Pub Date : 2024-10-07DOI: 10.1016/j.camwa.2024.09.018
Thomas Apel, Philipp Zilk
The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their -continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements in the physical domain and to omit redundant degrees of freedom in the vicinity of the singularity. Numerical results validate the effectiveness of hierarchical mesh grading for simulating eigenfunctions of low and high regularity.
{"title":"Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties and graded meshes","authors":"Thomas Apel, Philipp Zilk","doi":"10.1016/j.camwa.2024.09.018","DOIUrl":"10.1016/j.camwa.2024.09.018","url":null,"abstract":"<div><div>The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements in the physical domain and to omit redundant degrees of freedom in the vicinity of the singularity. Numerical results validate the effectiveness of hierarchical mesh grading for simulating eigenfunctions of low and high regularity.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1016/j.camwa.2024.09.030
Lele Wang, Xin Liao, Can Chen
In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the -norm directly, which represents an advancement over the -norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.
本文介绍了非线性正则化长波（RLW）方程的两种线性化二阶能量守恒方案，并利用符合有限元法（FEM）给出了无条件超近和超收敛结果。首先，通过对非线性项的巧妙分解，建立了两个线性化的二阶全离散方案。与之前的非线性方法相比，这些方案大大减少了迭代次数，提高了计算效率；此外，它们还节约了能量，并直接确保了数值解在 H1 规范下的有界性，这与之前研究中报告的 L∞ 规范有界性相比是一个进步。其次，基于 FE 解的有界性、Ritz 投影算子和线性三角形元素的高精度结果，在不限制时间步长 Δt 和空间网格大小 h 之间的比率的情况下，推导出超逼近和超收敛的误差估计值。
{"title":"Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation","authors":"Lele Wang, Xin Liao, Can Chen","doi":"10.1016/j.camwa.2024.09.030","DOIUrl":"10.1016/j.camwa.2024.09.030","url":null,"abstract":"<div><div>In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm directly, which represents an advancement over the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δ<em>t</em> and spatial mesh size <em>h</em>. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421789","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 : 2024-10-04DOI: 10.1016/j.camwa.2024.09.029
Shaolin Ji, Linlin Zhu
In this paper, we construct the reweighted Nadaraya–Watson estimators of the infinitesimal moments for the volatility process of the stochastic volatility models, with the application of the threshold estimator of the unobserved volatility process. Our model includes jumps in both the underlying asset price and its volatility process. We derive the asymptotic properties of the estimators under the infill and long span assumptions. The results are useful for identification of the process. The finite-sample performance of the estimators is studied through Monte Carlo simulation.
{"title":"Reweighted Nadaraya–Watson estimation of stochastic volatility jump-diffusion models","authors":"Shaolin Ji, Linlin Zhu","doi":"10.1016/j.camwa.2024.09.029","DOIUrl":"10.1016/j.camwa.2024.09.029","url":null,"abstract":"<div><div>In this paper, we construct the reweighted Nadaraya–Watson estimators of the infinitesimal moments for the volatility process of the stochastic volatility models, with the application of the threshold estimator of the unobserved volatility process. Our model includes jumps in both the underlying asset price and its volatility process. We derive the asymptotic properties of the estimators under the infill and long span assumptions. The results are useful for identification of the process. The finite-sample performance of the estimators is studied through Monte Carlo simulation.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421788","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 : 2024-10-04DOI: 10.1016/j.camwa.2024.09.033
Jyoti Jaglan , Vikas Maurya , Ankit Singh , Vivek S. Yadav , Manoj K. Rajpoot
This study presents an energy preserving partially implicit scheme for the simulation of wave propagation in homogeneous and heterogeneous mediums. Despite its implicit nature, the developed scheme does not require any explicit numerical or analytical inversion of the coefficient matrix. Theoretical analysis and numerical experiments are performed to validate the energy preserving properties of the fully-discrete scheme. Convergence analysis is also performed to assess the rate of convergence of the developed scheme. The efficiency and accuracy of the developed scheme are validated by numerical solutions of wave propagation in layered heterogeneous mediums. Furthermore, simulations of soliton propagation following nonlinear sine-Gordon and Klein-Gordon equations in homogeneous and heterogeneous mediums are discussed. Numerical solutions are also compared with the results available in the literature. The present method accurately resolves the physical characteristics of the chosen problems, competing well with existing multi-stage time-integration methods. Moreover, it has significantly lower computational complexity than the four-stage, fourth-order Runge-Kutta-Nyström method.
