The present study explores the nanofluid boundary layer flow over a stretching sheet with the combined influence of the double diffusive effects of thermophoresis and Brownian motion effects. For the purpose of transforming nonlinear partial differential equations into the linear united ordinary differential equation method, the similarity transformation technique is used. The Runge–Kutta–Fehlberg method was used to solve the equations of flow, along with sufficient boundary conditions. The effect on hydrodynamic, thermal and solutes boundary layers of a number of related parameters is investigated and the effects are graphically displayed. In conclusion, a strong agreement between the current numerical findings and the previous literature results is sought.
{"title":"Double diffusive effects on nanofluid flow toward a permeable stretching surface in presence of Thermophoresis and Brownian motion effects: A numerical study","authors":"V. V. L. Deepthi, V. K. Narla, R. Srinivasa Raju","doi":"10.1002/num.23086","DOIUrl":"https://doi.org/10.1002/num.23086","url":null,"abstract":"The present study explores the nanofluid boundary layer flow over a stretching sheet with the combined influence of the double diffusive effects of thermophoresis and Brownian motion effects. For the purpose of transforming nonlinear partial differential equations into the linear united ordinary differential equation method, the similarity transformation technique is used. The Runge–Kutta–Fehlberg method was used to solve the equations of flow, along with sufficient boundary conditions. The effect on hydrodynamic, thermal and solutes boundary layers of a number of related parameters is investigated and the effects are graphically displayed. In conclusion, a strong agreement between the current numerical findings and the previous literature results is sought.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139499728","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 provide improved uniform error estimates for the time-splitting Fourier pseudo-spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter
{"title":"Improved error estimates of the time-splitting methods for the long-time dynamics of the Klein–Gordon–Dirac system with the small coupling constant","authors":"Jiyong Li","doi":"10.1002/num.23084","DOIUrl":"https://doi.org/10.1002/num.23084","url":null,"abstract":"We provide improved uniform error estimates for the time-splitting Fourier pseudo-spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter <mjx-container ctxtmenu_counter=\"0\" jax=\"CHTML\" style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" data-semantic-complexity=\"5.5\" location=\"graphic/num23084-math-0001.png\"><mjx-semantics data-semantic-complexity=\"5.5\"><mjx-maction data-collapsible=\"true\" data-semantic-complexity=\"1.5\" toggle=\"2\"><mjx-mrow data-semantic-children=\"0,8\" data-semantic-complexity=\"16\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"element\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-complexity=\"1\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-complexity=\"1\" data-semantic- data-semantic-operator=\"infixop,∈\" data-semantic-parent=\"9\" data-semantic-role=\"element\" data-semantic-type=\"operator\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"7\" data-semantic-complexity=\"11\" data-semantic-content=\"2,6\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic-complexity=\"1\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3,4,5\" data-semantic-complexity=\"6\" data-semantic-content=\"4\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-complexity=\"1\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-complexity=\"1\" data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"7\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-complexity=\"1\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic-complexity=\"1\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"8\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-maction></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:num:media:num23084:num23084-math-0001\" data-semantic-complexity=\"5.5\" display=\"inline\" location=\"graphic/num23084-math-0001.png\" overflow=\"s","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138575381","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}
Ruben Caraballo, Chansophea Wathanak In, Alberto F. Martín, Ricardo Ruiz-Baier
In this article, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in a variety of regimes, including the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification and testing of robustness of block-diagonal preconditioners with respect to model parameters.
{"title":"Robust finite element methods and solvers for the Biot–Brinkman equations in vorticity form","authors":"Ruben Caraballo, Chansophea Wathanak In, Alberto F. Martín, Ricardo Ruiz-Baier","doi":"10.1002/num.23083","DOIUrl":"https://doi.org/10.1002/num.23083","url":null,"abstract":"In this article, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in a variety of regimes, including the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification and testing of robustness of block-diagonal preconditioners with respect to model parameters.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532565","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 designing efficient iteration algorithms for nonequilibrium three-temperature heat conduction equations, which are used to formulate the radiative energy transport problem. Based on the framework of relaxation iteration, we design a new accelerated iteration algorithm by reasonable approximation of the Jacobi matrix, according to the characteristics of the discrete scheme for the three-temperature equations. Adopting the iteration framework, we analyze the advantages and disadvantages of several iteration algorithms commonly used in practice and the new iteration algorithm. Finally, we compare the new iteration algorithm with some other iteration algorithms by solving several nonlinear models, and show that the new algorithm can achieve significant acceleration effect.
{"title":"Iteration acceleration methods for solving three-temperature heat conduction equations on distorted meshes","authors":"Yunlong Yu, Xingding Chen, Yanzhong Yao","doi":"10.1002/num.23085","DOIUrl":"https://doi.org/10.1002/num.23085","url":null,"abstract":"This article focuses on designing efficient iteration algorithms for nonequilibrium three-temperature heat conduction equations, which are used to formulate the radiative energy transport problem. Based on the framework of relaxation iteration, we design a new accelerated iteration algorithm by reasonable approximation of the Jacobi matrix, according to the characteristics of the discrete scheme for the three-temperature equations. Adopting the iteration framework, we analyze the advantages and disadvantages of several iteration algorithms commonly used in practice and the new iteration algorithm. Finally, we compare the new iteration algorithm with some other iteration algorithms by solving several nonlinear models, and show that the new algorithm can achieve significant acceleration effect.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532564","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 Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi-discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function-finite difference method. The convergence rate and stability of the fully-discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme.
