Pub Date : 2025-02-05DOI: 10.1016/j.camwa.2025.01.021
Sarvesh Kumar , Devika Shylaja
This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the conforming virtual element method for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented.
{"title":"Nonconforming virtual element method for an incompressible miscible displacement problem in porous media","authors":"Sarvesh Kumar , Devika Shylaja","doi":"10.1016/j.camwa.2025.01.021","DOIUrl":"10.1016/j.camwa.2025.01.021","url":null,"abstract":"<div><div>This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the <span><math><mi>H</mi><mo>(</mo><mrow><mi>div</mi></mrow><mo>)</mo></math></span> conforming virtual element method for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 153-179"},"PeriodicalIF":2.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143125106","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 : 2025-02-05DOI: 10.1016/j.camwa.2025.01.024
Zhiwei Yang , Jie Ma , Ning Du , Hong Wang
We develop a peridynamic model, called non-local bond-based linear peridynamics model, for viscoelastic materials based on the fractional derivatives, which captures power-law responses and so provides a very competitive description of the mechanical vibrations. We accordingly derive a numerical scheme, efficient collocation method, to simulate the viscoelastic nonlocal model. Finally, we investigate the performance of the peridynamics model, which shows the utility of the new model and the efficiency of the fast scheme.
{"title":"A bond-based linear peridynamic model for viscoelastic materials and its efficient collocation method","authors":"Zhiwei Yang , Jie Ma , Ning Du , Hong Wang","doi":"10.1016/j.camwa.2025.01.024","DOIUrl":"10.1016/j.camwa.2025.01.024","url":null,"abstract":"<div><div>We develop a peridynamic model, called non-local bond-based linear peridynamics model, for viscoelastic materials based on the fractional derivatives, which captures power-law responses and so provides a very competitive description of the mechanical vibrations. We accordingly derive a numerical scheme, efficient collocation method, to simulate the viscoelastic nonlocal model. Finally, we investigate the performance of the peridynamics model, which shows the utility of the new model and the efficiency of the fast scheme.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 121-136"},"PeriodicalIF":2.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143102662","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 : 2025-02-05DOI: 10.1016/j.camwa.2025.01.025
Wensheng Li , Chuncheng Wang , Hanting Guan , Jian Wang , Jie Yang , Chao Zhang , Dacheng Tao
Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs), but there still remain some challenges in the application of PINNs, for example, how to exhaustively utilize a small size of (usually very few) labeled samples, which are the exact solutions to the PDEs or their high-accuracy approximations, to improve the accuracy and the training efficiency. In this paper, we propose the generative adversarial physics-informed neural networks (GA-PINNs), which integrate the generative adversarial (GA) mechanism with original PINNs, to improve the performance of PINNs by exploiting a small size of labeled samples. The numerical experiments show that, compared with the original PINNs equipped with an additive loss computed on these labeled samples, GA-PINNs can more effectively utilize the small size of labeled samples when solving forward and inverse problems. As a generalization of GA-PINNs, we also combine the GA mechanism with the deep Ritz method (DRM) and the deep Galerkin method (DGM) to form GA-DRM and GA-DGM, respectively. The experimental results validate their superiority as well.
{"title":"Generative adversarial physics-informed neural networks for solving forward and inverse problem with small labeled samples","authors":"Wensheng Li , Chuncheng Wang , Hanting Guan , Jian Wang , Jie Yang , Chao Zhang , Dacheng Tao","doi":"10.1016/j.camwa.2025.01.025","DOIUrl":"10.1016/j.camwa.2025.01.025","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs), but there still remain some challenges in the application of PINNs, for example, how to exhaustively utilize a small size of (usually very few) labeled samples, which are the exact solutions to the PDEs or their high-accuracy approximations, to improve the accuracy and the training efficiency. In this paper, we propose the generative adversarial physics-informed neural networks (GA-PINNs), which integrate the generative adversarial (GA) mechanism with original PINNs, to improve the performance of PINNs by exploiting a small size of labeled samples. The numerical experiments show that, compared with the original PINNs equipped with an additive loss computed on these labeled samples, GA-PINNs can more effectively utilize the small size of labeled samples when solving forward and inverse problems. As a generalization of GA-PINNs, we also combine the GA mechanism with the deep Ritz method (DRM) and the deep Galerkin method (DGM) to form GA-DRM and GA-DGM, respectively. The experimental results validate their superiority as well.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 98-120"},"PeriodicalIF":2.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143102661","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 : 2025-02-05DOI: 10.1016/j.camwa.2025.01.022
Tiantian Bao, Xiufang Feng
This paper reports a class of new hybrid compact finite-difference schemes with high-order accuracy for solving the Helmholtz equation in two and three dimensions. The innovation of the scheme is to hybridize explicit and implicit compact schemes to deal with the solution and its first- and second-order derivatives, and to solve it by a step-by-step coupled iterative method. In response to the inefficiency of serial algorithms in solving the Helmholtz equation with large wavenumber and the issue of memory overflow on a single processor making computation infeasible because of excessively large computational scale, a parallel algorithm is proposed based on the Message Passing Interface (MPI) environment. Truncation error analysis demonstrates sixth-order accuracy for the proposed scheme, and numerical experiments confirm the theoretical sixth-order accuracy for problems with variable and large wavenumber. Also, the MPI-based parallel algorithm exhibits great parallel speedup and enables the tackling of large-scale problems that are beyond the reach of serial algorithms.