{"title":"Acoustic and soliton propagation using fully-discrete energy preserving partially implicit scheme in homogeneous and heterogeneous mediums","authors":"Jyoti Jaglan , Vikas Maurya , Ankit Singh , Vivek S. Yadav , Manoj K. Rajpoot","doi":"10.1016/j.camwa.2024.09.033","DOIUrl":"10.1016/j.camwa.2024.09.033","url":null,"abstract":"<div><div>This study presents an energy preserving partially implicit scheme for the simulation of wave propagation in homogeneous and heterogeneous mediums. Despite its implicit nature, the developed scheme does not require any explicit numerical or analytical inversion of the coefficient matrix. Theoretical analysis and numerical experiments are performed to validate the energy preserving properties of the fully-discrete scheme. Convergence analysis is also performed to assess the rate of convergence of the developed scheme. The efficiency and accuracy of the developed scheme are validated by numerical solutions of wave propagation in layered heterogeneous mediums. Furthermore, simulations of soliton propagation following nonlinear sine-Gordon and Klein-Gordon equations in homogeneous and heterogeneous mediums are discussed. Numerical solutions are also compared with the results available in the literature. The present method accurately resolves the physical characteristics of the chosen problems, competing well with existing multi-stage time-integration methods. Moreover, it has significantly lower computational complexity than the four-stage, fourth-order Runge-Kutta-Nyström method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421819","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 : 2024-10-04DOI: 10.1016/j.camwa.2024.09.028
Thomas Bellotti
Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.
{"title":"The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes","authors":"Thomas Bellotti","doi":"10.1016/j.camwa.2024.09.028","DOIUrl":"10.1016/j.camwa.2024.09.028","url":null,"abstract":"<div><div>Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a <em>Jin-Xin</em> approximation and high-order splitting formulæ.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421913","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 : 2024-10-03DOI: 10.1016/j.camwa.2024.09.034
Rodrigo L.R. Madureira , Mauro A. Rincon , Ricardo F. Apolaya , Bruno A. Carmo
Existence, uniqueness, energy decay, and approximate numerical solution for the nonlinear wave equation with dynamic control at the boundary is being studied in this work. The theoretical analysis of the problem will be conducted using the Faedo-Galerkin method and compactness results. To obtain the approximate numerical solution, a combined approach of the finite element method and a finite difference method will be employed, known as the linearized Crank-Nicolson Galerkin method. This method optimizes the calculations and preserves the quadratic order of convergence in time. Finally, numerical experiments are performed, and tables and graphs are presented to illustrate the theoretical convergence rates and demonstrate the consistency between theoretical and numerical results.
{"title":"Control of a nonlinear wave equation with a dynamic boundary condition","authors":"Rodrigo L.R. Madureira , Mauro A. Rincon , Ricardo F. Apolaya , Bruno A. Carmo","doi":"10.1016/j.camwa.2024.09.034","DOIUrl":"10.1016/j.camwa.2024.09.034","url":null,"abstract":"<div><div>Existence, uniqueness, energy decay, and approximate numerical solution for the nonlinear wave equation with dynamic control at the boundary is being studied in this work. The theoretical analysis of the problem will be conducted using the Faedo-Galerkin method and compactness results. To obtain the approximate numerical solution, a combined approach of the finite element method and a finite difference method will be employed, known as the linearized Crank-Nicolson Galerkin method. This method optimizes the calculations and preserves the quadratic order of convergence in time. Finally, numerical experiments are performed, and tables and graphs are presented to illustrate the theoretical convergence rates and demonstrate the consistency between theoretical and numerical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421296","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 : 2024-10-03DOI: 10.1016/j.camwa.2024.09.014
Dimitrios S. Lazaridis, Konstantinos A. Draziotis, Nikolaos L. Tsitsas
Block matrices with simultaneously diagonalizable blocks arise in diverse application areas, including, e.g., numerical methods for engineering based on partial differential equations as well as network synchronization, cryptography and control theory. In the present paper, we develop a parallel algorithm for the inversion of block matrices with simultaneously-diagonalizable blocks of order n. First, a sequential version of the algorithm is presented and its computational complexity is determined. Then, a parallelization of the algorithm is implemented and analyzed. The complexity of the derived parallel algorithm is expressed as a function of m and n as well as of the number μ of utilized CPU threads. Results of numerical experiments demonstrate the CPU time superiority of the parallel algorithm versus the respective sequential version and a standard inversion method applied to the original block matrix. An efficient parallelizable procedure to compute the determinants of such block matrices is also described. Numerical examples are presented for using the developed serial and parallel inversion algorithms for boundary-value problems involving transmission problems for the Helmholtz partial differential equation in piecewise homogeneous media.
