Pub Date : 2025-01-23DOI: 10.1016/j.apnum.2025.01.010
Qifeng Zhang , Haiyan Cao , Hongyu Qin
In this paper, two classes of linearized difference schemes are presented for the one and two-dimensional pseudo parabolic equations with logarithmic nonlinearity. These schemes are derived based on the implicit-explicit second-order backward differential formula (BDF2) for the temporal discretization and fourth-order compact/second-order difference schemes for spatial discretization. With the help of the truncation function method and regularization technique, error estimates for the logarithmic nonlinear term are handled rigorously. As a result, the convergence of the fully-discrete schemes is obtained based on the energy argument. Extensive numerical examples are presented to confirm the theoretical results.
{"title":"Error estimates of implicit-explicit compact BDF2 schemes for the pseudo parabolic equations with logarithmic nonlinearity","authors":"Qifeng Zhang , Haiyan Cao , Hongyu Qin","doi":"10.1016/j.apnum.2025.01.010","DOIUrl":"10.1016/j.apnum.2025.01.010","url":null,"abstract":"<div><div>In this paper, two classes of linearized difference schemes are presented for the one and two-dimensional pseudo parabolic equations with logarithmic nonlinearity. These schemes are derived based on the implicit-explicit second-order backward differential formula (BDF2) for the temporal discretization and fourth-order compact/second-order difference schemes for spatial discretization. With the help of the truncation function method and regularization technique, error estimates for the logarithmic nonlinear term are handled rigorously. As a result, the convergence of the fully-discrete schemes is obtained based on the energy argument. Extensive numerical examples are presented to confirm the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 135-154"},"PeriodicalIF":2.2,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-23DOI: 10.1016/j.apnum.2025.01.014
Haifeng Wang, Yan Wang, Hong Zhang, Songhe Song
In this paper, we develop a second-order accurate scheme for the extended Fisher–Kolmogorov (EFK) equation and investigate its global-in-time energy stability and convergence. The proposed scheme uses the Fourier spectral collocation method in space and the stabilization Runge–Kutta–Munthe–Kaas-2e (RKMK2e) method for temporal approximation. To demonstrate the global-in-time energy stability of the proposed scheme, we first verify that, under the assumption that all numerical solutions are uniformly bounded, the scheme is energy stable when using a sufficiently large stabilization parameter. Then, to establish the uniform-in-time boundedness of the numerical solutions, we fully utilize the nonlinear operator estimates and discrete Sobolev embedding in each stage of the scheme. Moreover, we conduct an optimal rate convergence analysis with a sufficient regularity assumption for the exact solution. Several numerical examples are presented to validate the accuracy, computational efficiency, and energy stability of the proposed scheme.
{"title":"Energy stability and error estimate of the RKMK2e scheme for the extended Fisher–Kolmogorov equation","authors":"Haifeng Wang, Yan Wang, Hong Zhang, Songhe Song","doi":"10.1016/j.apnum.2025.01.014","DOIUrl":"10.1016/j.apnum.2025.01.014","url":null,"abstract":"<div><div>In this paper, we develop a second-order accurate scheme for the extended Fisher–Kolmogorov (EFK) equation and investigate its global-in-time energy stability and convergence. The proposed scheme uses the Fourier spectral collocation method in space and the stabilization Runge–Kutta–Munthe–Kaas-2e (RKMK2e) method for temporal approximation. To demonstrate the global-in-time energy stability of the proposed scheme, we first verify that, under the assumption that all numerical solutions are uniformly bounded, the scheme is energy stable when using a sufficiently large stabilization parameter. Then, to establish the uniform-in-time boundedness of the numerical solutions, we fully utilize the nonlinear operator estimates and discrete Sobolev embedding in each stage of the scheme. Moreover, we conduct an optimal rate convergence analysis with a sufficient regularity assumption for the exact solution. Several numerical examples are presented to validate the accuracy, computational efficiency, and energy stability of the proposed scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 60-76"},"PeriodicalIF":2.2,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-23DOI: 10.1016/j.apnum.2025.01.013
Tapas Roy, Dilip Kumar Maiti
Our focus is on refining the Variational Iteration Method (VIM), which will henceforth be called R-VIM. We begin by introducing a general linear operator and consequently the Lagrange's multiplier; then, minimizing the residual error, we subsequently find the optimal linear operator as well as the best fitting Lagrange's multiplier for a given differential equation. By means of Banach Fixed Point Theorem, the necessary conditions for the convergence of the solutions of VIM, Optimal VIM, and R-VIM are established involving the Lagrange's multiplier; it has been demonstrated (and subsequently numerically certified) that solutions of VIM and Optimal VIM locally converge. While R-VIM solutions, with the aid of necessary and sufficient criteria, have been shown to converge globally for appropriate values of the unknown parameters involved in the general linear operator. We make sure that the convergence of the VIM solution depends critically on the optimal selection of the linear operator and, consequently, the best suited Lagrange's multiplier.
