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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Shaoshuai Chu, Olyana A. Kovyrkina, Alexander Kurganov, Vladimir V. Ostapenko
Abstract We study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.
{"title":"Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes","authors":"Shaoshuai Chu, Olyana A. Kovyrkina, Alexander Kurganov, Vladimir V. Ostapenko","doi":"10.1002/num.23053","DOIUrl":"https://doi.org/10.1002/num.23053","url":null,"abstract":"Abstract We study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135916102","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}
We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.
{"title":"Behavior of Lagrange‐Galerkin solutions to the Navier‐Stokes problem for small time increment","authors":"M. Tabata, Shinya Uchiumi","doi":"10.1002/num.23051","DOIUrl":"https://doi.org/10.1002/num.23051","url":null,"abstract":"We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48447654","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 finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.
{"title":"A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes","authors":"Huifang Zhou, Yuanyuan Liu, Z. Sheng","doi":"10.1002/num.23050","DOIUrl":"https://doi.org/10.1002/num.23050","url":null,"abstract":"In this article, we present a finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46724969","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 is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ with time step τ$$ tau $$ and mesh size h$$ h $$ in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete L2$$ {L}^2 $$ norm or H1$$ {H}^1 $$ norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.
{"title":"A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions","authors":"Jianfeng Liu, Q. Tang, Ting-chun Wang","doi":"10.1002/num.23042","DOIUrl":"https://doi.org/10.1002/num.23042","url":null,"abstract":"This article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ with time step τ$$ tau $$ and mesh size h$$ h $$ in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete L2$$ {L}^2 $$ norm or H1$$ {H}^1 $$ norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46907877","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, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. H2$$ {H}^2 $$ ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in H1$$ {H}^1 $$ and L2$$ {L}^2 $$ ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes h=𝒪(H2) . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.
{"title":"Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations","authors":"X. Chu, Chuanjun Chen, T. Zhang","doi":"10.1002/num.23043","DOIUrl":"https://doi.org/10.1002/num.23043","url":null,"abstract":"In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. H2$$ {H}^2 $$ ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in H1$$ {H}^1 $$ and L2$$ {L}^2 $$ ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes h=𝒪(H2) . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44329352","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 deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ varepsilon $$ ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ varepsilon $$ ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.
{"title":"Parameter robust higher‐order finite difference method for convection‐diffusion problem with time delay","authors":"Sanjaya Sahoo, Vikas Gupta","doi":"10.1002/num.23039","DOIUrl":"https://doi.org/10.1002/num.23039","url":null,"abstract":"This paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ varepsilon $$ ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ varepsilon $$ ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44388556","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：