具有可同时对角的块的块矩阵出现在多种应用领域，包括基于偏微分方程的工程数值方法以及网络同步、密码学和控制理论等。在本文中，我们开发了一种用于反演 m×m 块矩阵的并行算法，该矩阵具有阶数为 n 的可同时对角化的块。然后，实现并分析了该算法的并行化。得出的并行算法的复杂度是 m 和 n 以及使用的 CPU 线程数 μ 的函数。数值实验结果表明，并行算法在 CPU 时间上优于相应的顺序版本和应用于原始块矩阵的标准反演方法。此外，还介绍了计算此类块矩阵行列式的高效可并行程序。还介绍了使用所开发的串行和并行反演算法处理涉及片状均质介质中亥姆霍兹偏微分方程传输问题的边界值问题的数值示例。
{"title":"A parallel algorithm for the inversion of matrices with simultaneously diagonalizable blocks","authors":"Dimitrios S. Lazaridis, Konstantinos A. Draziotis, Nikolaos L. Tsitsas","doi":"10.1016/j.camwa.2024.09.014","DOIUrl":"10.1016/j.camwa.2024.09.014","url":null,"abstract":"<div><div>Block matrices with simultaneously diagonalizable blocks arise in diverse application areas, including, e.g., numerical methods for engineering based on partial differential equations as well as network synchronization, cryptography and control theory. In the present paper, we develop a parallel algorithm for the inversion of <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> block matrices with simultaneously-diagonalizable blocks of order <em>n</em>. First, a sequential version of the algorithm is presented and its computational complexity is determined. Then, a parallelization of the algorithm is implemented and analyzed. The complexity of the derived parallel algorithm is expressed as a function of <em>m</em> and <em>n</em> as well as of the number <em>μ</em> of utilized CPU threads. Results of numerical experiments demonstrate the CPU time superiority of the parallel algorithm versus the respective sequential version and a standard inversion method applied to the original block matrix. An efficient parallelizable procedure to compute the determinants of such block matrices is also described. Numerical examples are presented for using the developed serial and parallel inversion algorithms for boundary-value problems involving transmission problems for the Helmholtz partial differential equation in piecewise homogeneous media.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421917","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 : 2024-10-03DOI: 10.1016/j.camwa.2024.09.027
Satyajith Bommana Boyana , Thomas Lewis , Sijing Liu , Yi Zhang
In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.
{"title":"Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem","authors":"Satyajith Bommana Boyana , Thomas Lewis , Sijing Liu , Yi Zhang","doi":"10.1016/j.camwa.2024.09.027","DOIUrl":"10.1016/j.camwa.2024.09.027","url":null,"abstract":"<div><div>In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419164","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 : 2024-10-01DOI: 10.1016/j.camwa.2024.09.024
Wei Liu , Zhifeng Wang , Gexian Fan , Yingxue Song
In this paper, a dimensionally reduced model is introduced to express the solute transport in the porous media containing with intersecting fractures, in which the fractures are treated as dimensionally reduced manifolds with respect to the dimensions of surrounding media. The transmission conditions can be used to describe the physical behavior of concentration and flux. We construct a hybrid-dimensional finite volume method involving BDF2 time discretization and modified upwind scheme for advection-dominated diffusion model. Fully space-time second-order convergence rate is deduced on the staggered nonuniform grids based on the error estimates of coupling terms. The numerical tests are presented to show that the proposed finite volume method can handle reduced model in porous media with multiple L-shaped, crossing and bifurcated fractures efficiently and flexibly. In addition, the Lagrange multiplier approach is developed to construct bound preserving schemes for dimensionally reduced advection-dominated diffusion model in intersecting fractured porous media.