{"title":"A radial basis function (RBF)-finite difference method for solving improved Boussinesq model with error estimation and description of solitary waves","authors":"Mostafa Abbaszadeh, AliReza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej","doi":"10.1002/num.23077","DOIUrl":"https://doi.org/10.1002/num.23077","url":null,"abstract":"The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi-discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function-finite difference method. The convergence rate and stability of the fully-discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532623","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 work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three-dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high-dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order , where , , and denote the time-space step sizes, respectively, and the second scheme converges to the space-time second-order accuracy. Finally, the numerical results verify the correctness of the theoretical analysis and show that the method of this article is competitive with the existing research work.
{"title":"Efficient and accurate temporal second-order numerical methods for multidimensional multi-term integrodifferential equations with the Abel kernels","authors":"Mingchao Zhao, Hao Chen, Kexin Li","doi":"10.1002/num.23082","DOIUrl":"https://doi.org/10.1002/num.23082","url":null,"abstract":"This work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three-dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high-dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order <math altimg=\"urn:x-wiley:num:media:num23082:num23082-math-0001\" display=\"inline\" location=\"graphic/num23082-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi>τ</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msup>\u0000<mo>+</mo>\u0000<msubsup>\u0000<mrow>\u0000<mi>h</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>1</mn>\u0000</mrow>\u0000<mrow>\u0000<mn>4</mn>\u0000</mrow>\u0000</msubsup>\u0000<mo>+</mo>\u0000<msubsup>\u0000<mrow>\u0000<mi>h</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000<mrow>\u0000<mn>4</mn>\u0000</mrow>\u0000</msubsup>\u0000</mrow>\u0000$$ {tau}^2+{h}_1^4+{h}_2^4 $$</annotation>\u0000</semantics></math>, where <math altimg=\"urn:x-wiley:num:media:num23082:num23082-math-0002\" display=\"inline\" location=\"graphic/num23082-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>τ</mi>\u0000</mrow>\u0000$$ tau $$</annotation>\u0000</semantics></math>, <math altimg=\"urn:x-wiley:num:media:num23082:num23082-math-0003\" display=\"inline\" location=\"graphic/num23082-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>h</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {h}_1 $$</annotation>\u0000</semantics></math>, and <math altimg=\"urn:x-wiley:num:media:num23082:num23082-math-0004\" display=\"inline\" location=\"graphic/num23082-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>h</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {h}_2 $$</annotation>\u0000</semantics></math> denote the time-space step sizes, respectively, and the second scheme converges to the space-time second-order accuracy. Finally, the numerical results verify the correctness of the theoretical analysis and show that the method of this article is competitive with the existing research work.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532554","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 Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM.
{"title":"Unfitted generalized finite element methods for Dirichlet problems without penalty or stabilization","authors":"Qinghui Zhang","doi":"10.1002/num.23081","DOIUrl":"https://doi.org/10.1002/num.23081","url":null,"abstract":"Abstract Unfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292259","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 article presents the stable miscible displacement problem in fractured porous media, and finite element discretization is constructed for this reduced model. The transmission interface conditions presented in this article enable us to derive a stability result and conduct the case where the pressure and concentration are both discontinuous across the fracture. The error estimates for and norm are established under the assumption of regular solutions. We perform some numerical examples to verify the theoretical analysis. Last, some unsteady physical experiments, more realistic test cases, are presented to prove the validity of the model and method.
{"title":"Numerical approximation for hybrid‐dimensional flow and transport in fractured porous media","authors":"Jijing Zhao, Hongxing Rui","doi":"10.1002/num.23080","DOIUrl":"https://doi.org/10.1002/num.23080","url":null,"abstract":"Abstract This article presents the stable miscible displacement problem in fractured porous media, and finite element discretization is constructed for this reduced model. The transmission interface conditions presented in this article enable us to derive a stability result and conduct the case where the pressure and concentration are both discontinuous across the fracture. The error estimates for and norm are established under the assumption of regular solutions. We perform some numerical examples to verify the theoretical analysis. Last, some unsteady physical experiments, more realistic test cases, are presented to prove the validity of the model and method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135725561","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 In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.
{"title":"Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem","authors":"Hu Li, Jin Huang","doi":"10.1002/num.23079","DOIUrl":"https://doi.org/10.1002/num.23079","url":null,"abstract":"Abstract In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136069518","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 In this article, two numerical schemes are designed and analyzed for the distributed‐order non‐Fickian flow. Two different processing techniques are applied to deal with the time distributed‐order derivative for the constructed two schemes, while the classical block‐centered finite difference method is used in spatial discretization. To be precise, one adopts the standard numerical scheme called SD scheme in the temporal direction, and the other utilizes an efficient method called EF scheme. We derive the stabilities of the two schemes rigorously. The convergence result of the SD scheme for pressure and velocity is . However, to get a faster computing speed, the super parameter is needed for the EF scheme, which leads to the accuracy is . Finally, some numerical experiments are carried out to verify the theoretical analysis.
{"title":"Error analyses on block‐centered finite difference schemes for distributed‐order non‐Fickian flow","authors":"Xuan Zhao, Ziyan Li","doi":"10.1002/num.23078","DOIUrl":"https://doi.org/10.1002/num.23078","url":null,"abstract":"Abstract In this article, two numerical schemes are designed and analyzed for the distributed‐order non‐Fickian flow. Two different processing techniques are applied to deal with the time distributed‐order derivative for the constructed two schemes, while the classical block‐centered finite difference method is used in spatial discretization. To be precise, one adopts the standard numerical scheme called SD scheme in the temporal direction, and the other utilizes an efficient method called EF scheme. We derive the stabilities of the two schemes rigorously. The convergence result of the SD scheme for pressure and velocity is . However, to get a faster computing speed, the super parameter is needed for the EF scheme, which leads to the accuracy is . Finally, some numerical experiments are carried out to verify the theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135315526","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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