Our bird's eye view would be to differentiate between the contributions of the homotopy equation in homotopy-based approaches and Lagrange's multiplier-based correction functional of the VIM for the fastest convergence. Furthermore, in order to achieve fast convergence, we additionally examine if the optimal VIM's regulating parameter offers any further benefit above proposed R-VIM. We aim to apply R-VIM to problems with multiple solutions to ensure that all solutions are accurately identified. Our proposed method has been proven superior and more widely applicable through both theoretical and numerical illustrations. However, it also has its limits. Hence, we suggest the implementation of the multistep R-VIM, referred to as MR-VIM, as a means to solve chaotic dynamical systems.
{"title":"Refined variational iteration method through optimal linear operator and Lagrange multiplier: Local/global convergence","authors":"Tapas Roy, Dilip Kumar Maiti","doi":"10.1016/j.apnum.2025.01.013","DOIUrl":"10.1016/j.apnum.2025.01.013","url":null,"abstract":"<div><div>Our focus is on refining the Variational Iteration Method (VIM), which will henceforth be called R-VIM. We begin by introducing a general linear operator and consequently the Lagrange's multiplier; then, minimizing the residual error, we subsequently find the optimal linear operator as well as the best fitting Lagrange's multiplier for a given differential equation. By means of Banach Fixed Point Theorem, the necessary conditions for the convergence of the solutions of VIM, Optimal VIM, and R-VIM are established involving the Lagrange's multiplier; it has been demonstrated (and subsequently numerically certified) that solutions of VIM and Optimal VIM locally converge. While R-VIM solutions, with the aid of necessary and sufficient criteria, have been shown to converge globally for appropriate values of the unknown parameters involved in the general linear operator. We make sure that the convergence of the VIM solution depends critically on the optimal selection of the linear operator and, consequently, the best suited Lagrange's multiplier.</div><div>Our bird's eye view would be to differentiate between the contributions of the homotopy equation in homotopy-based approaches and Lagrange's multiplier-based correction functional of the VIM for the fastest convergence. Furthermore, in order to achieve fast convergence, we additionally examine if the optimal VIM's regulating parameter offers any further benefit above proposed R-VIM. We aim to apply R-VIM to problems with multiple solutions to ensure that all solutions are accurately identified. Our proposed method has been proven superior and more widely applicable through both theoretical and numerical illustrations. However, it also has its limits. Hence, we suggest the implementation of the multistep R-VIM, referred to as MR-VIM, as a means to solve chaotic dynamical systems.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 1-28"},"PeriodicalIF":2.2,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.apnum.2025.01.009
Jingjing Xiao , Desong Kong
Coupling the averaged L1 scheme and the Crank–Nicolson scheme for temporal derivatives, we study Maxwell's equations in the Cole–Cole dispersive medium. A rigorous analysis is carried out to show that the proposed scheme is unconditionally stable and has a second-order convergence in time for sufficiently smooth solutions. We avoid using the mathematical induction method, which simplifies the analysis than the existing schemes. A fully discrete scheme with a finite difference method at Yee's grid is proposed. Numerical experiments are carefully designed to illustrate our theoretical analysis.