{"title":"A new parallel algorithm with high-order finite difference scheme for solving the Helmholtz equation in two and three dimensions","authors":"Tiantian Bao, Xiufang Feng","doi":"10.1016/j.camwa.2025.01.022","DOIUrl":"10.1016/j.camwa.2025.01.022","url":null,"abstract":"<div><div>This paper reports a class of new hybrid compact finite-difference schemes with high-order accuracy for solving the Helmholtz equation in two and three dimensions. The innovation of the scheme is to hybridize explicit and implicit compact schemes to deal with the solution and its first- and second-order derivatives, and to solve it by a step-by-step coupled iterative method. In response to the inefficiency of serial algorithms in solving the Helmholtz equation with large wavenumber and the issue of memory overflow on a single processor making computation infeasible because of excessively large computational scale, a parallel algorithm is proposed based on the Message Passing Interface (MPI) environment. Truncation error analysis demonstrates sixth-order accuracy for the proposed scheme, and numerical experiments confirm the theoretical sixth-order accuracy for problems with variable and large wavenumber. Also, the MPI-based parallel algorithm exhibits great parallel speedup and enables the tackling of large-scale problems that are beyond the reach of serial algorithms.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 71-97"},"PeriodicalIF":2.9,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143102663","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 : 2025-02-04DOI: 10.1016/j.camwa.2025.01.034
M. Arrutselvi , E. Natarajan , S. Natarajan
The virtual element method (VEM) was proposed for the nonlinear convection-diffusion-reaction problem in [5]. Using projection operators, a computable VEM discrete scheme was derived and the existence of the solution was proved. However, even when higher order elements were introduced, the SUPG framework shows spurious oscillations in the crosswind direction. In this paper, we propose, in the context of VEM, a shock capture technique inspired by the work of [30] to provide a stable and more robust solution technique that does not exhibit numerical oscillations when higher order elements are employed. For the proposed framework, convergence analysis is performed and optimal order error estimates are derived in the energy norm. Numerical experiments are computed to show the performance of this technique and to validate the theoretical results obtained.
{"title":"Numerical solution of nonlinear convection-diffusion-reaction equation using a stabilized virtual element method","authors":"M. Arrutselvi , E. Natarajan , S. Natarajan","doi":"10.1016/j.camwa.2025.01.034","DOIUrl":"10.1016/j.camwa.2025.01.034","url":null,"abstract":"<div><div>The virtual element method (VEM) was proposed for the nonlinear convection-diffusion-reaction problem in <span><span>[5]</span></span>. Using projection operators, a computable VEM discrete scheme was derived and the existence of the solution was proved. However, even when higher order elements were introduced, the SUPG framework shows spurious oscillations in the crosswind direction. In this paper, we propose, in the context of VEM, a shock capture technique inspired by the work of <span><span>[30]</span></span> to provide a stable and more robust solution technique that does not exhibit numerical oscillations when higher order elements are employed. For the proposed framework, convergence analysis is performed and optimal order error estimates are derived in the energy norm. Numerical experiments are computed to show the performance of this technique and to validate the theoretical results obtained.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 46-70"},"PeriodicalIF":2.9,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143102664","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 : 2025-02-01DOI: 10.1016/j.camwa.2024.12.003
Jiliang Cao , Wansheng Wang , Aiguo Xiao
In this paper, we study a posteriori error estimates of the fast scheme for time discretization of time fractional parabolic differential equations. To overcome the huge workload caused by the nonlocality of fractional derivative, a fast algorithm is applied to the construction of the scheme. Employing the numerical solution obtained by the fast scheme, a piecewise continuous function approximating the exact solution is constructed. Then, by exploring the error equations, a posteriori error estimates are obtained in different norms, which depend only on the discretization parameters and the data of the problems. Various numerical experiments for the fractional parabolic equations with smooth or nonsmooth exact solutions on different time meshes, including the frequently-used graded mesh, are carried out to verify and complement our theoretical results. Based on the obtained a posteriori error estimates, a time adaptive algorithm is proposed to reduce the computational cost substantially and provides efficient error control for the solution.
