unconditional and optimal. By utilizing the anti‐symmetric structure of the coupling terms and the existence of the extra dissipative term, and the negative‐norm estimate for the mixed Poisson projection, we establish the unconditional and optimal error estimates for all the variables. Numerical tests are presented to illustrate our theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972503","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}
Linear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.
{"title":"Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type","authors":"Yongseok Jang, Simon Shaw","doi":"10.1002/num.23107","DOIUrl":"https://doi.org/10.1002/num.23107","url":null,"abstract":"Linear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, <jats:italic>symmetric interior penalty Galerkin method</jats:italic> (SIPG) for spatial discretization, and the implicit finite difference schemes in time, <jats:italic>Crank–Nicolson method</jats:italic>. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881626","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 investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.
{"title":"A wavelet collocation method for fractional Black–Scholes equations by subdiffusive model","authors":"Davood Damircheli, Mohsen Razzaghi","doi":"10.1002/num.23103","DOIUrl":"https://doi.org/10.1002/num.23103","url":null,"abstract":"In this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827489","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 paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.
{"title":"Error estimates for completely discrete FEM in energy‐type and weaker norms","authors":"Lutz Angermann, Peter Knabner, Andreas Rupp","doi":"10.1002/num.23106","DOIUrl":"https://doi.org/10.1002/num.23106","url":null,"abstract":"The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827488","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 generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.
{"title":"Generalized log orthogonal functions spectral collocation method for two dimensional weakly singular Volterra integral equations of the second kind","authors":"Qiumei Huang, Min Wang","doi":"10.1002/num.23105","DOIUrl":"https://doi.org/10.1002/num.23105","url":null,"abstract":"In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827597","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 focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.
本文主要研究制度转换下美式期权估值的数值算法。此类期权的价格满足一组无界域上的互补抛物线问题。根据我们以往的经验，定价模型可以截断为有界域上的线性互补问题（LCP）。此外，我们还将 LCP 转化为等效的变分问题（VP），并采用欧拉有限元法对 VP 进行离散化处理。由于离散化系统中的变分矩阵是 P 矩阵，因此我们提出了一种原始双主动集（PDAS）算法来有效评估期权价格。作为 PDAS 的特长，无需更多计算成本即可获得所有制度下的最优行使边界。最后，我们进行了数值模拟，以检验我们提出的算法的性能，并将其与现有方法进行比较。
{"title":"Primal‐dual active set algorithm for valuating American options under regime switching","authors":"Haiming Song, Jingbo Xu, Jinda Yang, Yutian Li","doi":"10.1002/num.23104","DOIUrl":"https://doi.org/10.1002/num.23104","url":null,"abstract":"This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628089","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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