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":"53 1","pages":""},"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":"25 1","pages":""},"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":" 12","pages":"0"},"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":"51 12","pages":"0"},"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":"684 ","pages":"0"},"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":"83 1","pages":"0"},"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}
Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.
{"title":"Richardson extrapolation method for solving the Riesz space fractional diffusion problem","authors":"Ren‐jun Qi, Zhi‐zhong Sun","doi":"10.1002/num.23076","DOIUrl":"https://doi.org/10.1002/num.23076","url":null,"abstract":"Abstract The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730881","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 We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.
{"title":"A semi‐Lagrangian ε$$ varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate","authors":"Yaowen Lu, Duy‐Minh Dang","doi":"10.1002/num.23075","DOIUrl":"https://doi.org/10.1002/num.23075","url":null,"abstract":"Abstract We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729697","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 An efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.
{"title":"Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian","authors":"Yanming Zhang, Yu Li, Yuexin Yu, Wansheng Wang","doi":"10.1002/num.23074","DOIUrl":"https://doi.org/10.1002/num.23074","url":null,"abstract":"Abstract An efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969413","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 A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.
{"title":"Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation","authors":"Hanzhang Hu, Yanping Chen, Jianwei Zhou","doi":"10.1002/num.23073","DOIUrl":"https://doi.org/10.1002/num.23073","url":null,"abstract":"Abstract A two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093531","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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