{"title":"Adaptive fast L1 − 2 scheme for solving time fractional parabolic problems","authors":"Jiliang Cao , Wansheng Wang , Aiguo Xiao","doi":"10.1016/j.camwa.2024.12.003","DOIUrl":"10.1016/j.camwa.2024.12.003","url":null,"abstract":"<div><div>In this paper, we study a posteriori error estimates of the fast <span><math><mi>L</mi><mn>1</mn><mo>−</mo><mn>2</mn></math></span> scheme for time discretization of time fractional parabolic differential equations. To overcome the huge workload caused by the nonlocality of fractional derivative, a fast algorithm is applied to the construction of the <span><math><mi>L</mi><mn>1</mn><mo>−</mo><mn>2</mn></math></span> scheme. Employing the numerical solution obtained by the fast <span><math><mi>L</mi><mn>1</mn><mo>−</mo><mn>2</mn></math></span> scheme, a piecewise continuous function approximating the exact solution is constructed. Then, by exploring the error equations, a posteriori error estimates are obtained in different norms, which depend only on the discretization parameters and the data of the problems. Various numerical experiments for the fractional parabolic equations with smooth or nonsmooth exact solutions on different time meshes, including the frequently-used graded mesh, are carried out to verify and complement our theoretical results. Based on the obtained a posteriori error estimates, a time adaptive algorithm is proposed to reduce the computational cost substantially and provides efficient error control for the solution.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"179 ","pages":"Pages 59-76"},"PeriodicalIF":2.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142804472","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}
This paper presents an effective nonlocal finite element method (FEM) for investigating the free vibration behavior of sigmoid functionally graded material (S-FGM) nanoshells using nonlocal elasticity theory. The effective nonlocal parameters via third-order shear deformation theory (TSDT) are varied along the thickness of the nanoshells following the sigmoid function. In this study, two different sigmoid functions FGM (S1-FGM and S2-FGM) are considered for the ceramic volume fraction. For S1-FGM, the top and bottom surfaces are ceramic and metal, respectively, whereas the middle surface has the average properties of its constituent materials. In order to increase the stiffness of S1-FGM, ceramic and metal are used at the bottom and midplane surfaces, respectively, to form S2-FGM, which is used to investigate and compare with S1-FGM. The governing equation of the S-FGM nanoshells is formulated based on Hamilton's principle. The numerical results are obtained by finite element method (FEM) with a nine-node quadrilateral (Q9) Lagrangian element and are in close agreement with the published results. The numerical investigation indicates that the frequency parameter decreases with increasing nonlocal parameters. The frequency parameters of S1-FGM nanoshells decrease slowly when the sigmoid material index increases, whereas the frequency parameters of the S2-FGM shells increase quickly (0 ≤ n ≤ 1). then slowly as the sigmoid material index increases. Finally, the effects of the geometrical parameters of the S-FGM nanoshells accounting for the effective nonlocal parameters on the non-dimensional frequency parameter are investigated.