{"title":"An unconditionally stable second-order scheme for Maxwell's equations in the Cole–Cole dispersive medium","authors":"Jingjing Xiao , Desong Kong","doi":"10.1016/j.apnum.2025.01.009","DOIUrl":"10.1016/j.apnum.2025.01.009","url":null,"abstract":"<div><div>Coupling the averaged L1 scheme and the Crank–Nicolson scheme for temporal derivatives, we study Maxwell's equations in the Cole–Cole dispersive medium. A rigorous analysis is carried out to show that the proposed scheme is unconditionally stable and has a second-order convergence in time for sufficiently smooth solutions. We avoid using the mathematical induction method, which simplifies the analysis than the existing schemes. A fully discrete scheme with a finite difference method at Yee's grid is proposed. Numerical experiments are carefully designed to illustrate our theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 211-227"},"PeriodicalIF":2.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.apnum.2025.01.012
Houchao Zhang, Junjun Wang, Xueran Gong
This paper is concerned with the construction and analysis of two structure-preserving finite element approximation schemes for solving one and two dimensional coupled Schrödinger-Boussinesq (SBq) equations. Firstly, two finite element approximation schemes are developed and the total mass and energy preserving properties of the proposed schemes are demonstrated in the discrete sense. Secondly, by use of the innovative cut-off function method, an auxiliary fully-discrete system is established, which leads to the unique solvability with the Brouwer's fixed-pointed theorem and the optimal error estimates for the proposed nonlinear scheme. Finally, linearized iterative algorithms are showed rigorously and extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical methods.
{"title":"Optimal L2 error estimates of two structure-preserving finite element methods for Schrödinger-Boussinesq equations","authors":"Houchao Zhang, Junjun Wang, Xueran Gong","doi":"10.1016/j.apnum.2025.01.012","DOIUrl":"10.1016/j.apnum.2025.01.012","url":null,"abstract":"<div><div>This paper is concerned with the construction and analysis of two structure-preserving finite element approximation schemes for solving one and two dimensional coupled Schrödinger-Boussinesq (SBq) equations. Firstly, two finite element approximation schemes are developed and the total mass and energy preserving properties of the proposed schemes are demonstrated in the discrete sense. Secondly, by use of the innovative cut-off function method, an auxiliary fully-discrete system is established, which leads to the unique solvability with the Brouwer's fixed-pointed theorem and the optimal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimates for the proposed nonlinear scheme. Finally, linearized iterative algorithms are showed rigorously and extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 193-210"},"PeriodicalIF":2.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-22DOI: 10.1016/j.apnum.2025.01.011
Akbar Shirilord, Mehdi Dehghan
This paper explores advanced gradient descent-based parameter-free methods for solving coupled matrix equations and examines their applications in dynamical systems. We focus on the coupled matrix equations where are given matrices, and are the unknown matrices to be determined. We propose a novel gradient descent-based approach with parameter-free, enhancing convergence through an accelerated technique related to momentum methods. A comprehensive analysis of the convergence and characteristics of these methods is provided. Our convergence analysis demonstrates that if the spectrum of a block matrix constructed from the matrices A, B, D, and E is confined within a horizontal ellipse in the complex plane, centered at with a major axis length of 3 and a minor axis length of 1, then the accelerated momentum method will converge to the exact solution of the discussed model. The numerical results indicate that proposed methods significantly improve efficiency, showing faster convergence and reduced computational time compared to traditional approaches. Additionally, we apply these methods to linear dynamic systems, demonstrating their effectiveness in real-world scenarios.
{"title":"Gradient descent-based parameter-free methods for solving coupled matrix equations and studying an application in dynamical systems","authors":"Akbar Shirilord, Mehdi Dehghan","doi":"10.1016/j.apnum.2025.01.011","DOIUrl":"10.1016/j.apnum.2025.01.011","url":null,"abstract":"<div><div>This paper explores advanced gradient descent-based parameter-free methods for solving coupled matrix equations and examines their applications in dynamical systems. We focus on the coupled matrix equations<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>A</mi><mi>X</mi><mo>+</mo><mi>Y</mi><mi>B</mi><mo>=</mo><mi>C</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>D</mi><mi>X</mi><mo>+</mo><mi>Y</mi><mi>E</mi><mo>=</mo><mi>F</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>A</mi><mo>,</mo><mi>D</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>,</mo><mi>B</mi><mo>,</mo><mi>E</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> are given matrices, and <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> are the unknown matrices to be determined. We propose a novel gradient descent-based approach with parameter-free, enhancing convergence through an accelerated technique related to momentum methods. A comprehensive analysis of the convergence and characteristics of these methods is provided. Our convergence analysis demonstrates that if the spectrum of a block matrix constructed from the matrices <em>A</em>, <em>B</em>, <em>D</em>, and <em>E</em> is confined within a horizontal ellipse in the complex plane, centered at <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span> with a major axis length of 3 and a minor axis length of 1, then the accelerated momentum method will converge to the exact solution of the discussed model. The numerical results indicate that proposed methods significantly improve efficiency, showing faster convergence and reduced computational time compared to traditional approaches. Additionally, we apply these methods to linear dynamic systems, demonstrating their effectiveness in real-world scenarios.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 29-59"},"PeriodicalIF":2.2,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-17DOI: 10.1016/j.apnum.2025.01.007
Sergio Caucao , Gabriel N. Gatica , Luis F. Gatica
We consider a Banach spaces-based mixed variational formulation that has been recently proposed for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations, and develop a reliable and efficient residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, we utilize the global inf-sup condition of the problem, combined with appropriate small data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and the local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local -spaces, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. In particular, the case of flow through a 2D porous medium with fracture networks is considered.
