Pub Date : 2025-03-31DOI: 10.1016/j.jcp.2025.113973
Songzheng Fan , Jiaxian Qin , Yidao Dong , Yi Jiang , Xiaogang Deng
Machine learning-based techniques have been introduced to help enhance the performance of high-order shock-capturing schemes in recent years. In this work, a novel neural network is devised to address the accuracy reduction issue faced by previous machine learning-based schemes. By fully leveraging the features of multi-resolution strategy, optimal accuracy of the original numerical scheme can be formally preserved at all grid levels by the proposed WCNS3-MR-NN scheme. Meanwhile, the present scheme is designed to achieve high-resolution property and robust shock-capturing ability simultaneously. Analysis and numerical experiments are presented for validation. The results confirm that WCNS3-MR-NN maintains its optimal accuracy even at the presence of extreme points, and demonstrates excellent performance across a wide range of benchmark cases.
{"title":"WCNS3-MR-NN: A machine learning-based shock-capturing scheme with accuracy-preserving and high-resolution properties","authors":"Songzheng Fan , Jiaxian Qin , Yidao Dong , Yi Jiang , Xiaogang Deng","doi":"10.1016/j.jcp.2025.113973","DOIUrl":"10.1016/j.jcp.2025.113973","url":null,"abstract":"<div><div>Machine learning-based techniques have been introduced to help enhance the performance of high-order shock-capturing schemes in recent years. In this work, a novel neural network is devised to address the accuracy reduction issue faced by previous machine learning-based schemes. By fully leveraging the features of multi-resolution strategy, optimal accuracy of the original numerical scheme can be formally preserved at all grid levels by the proposed WCNS3-MR-NN scheme. Meanwhile, the present scheme is designed to achieve high-resolution property and robust shock-capturing ability simultaneously. Analysis and numerical experiments are presented for validation. The results confirm that WCNS3-MR-NN maintains its optimal accuracy even at the presence of extreme points, and demonstrates excellent performance across a wide range of benchmark cases.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113973"},"PeriodicalIF":3.8,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143745788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-28DOI: 10.1016/j.jcp.2025.113970
Nanyi Zheng , Daniel Hayes , Andrew Christlieb , Jing-Mei Qiu
High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear advection and nonlinear Vlasov-Poisson systems without dimensional splitting. The proposed method leverages the semi-Lagrangian approach to allow for significantly larger time steps while also exploiting the low-rank structure of the solution. This is achieved through cross approximation of matrices, also referred to as CUR or pseudo-skeleton approximation, where representative columns and rows are selected using specific strategies. To maintain numerical stability and ensure local mass conservation, we apply singular value truncation and a mass-conservative projection following the cross approximation of the updated solution. The computational complexity of our method scales linearly with the mesh size N per dimension, compared to the complexity of traditional full-rank methods per time step. The algorithm is extended to handle nonlinear Vlasov-Poisson systems using a Runge-Kutta exponential integrator. Moreover, we evolve the macroscopic conservation laws for charge densities implicitly, enabling the use of large time steps that align with the semi-Lagrangian solver. We also perform a mass-conservative correction to ensure that the adaptive rank solution preserves macroscopic charge density conservation. To validate the efficiency and effectiveness of our method, we conduct a series of benchmark tests on both linear advection and nonlinear Vlasov-Poisson systems. The proposed algorithm will have the potential in overcoming the curse of dimensionality for beyond 2D high dimensional problems, which is the subject of our future work.
{"title":"A semi-Lagrangian adaptive-rank (SLAR) method for linear advection and nonlinear Vlasov-Poisson system","authors":"Nanyi Zheng , Daniel Hayes , Andrew Christlieb , Jing-Mei Qiu","doi":"10.1016/j.jcp.2025.113970","DOIUrl":"10.1016/j.jcp.2025.113970","url":null,"abstract":"<div><div>High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear advection and nonlinear Vlasov-Poisson systems without dimensional splitting. The proposed method leverages the semi-Lagrangian approach to allow for significantly larger time steps while also exploiting the low-rank structure of the solution. This is achieved through cross approximation of matrices, also referred to as CUR or pseudo-skeleton approximation, where representative columns and rows are selected using specific strategies. To maintain numerical stability and ensure local mass conservation, we apply singular value truncation and a mass-conservative projection following the cross approximation of the updated solution. The computational complexity of our method scales linearly with the mesh size <em>N</em> per dimension, compared to the <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> complexity of traditional full-rank methods per time step. The algorithm is extended to handle nonlinear Vlasov-Poisson systems using a Runge-Kutta exponential integrator. Moreover, we evolve the macroscopic conservation laws for charge densities implicitly, enabling the use of large time steps that align with the semi-Lagrangian solver. We also perform a mass-conservative correction to ensure that the adaptive rank solution preserves macroscopic charge density conservation. To validate the efficiency and effectiveness of our method, we conduct a series of benchmark tests on both linear advection and nonlinear Vlasov-Poisson systems. The proposed algorithm will have the potential in overcoming the curse of dimensionality for beyond 2D high dimensional problems, which is the subject of our future work.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113970"},"PeriodicalIF":3.8,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-27DOI: 10.1016/j.jcp.2025.113955
Jonas Beddrich, Stephan B. Lunowa, Barbara Wohlmuth
We address the numerical challenge of solving the Hookean-type time-fractional Navier–Stokes–Fokker–Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two d-dimensional spaces in the turbulent regime. Due to its high dimensionality, the non-locality with respect to time, and the resolution required to resolve turbulent flow, this problem is highly demanding.