本文引入了一个降维模型来表达含有相交裂缝的多孔介质中的溶质传输，其中裂缝被视为相对于周围介质尺寸的降维流形。传输条件可用于描述浓度和通量的物理行为。我们为平流主导的扩散模型构建了一种涉及 BDF2 时间离散化和修正上风方案的混合维有限体积方法。根据耦合项的误差估计，在交错非均匀网格上推导出完全时空二阶收敛率。数值测试表明，所提出的有限体积方法可以高效灵活地处理多孔介质中具有多条 L 形、交叉和分叉裂缝的简化模型。此外，还开发了拉格朗日乘法器方法，用于在相交断裂多孔介质中构建降维平流主导扩散模型的保界方案。
{"title":"Modified upwind finite volume scheme with second-order Lagrange multiplier method for dimensionally reduced transport model in intersecting fractured porous media","authors":"Wei Liu , Zhifeng Wang , Gexian Fan , Yingxue Song","doi":"10.1016/j.camwa.2024.09.024","DOIUrl":"10.1016/j.camwa.2024.09.024","url":null,"abstract":"<div><div>In this paper, a dimensionally reduced model is introduced to express the solute transport in the porous media containing with intersecting fractures, in which the fractures are treated as dimensionally reduced manifolds with respect to the dimensions of surrounding media. The transmission conditions can be used to describe the physical behavior of concentration and flux. We construct a hybrid-dimensional finite volume method involving BDF2 time discretization and modified upwind scheme for advection-dominated diffusion model. Fully space-time second-order convergence rate is deduced on the staggered nonuniform grids based on the error estimates of coupling terms. The numerical tests are presented to show that the proposed finite volume method can handle reduced model in porous media with multiple L-shaped, crossing and bifurcated fractures efficiently and flexibly. In addition, the Lagrange multiplier approach is developed to construct bound preserving schemes for dimensionally reduced advection-dominated diffusion model in intersecting fractured porous media.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419163","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 : 2024-09-27DOI: 10.1016/j.camwa.2024.09.010
Marcin Nowak , Eduardo Divo , Tomasz Borkowski , Ewelina Marciniak , Marek Rojczyk , Ryszard Białecki
This research presents a numerical model dedicated for virtual patient diagnostics in the field of synthetic valve implantation. The model operates based on computational fluid dynamics solver with implemented rigid body motion solver. Characteristic indicators related to the prosthetic valve were determined to assess the correctness of cardiac system operation after implantation. A novel approach for dynamic time discretization was developed for reliable and time-efficient calculation. The solver efficiency and computational savings due to application of the developed time-discretization scheme is discussed. Numerical results were validated using experimental data acquired from a test rig, including mass flow meter, pressure transducers, and valve holder designed for this purpose. Multivariant analysis of the model constant was performed towards different levels of the valve resistance to motion. The in-house algorithm was prepared to automatically determine the prosthetic valve position from fast camera images.
{"title":"Flow through a prosthetic mechanical aortic valve: Numerical model and experimental study","authors":"Marcin Nowak , Eduardo Divo , Tomasz Borkowski , Ewelina Marciniak , Marek Rojczyk , Ryszard Białecki","doi":"10.1016/j.camwa.2024.09.010","DOIUrl":"10.1016/j.camwa.2024.09.010","url":null,"abstract":"<div><div>This research presents a numerical model dedicated for virtual patient diagnostics in the field of synthetic valve implantation. The model operates based on computational fluid dynamics solver with implemented rigid body motion solver. Characteristic indicators related to the prosthetic valve were determined to assess the correctness of cardiac system operation after implantation. A novel approach for dynamic time discretization was developed for reliable and time-efficient calculation. The solver efficiency and computational savings due to application of the developed time-discretization scheme is discussed. Numerical results were validated using experimental data acquired from a test rig, including mass flow meter, pressure transducers, and valve holder designed for this purpose. Multivariant analysis of the model constant was performed towards different levels of the valve resistance to motion. The in-house algorithm was prepared to automatically determine the prosthetic valve position from fast camera images.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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