{"title":"A posteriori error analysis of a mixed finite element method for the stationary convective Brinkman–Forchheimer problem","authors":"Sergio Caucao , Gabriel N. Gatica , Luis F. Gatica","doi":"10.1016/j.apnum.2025.01.007","DOIUrl":"10.1016/j.apnum.2025.01.007","url":null,"abstract":"<div><div>We consider a Banach spaces-based mixed variational formulation that has been recently proposed for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations, and develop a reliable and efficient residual-based <em>a posteriori</em> error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, we utilize the global inf-sup condition of the problem, combined with appropriate small data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and the local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-spaces, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. In particular, the case of flow through a 2D porous medium with fracture networks is considered.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 158-178"},"PeriodicalIF":2.2,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-16DOI: 10.1016/j.apnum.2025.01.008
Yuling Guo , Zhongqing Wang , Chao Zhang
In this paper, we introduce a robust mapping Legendre spectral-Galerkin method for addressing elastic problems in simply-connected, fan-shaped domains with curved boundaries. By employing a polar coordinate transformation, we map the fan-shaped domain onto a rectangle, which transforms the original elastic equation into a variable coefficient equation. We then develop a Legendre spectral-Galerkin scheme for this variable coefficient equation. Additionally, we demonstrate the optimal convergence of the numerical solution in the -norm as the Lamé coefficient λ remains bounded. Numerical examples illustrate the high accuracy and robustness of our method, even as λ approaches infinity.
{"title":"A robust mapping spectral method for elastic equations in curved fan-shaped domains","authors":"Yuling Guo , Zhongqing Wang , Chao Zhang","doi":"10.1016/j.apnum.2025.01.008","DOIUrl":"10.1016/j.apnum.2025.01.008","url":null,"abstract":"<div><div>In this paper, we introduce a robust mapping Legendre spectral-Galerkin method for addressing elastic problems in simply-connected, fan-shaped domains with curved boundaries. By employing a polar coordinate transformation, we map the fan-shaped domain onto a rectangle, which transforms the original elastic equation into a variable coefficient equation. We then develop a Legendre spectral-Galerkin scheme for this variable coefficient equation. Additionally, we demonstrate the optimal convergence of the numerical solution in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm as the Lamé coefficient <em>λ</em> remains bounded. Numerical examples illustrate the high accuracy and robustness of our method, even as <em>λ</em> approaches infinity.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 144-157"},"PeriodicalIF":2.2,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-15DOI: 10.1016/j.apnum.2025.01.005
Meng Li , Dan Wang , Junjun Wang , Xiaolong Zhao
In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter , the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) [15] and Liao et al. (2021) [29].
{"title":"Variable-time-step weighted IMEX FEMs for nonlinear evolution equations","authors":"Meng Li , Dan Wang , Junjun Wang , Xiaolong Zhao","doi":"10.1016/j.apnum.2025.01.005","DOIUrl":"10.1016/j.apnum.2025.01.005","url":null,"abstract":"<div><div>In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter <span><math><mi>θ</mi><mo>=</mo><mn>1</mn></math></span>, the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) <span><span>[15]</span></span> and Liao et al. (2021) <span><span>[29]</span></span>.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 123-143"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-15DOI: 10.1016/j.apnum.2025.01.006
Fengna Yan , Yinhua Xia
We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.
{"title":"Analysis of average bound preserving time-implicit discretizations for convection-diffusion-reaction equation","authors":"Fengna Yan , Yinhua Xia","doi":"10.1016/j.apnum.2025.01.006","DOIUrl":"10.1016/j.apnum.2025.01.006","url":null,"abstract":"<div><div>We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 103-122"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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