To overcome these challenges, we employ the Hermite spectral method for the configuration space of the Fokker–Planck equation, reducing the problem to a purely macroscopic model. Considering scenarios for available analytical solutions, we prove the existence of a Hermite scaling parameter, which exactly reproduces the analytical polymer stress tensor. With this choice, the macroscopic system is equivalent to solving the coupled micro-macro system. We apply second-order time integration and extrapolation of the coupling terms, achieving, for the first time, convergence rates for the fully coupled time-fractional system independent of the order of the time-fractional derivative.
Our efficient implementation of the numerical scheme allows turbulent simulations of dilute polymeric fluids with memory effects in two and three dimensions. Numerical simulations show that memory effects weaken the drag-reducing effect of added polymer molecules in the turbulent flow regime.
{"title":"Numerical simulation of dilute polymeric fluids with memory effects in the turbulent flow regime","authors":"Jonas Beddrich, Stephan B. Lunowa, Barbara Wohlmuth","doi":"10.1016/j.jcp.2025.113955","DOIUrl":"10.1016/j.jcp.2025.113955","url":null,"abstract":"<div><div>We address the numerical challenge of solving the Hookean-type time-fractional Navier–Stokes–Fokker–Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two <em>d</em>-dimensional spaces in the turbulent regime. Due to its high dimensionality, the non-locality with respect to time, and the resolution required to resolve turbulent flow, this problem is highly demanding.</div><div>To overcome these challenges, we employ the Hermite spectral method for the configuration space of the Fokker–Planck equation, reducing the problem to a purely macroscopic model. Considering scenarios for available analytical solutions, we prove the existence of a Hermite scaling parameter, which exactly reproduces the analytical polymer stress tensor. With this choice, the macroscopic system is equivalent to solving the coupled micro-macro system. We apply second-order time integration and extrapolation of the coupling terms, achieving, for the first time, convergence rates for the fully coupled time-fractional system independent of the order of the time-fractional derivative.</div><div>Our efficient implementation of the numerical scheme allows turbulent simulations of dilute polymeric fluids with memory effects in two and three dimensions. Numerical simulations show that memory effects weaken the drag-reducing effect of added polymer molecules in the turbulent flow regime.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113955"},"PeriodicalIF":3.8,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-27DOI: 10.1016/j.jcp.2025.113959
Liyan Luo , Qi Li , Fei Fei , Lei Wu
A deterministic-stochastic coupling scheme is developed for simulating rarefied gas flows, where the key process is the alternative solving of the macroscopic synthetic equations [Su et al. (2020) [22]] and the mesoscopic equation via the asymptotic-preserving time-relaxed Monte Carlo scheme [Fei (2023) [19]]. Firstly, the macroscopic synthetic equations are exactly derived from the Boltzmann equation, incorporating not only the Newtonian viscosity and Fourier thermal conduction laws but also higher-order constitutive relations that capture rarefaction effects; the latter are extracted from the stochastic solver over a defined sampling interval. Secondly, the macroscopic synthetic equations, with the initial field extracted from the stochastic solver over the same sampling interval, are solved to the steady state or over a certain iteration steps. Finally, the simulation particles in the stochastic solver are updated to match the density, velocity, and temperature obtained from the macroscopic synthetic equations. Moreover, simulation particles in the subsequent interval will be partly sampled according to the solutions of macroscopic synthetic equations. As a result, our coupling strategy enhances the asymptotic-preserving characteristic of the stochastic solver and substantially accelerates convergence towards the steady state. Several numerical tests are performed, and it is found that our method can reduce the computational cost in the near-continuum flow regime by two orders of magnitude compared to the direct simulation Monte Carlo method.
