The P1$$ {P}_1 $$ –nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non‐invertible matrices. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensions is provided.
{"title":"P1$$ {P}_1 $$ –Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis","authors":"Jaeryun Yim, D. Sheen","doi":"10.1002/num.23023","DOIUrl":"https://doi.org/10.1002/num.23023","url":null,"abstract":"The P1$$ {P}_1 $$ –nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non‐invertible matrices. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensions is provided.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3725 - 3753"},"PeriodicalIF":3.9,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47730678","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 development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.
{"title":"A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media","authors":"Da Li, Keran Li, Wenyuan Liao","doi":"10.1002/num.23036","DOIUrl":"https://doi.org/10.1002/num.23036","url":null,"abstract":"In this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4062 - 4086"},"PeriodicalIF":3.9,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45357743","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 nonlinear two dimensional Zakharov‐Kuznetsov modified equal width equation investigated under the observation of extended modified rational expansion method and determined the multiple solitary wave solutions. The interested and important things in this work is the multiple solitary wave solutions which have various kinds of physical structure including anti‐kink soliton, travelling wave solutions, bright soliton, kink soliton, dark soliton, kink bright and dark solitons, anti‐kink bright and dark solitons. In our knowledge investigated various kinds of solitary solutions found first time under one method in the existing literatures. The constructed multiple solutions for nonlinear ZK‐MEW equation will play keen role in the investigation of different physical structure in nonlinear sciences. The investigated work prove that applied method is very efficient, reliable, and powerful.
{"title":"Structure of analytical and symbolic computational approach of multiple solitary wave solutions for nonlinear Zakharov‐Kuznetsov modified equal width equation","authors":"M. Iqbal, A. Seadawy, D. Lu, Zhengdi Zhang","doi":"10.1002/num.23033","DOIUrl":"https://doi.org/10.1002/num.23033","url":null,"abstract":"The nonlinear two dimensional Zakharov‐Kuznetsov modified equal width equation investigated under the observation of extended modified rational expansion method and determined the multiple solitary wave solutions. The interested and important things in this work is the multiple solitary wave solutions which have various kinds of physical structure including anti‐kink soliton, travelling wave solutions, bright soliton, kink soliton, dark soliton, kink bright and dark solitons, anti‐kink bright and dark solitons. In our knowledge investigated various kinds of solitary solutions found first time under one method in the existing literatures. The constructed multiple solutions for nonlinear ZK‐MEW equation will play keen role in the investigation of different physical structure in nonlinear sciences. The investigated work prove that applied method is very efficient, reliable, and powerful.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3987 - 4006"},"PeriodicalIF":3.9,"publicationDate":"2023-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42140541","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 nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay.
{"title":"Energy stability of exponential time differencing schemes for the nonlocal Cahn‐Hilliard equation","authors":"Quan Zhou, Yabing Sun","doi":"10.1002/num.23035","DOIUrl":"https://doi.org/10.1002/num.23035","url":null,"abstract":"The nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4030 - 4058"},"PeriodicalIF":3.9,"publicationDate":"2023-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47263365","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 hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.
{"title":"Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations","authors":"Zhoushun Zheng, Jilong He, Changfa Du, Zhijian Ye","doi":"10.1002/num.23032","DOIUrl":"https://doi.org/10.1002/num.23032","url":null,"abstract":"In this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3964 - 3986"},"PeriodicalIF":3.9,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45199961","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 give a description of a technique to develop an a posteriori error estimator for the dual mixed methods, when applied to elliptic partial differential equations with non homogeneous mixed boundary conditions. The approach considers conforming finite elements for the discrete scheme, and a quasi‐Helmholtz decomposition result to obtain a residual a posteriori error estimator. After applying first a homogenization technique (for the Neumann boundary condition), we derive an a posteriori error estimator, which looks to be expensive to compute. This motivates the derivation of another a posteriori error estimator, that is fully computable. As a consequence, we establish the equivalence between the latter a posteriori error estimator and the natural norm of the error, that is, we prove the reliability and local efficiency of the aforementioned estimator. Finally, we report numerical examples showing the good properties of the estimator, in agreement with the theoretical results of this work.
{"title":"A note on a posteriori error analysis for dual mixed methods with mixed boundary conditions","authors":"T. Barrios, R. Bustinza, Camila Campos","doi":"10.1002/num.23029","DOIUrl":"https://doi.org/10.1002/num.23029","url":null,"abstract":"In this article, we give a description of a technique to develop an a posteriori error estimator for the dual mixed methods, when applied to elliptic partial differential equations with non homogeneous mixed boundary conditions. The approach considers conforming finite elements for the discrete scheme, and a quasi‐Helmholtz decomposition result to obtain a residual a posteriori error estimator. After applying first a homogenization technique (for the Neumann boundary condition), we derive an a posteriori error estimator, which looks to be expensive to compute. This motivates the derivation of another a posteriori error estimator, that is fully computable. As a consequence, we establish the equivalence between the latter a posteriori error estimator and the natural norm of the error, that is, we prove the reliability and local efficiency of the aforementioned estimator. Finally, we report numerical examples showing the good properties of the estimator, in agreement with the theoretical results of this work.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3897 - 3918"},"PeriodicalIF":3.9,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46289658","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 propose a second order, unconditionally stable artificial compression method for the fully evolutionary Stokes/Darcy and Navier‐Stokes/Darcy equations that model the coupling surface and groundwater flows. It uncouples the surface from the groundwater flow by the Crank‐Nicolson Leapfrog scheme for the discretization in time, and through the artificial compression method without artificial pressure boundary conditions to decouple the velocity and pressure of the incompressible flow. Finally, we have verified the stability and second‐order convergence of the algorithm from theoretical analysis and numerical experiments.