{"title":"Effective nonlocal finite element formulation for free vibration analysis of S-FGM doubly curved nanoshells based on linear strain–displacement relations using TSDT","authors":"Weeraphan Jiammeepreecha , Komkorn Chaidachatorn , Boonchai Phungpaingam , Karun Klaycham , Somchai Chucheepsakul","doi":"10.1016/j.camwa.2024.11.021","DOIUrl":"10.1016/j.camwa.2024.11.021","url":null,"abstract":"<div><div>This paper presents an effective nonlocal finite element method (FEM) for investigating the free vibration behavior of sigmoid functionally graded material (S-FGM) nanoshells using nonlocal elasticity theory. The effective nonlocal parameters via third-order shear deformation theory (TSDT) are varied along the thickness of the nanoshells following the sigmoid function. In this study, two different sigmoid functions FGM (S1-FGM and S2-FGM) are considered for the ceramic volume fraction. For S1-FGM, the top and bottom surfaces are ceramic and metal, respectively, whereas the middle surface has the average properties of its constituent materials. In order to increase the stiffness of S1-FGM, ceramic and metal are used at the bottom and midplane surfaces, respectively, to form S2-FGM, which is used to investigate and compare with S1-FGM. The governing equation of the S-FGM nanoshells is formulated based on Hamilton's principle. The numerical results are obtained by finite element method (FEM) with a nine-node quadrilateral (Q9) Lagrangian element and are in close agreement with the published results. The numerical investigation indicates that the frequency parameter decreases with increasing nonlocal parameters. The frequency parameters of S1-FGM nanoshells decrease slowly when the sigmoid material index increases, whereas the frequency parameters of the S2-FGM shells increase quickly (0 ≤ <em>n</em> ≤ 1). then slowly as the sigmoid material index increases. Finally, the effects of the geometrical parameters of the S-FGM nanoshells accounting for the effective nonlocal parameters on the non-dimensional frequency parameter are investigated.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"179 ","pages":"Pages 77-102"},"PeriodicalIF":2.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841383","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}
Behavioral non-equilibrium hyperbolic traffic models, derived from approximated car-following models with human factors, can lose their conservative form, rendering traditional flux-based numerical methods ineffective. This challenge also applies to the recently proposed behavioral continuum (non-equilibrium traffic model based on risk allostasis theory, that is, NET-RAT) model. This paper is focused on solving the Riemann problem and several other initial-value problems for the novel NET-RAT model in the non-conservative form by path-conservative central-upwind (PCCU) schemes. We design extensive numerical tests considering the unique behavioral properties of the NET-RAT model. The PCCU schemes are then applied to these tests and the obtained results demonstrate that major wave types are effectively and accurately captured. At the same time, the fifth-order scheme, which is constructed using an alternative weighted essentially non-oscillatory (A-WENO) approach, yields substantially sharper resolution than its second-order counterpart. The presented numerical study can facilitate the practical implementation of the NET-RAT model for real-world traffic.
{"title":"Numerical study of the non-conservative NET-RAT traffic flow model by path-conservative central-upwind schemes","authors":"Saeed Mohammadian , Zuduo Zheng , Shaoshuai Chu , Alexander Kurganov","doi":"10.1016/j.camwa.2024.12.014","DOIUrl":"10.1016/j.camwa.2024.12.014","url":null,"abstract":"<div><div>Behavioral non-equilibrium hyperbolic traffic models, derived from approximated car-following models with human factors, can lose their conservative form, rendering traditional flux-based numerical methods ineffective. This challenge also applies to the recently proposed behavioral continuum (non-equilibrium traffic model based on risk allostasis theory, that is, NET-RAT) model. This paper is focused on solving the Riemann problem and several other initial-value problems for the novel NET-RAT model in the non-conservative form by path-conservative central-upwind (PCCU) schemes. We design extensive numerical tests considering the unique behavioral properties of the NET-RAT model. The PCCU schemes are then applied to these tests and the obtained results demonstrate that major wave types are effectively and accurately captured. At the same time, the fifth-order scheme, which is constructed using an alternative weighted essentially non-oscillatory (A-WENO) approach, yields substantially sharper resolution than its second-order counterpart. The presented numerical study can facilitate the practical implementation of the NET-RAT model for real-world traffic.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"179 ","pages":"Pages 212-228"},"PeriodicalIF":2.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142911898","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 : 2025-02-01DOI: 10.1016/j.camwa.2024.12.011
Mostafa Abbaszadeh , Maryam Parvizi , Amirreza Khodadadian , Thomas Wick , Mehdi Dehghan
Meshless methods have become increasingly popular for solving a wide range of problems in both solid and fluid mechanics. In this study, we focus on a meshless numerical approach to solve the tropical Pacific Ocean model, which captures the horizontal velocity and layer thickness of ocean waves, using an advanced meshless Galerkin technique known as the reproducing kernel particle method (RKPM). To address the temporal component in this scheme, we apply a Crank-Nicolson finite difference method, resulting in a semi-discrete formulation. For spatial discretization, we use a kernel-based meshless Galerkin method that integrates the strengths of finite element methods with reproducing kernel particle approximations. We conduct a comprehensive stability analysis and provide an a priori estimate for the semi-discrete solution. Furthermore, we derive error estimates for both the semi-discrete and fully discrete solutions. Finally, we validate the theoretical findings and evaluate the method's efficiency through real-world test cases.
{"title":"A reproducing kernel particle method (RKPM) algorithm for solving the tropical Pacific Ocean model","authors":"Mostafa Abbaszadeh , Maryam Parvizi , Amirreza Khodadadian , Thomas Wick , Mehdi Dehghan","doi":"10.1016/j.camwa.2024.12.011","DOIUrl":"10.1016/j.camwa.2024.12.011","url":null,"abstract":"<div><div>Meshless methods have become increasingly popular for solving a wide range of problems in both solid and fluid mechanics. In this study, we focus on a meshless numerical approach to solve the tropical Pacific Ocean model, which captures the horizontal velocity and layer thickness of ocean waves, using an advanced meshless Galerkin technique known as the reproducing kernel particle method (RKPM). To address the temporal component in this scheme, we apply a Crank-Nicolson finite difference method, resulting in a semi-discrete formulation. For spatial discretization, we use a kernel-based meshless Galerkin method that integrates the strengths of finite element methods with reproducing kernel particle approximations. We conduct a comprehensive stability analysis and provide an a priori estimate for the semi-discrete solution. Furthermore, we derive error estimates for both the semi-discrete and fully discrete solutions. Finally, we validate the theoretical findings and evaluate the method's efficiency through real-world test cases.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"179 ","pages":"Pages 197-211"},"PeriodicalIF":2.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142911904","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 : 2025-02-01DOI: 10.1016/j.camwa.2024.12.002
Junlei Mu , Congshan Zhuo , Qingdian Zhang , Sha Liu , Chengwen Zhong
The high-order gas-kinetic scheme (HGKS) with WENO spatial reconstruction method has been extensively validated through numerous numerical experiments, demonstrating its superior accuracy, efficiency and robustness. In comparison to WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. This paper introduces two types of fifth-order HGKS utilizing TENO class schemes. One approach involves replacing the WENO scheme with the TENO scheme in the conventional WENO GKS. Both WENO and TENO schemes provide non-equilibrium state values at the cell interface. The slopes of the non-equilibrium state, as well as the values and slopes of the equilibrium state, are obtained through additional linear reconstruction. Another type of HGKS scheme named TENO GKS with adaptive order (TENO-A GKS), is similar to most multi-resolution HGKS schemes. Following a robust scale-separation procedure, a tailored ENO-like stencil selection strategy is proposed to ensure that high-order accuracy is maintained in smooth regions by selecting the candidate reconstruction from a large stencil, while enforcing the ENO property near discontinuities by adopting the candidate reconstruction from small smooth stencils. These TENO schemes include TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) and TENO schemes with dual ENO-like stencil selection (TENO-D). The HGKS scheme based on TENO-A reconstruction takes advantage of the large stencil to provide point values and slopes of the non-equilibrium states. By dynamically merging the reconstructed non-equilibrium slopes, the need for extra reconstruction of the equilibrium states at the beginning of each time step can be avoided. The simple TENO-A GKS scheme exhibits better robustness compared to the conventional WENO GKS or TENO GKS.
{"title":"High-order gas-kinetic scheme with TENO class reconstruction for the Euler and Navier-Stokes equations","authors":"Junlei Mu , Congshan Zhuo , Qingdian Zhang , Sha Liu , Chengwen Zhong","doi":"10.1016/j.camwa.2024.12.002","DOIUrl":"10.1016/j.camwa.2024.12.002","url":null,"abstract":"<div><div>The high-order gas-kinetic scheme (HGKS) with WENO spatial reconstruction method has been extensively validated through numerous numerical experiments, demonstrating its superior accuracy, efficiency and robustness. In comparison to WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. This paper introduces two types of fifth-order HGKS utilizing TENO class schemes. One approach involves replacing the WENO scheme with the TENO scheme in the conventional WENO GKS. Both WENO and TENO schemes provide non-equilibrium state values at the cell interface. The slopes of the non-equilibrium state, as well as the values and slopes of the equilibrium state, are obtained through additional linear reconstruction. Another type of HGKS scheme named TENO GKS with adaptive order (TENO-A GKS), is similar to most multi-resolution HGKS schemes. Following a robust scale-separation procedure, a tailored ENO-like stencil selection strategy is proposed to ensure that high-order accuracy is maintained in smooth regions by selecting the candidate reconstruction from a large stencil, while enforcing the ENO property near discontinuities by adopting the candidate reconstruction from small smooth stencils. These TENO schemes include TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) and TENO schemes with dual ENO-like stencil selection (TENO-D). The HGKS scheme based on TENO-A reconstruction takes advantage of the large stencil to provide point values and slopes of the non-equilibrium states. By dynamically merging the reconstructed non-equilibrium slopes, the need for extra reconstruction of the equilibrium states at the beginning of each time step can be avoided. The simple TENO-A GKS scheme exhibits better robustness compared to the conventional WENO GKS or TENO GKS.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"179 ","pages":"Pages 126-147"},"PeriodicalIF":2.9,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841380","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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