{"title":"An unconditionally stable artificial compression method for the time‐dependent groundwater‐surface water flows","authors":"Yi Qin, Yang Wang, Yanren Hou, Jian Li","doi":"10.1002/num.23022","DOIUrl":"https://doi.org/10.1002/num.23022","url":null,"abstract":"In this article, we propose a second order, unconditionally stable artificial compression method for the fully evolutionary Stokes/Darcy and Navier‐Stokes/Darcy equations that model the coupling surface and groundwater flows. It uncouples the surface from the groundwater flow by the Crank‐Nicolson Leapfrog scheme for the discretization in time, and through the artificial compression method without artificial pressure boundary conditions to decouple the velocity and pressure of the incompressible flow. Finally, we have verified the stability and second‐order convergence of the algorithm from theoretical analysis and numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3705 - 3724"},"PeriodicalIF":3.9,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48098459","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}
A stabilized mixed finite element method is proposed for solving the Maxwell eigenproblem in terms of the electric field and the multiplier. Using the Bochev‐Dohrmann‐Gunzburger stabilization, we introduce some ad hoc stabilizing parameters for stabilizing the kernel‐coercivity of the electric field and for stabilizing the inf‐sup condition of the multiplier. We show that the stabilized mixed method is stable and convergent, with applications to some lowest‐order edge elements on affine rectangular and cuboid mesh and on nonaffine quadrilateral mesh which fail in the classical methods. In particular, we prove the uniform convergence for guaranteeing spectral‐correct and spurious‐free discrete eigenmodes from the Babus̆ka‐Osborn spectral theory for compact operators. Numerical results have illustrated the performance of the stabilized method and confirmed the theoretical results obtained.
{"title":"A Bochev‐Dohrmann‐Gunzburger stabilized method for Maxwell eigenproblem","authors":"Zhijie Du, Huoyuan Duan, Can Wang, Qiuyu Zhang","doi":"10.1002/num.23026","DOIUrl":"https://doi.org/10.1002/num.23026","url":null,"abstract":"A stabilized mixed finite element method is proposed for solving the Maxwell eigenproblem in terms of the electric field and the multiplier. Using the Bochev‐Dohrmann‐Gunzburger stabilization, we introduce some ad hoc stabilizing parameters for stabilizing the kernel‐coercivity of the electric field and for stabilizing the inf‐sup condition of the multiplier. We show that the stabilized mixed method is stable and convergent, with applications to some lowest‐order edge elements on affine rectangular and cuboid mesh and on nonaffine quadrilateral mesh which fail in the classical methods. In particular, we prove the uniform convergence for guaranteeing spectral‐correct and spurious‐free discrete eigenmodes from the Babus̆ka‐Osborn spectral theory for compact operators. Numerical results have illustrated the performance of the stabilized method and confirmed the theoretical results obtained.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3811 - 3846"},"PeriodicalIF":3.9,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49211038","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, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in L2$$ {L}^2 $$ ‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.
{"title":"Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments","authors":"Yongtang Chen, Qi Wang","doi":"10.1002/num.23024","DOIUrl":"https://doi.org/10.1002/num.23024","url":null,"abstract":"In this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in L2$$ {L}^2 $$ ‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3754 - 3776"},"PeriodicalIF":3.9,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47013959","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 time‐fractional nonlinear stochastic fourth‐order reaction diffusion equation perturbed by the noise is paid close attention by the conforming finite element method in this paper. The semi‐ and fully discrete schemes are obtained. Further, the convergence orders of the semi‐ and fully discrete schemes in L2$$ {L}^2 $$ norm are given detailed proof. The numerical tests are gotten to verify the theoretical result.
{"title":"Conforming finite element method for the time‐fractional nonlinear stochastic fourth‐order reaction diffusion equation","authors":"Xinfei Liu, Xiaoyuan Yang","doi":"10.1002/num.23020","DOIUrl":"https://doi.org/10.1002/num.23020","url":null,"abstract":"The time‐fractional nonlinear stochastic fourth‐order reaction diffusion equation perturbed by the noise is paid close attention by the conforming finite element method in this paper. The semi‐ and fully discrete schemes are obtained. Further, the convergence orders of the semi‐ and fully discrete schemes in L2$$ {L}^2 $$ norm are given detailed proof. The numerical tests are gotten to verify the theoretical result.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"3657 - 3676"},"PeriodicalIF":3.9,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48023160","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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