{"title":"Boosting the convergence of DSMC by GSIS","authors":"Liyan Luo , Qi Li , Fei Fei , Lei Wu","doi":"10.1016/j.jcp.2025.113959","DOIUrl":"10.1016/j.jcp.2025.113959","url":null,"abstract":"<div><div>A deterministic-stochastic coupling scheme is developed for simulating rarefied gas flows, where the key process is the alternative solving of the macroscopic synthetic equations [Su et al. (2020) <span><span>[22]</span></span>] and the mesoscopic equation via the asymptotic-preserving time-relaxed Monte Carlo scheme [Fei (2023) <span><span>[19]</span></span>]. Firstly, the macroscopic synthetic equations are exactly derived from the Boltzmann equation, incorporating not only the Newtonian viscosity and Fourier thermal conduction laws but also higher-order constitutive relations that capture rarefaction effects; the latter are extracted from the stochastic solver over a defined sampling interval. Secondly, the macroscopic synthetic equations, with the initial field extracted from the stochastic solver over the same sampling interval, are solved to the steady state or over a certain iteration steps. Finally, the simulation particles in the stochastic solver are updated to match the density, velocity, and temperature obtained from the macroscopic synthetic equations. Moreover, simulation particles in the subsequent interval will be partly sampled according to the solutions of macroscopic synthetic equations. As a result, our coupling strategy enhances the asymptotic-preserving characteristic of the stochastic solver and substantially accelerates convergence towards the steady state. Several numerical tests are performed, and it is found that our method can reduce the computational cost in the near-continuum flow regime by two orders of magnitude compared to the direct simulation Monte Carlo method.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113959"},"PeriodicalIF":3.8,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-26DOI: 10.1016/j.jcp.2025.113957
Vedant Puri, Aviral Prakash, Levent Burak Kara, Yongjie Jessica Zhang
Reduced order modeling lowers the computational cost of solving PDEs by learning a low-dimensional spatial representation from data and dynamically evolving these representations using manifold projections of the governing equations. The commonly used linear subspace reduced-order models (ROMs) are often suboptimal for problems with a slow decay of Kolmogorov n-width, such as advection-dominated fluid flows at high Reynolds numbers. There has been a growing interest in nonlinear ROMs that use state-of-the-art representation learning techniques to accurately capture such phenomena with fewer degrees of freedom. We propose smooth neural field ROM (SNF-ROM), a nonlinear reduced order modeling framework that combines grid-free reduced representations with Galerkin projection. The SNF-ROM architecture constrains the learned ROM trajectories to a smoothly varying path, which proves beneficial in the dynamics evaluation when the reduced manifold is traversed in accordance with the governing PDEs. Furthermore, we devise robust regularization schemes to ensure the learned neural fields are smooth and differentiable. This allows us to compute physics-based dynamics of the reduced system nonintrusively with automatic differentiation and evolve the reduced system with classical time-integrators. SNF-ROM leads to fast offline training as well as enhanced accuracy and stability during the online dynamics evaluation. Numerical experiments reveal that SNF-ROM is able to accelerate the full-order computation by up to 199×. We demonstrate the efficacy of SNF-ROM on a range of advection-dominated linear and nonlinear PDE problems where we consistently outperform state-of-the-art ROMs.
{"title":"SNF-ROM: Projection-based nonlinear reduced order modeling with smooth neural fields","authors":"Vedant Puri, Aviral Prakash, Levent Burak Kara, Yongjie Jessica Zhang","doi":"10.1016/j.jcp.2025.113957","DOIUrl":"10.1016/j.jcp.2025.113957","url":null,"abstract":"<div><div>Reduced order modeling lowers the computational cost of solving PDEs by learning a low-dimensional spatial representation from data and dynamically evolving these representations using manifold projections of the governing equations. The commonly used linear subspace reduced-order models (ROMs) are often suboptimal for problems with a slow decay of Kolmogorov <em>n</em>-width, such as advection-dominated fluid flows at high Reynolds numbers. There has been a growing interest in nonlinear ROMs that use state-of-the-art representation learning techniques to accurately capture such phenomena with fewer degrees of freedom. We propose smooth neural field ROM (SNF-ROM), a nonlinear reduced order modeling framework that combines grid-free reduced representations with Galerkin projection. The SNF-ROM architecture constrains the learned ROM trajectories to a smoothly varying path, which proves beneficial in the dynamics evaluation when the reduced manifold is traversed in accordance with the governing PDEs. Furthermore, we devise robust regularization schemes to ensure the learned neural fields are smooth and differentiable. This allows us to compute physics-based dynamics of the reduced system nonintrusively with automatic differentiation and evolve the reduced system with classical time-integrators. SNF-ROM leads to fast offline training as well as enhanced accuracy and stability during the online dynamics evaluation. Numerical experiments reveal that SNF-ROM is able to accelerate the full-order computation by up to 199×. We demonstrate the efficacy of SNF-ROM on a range of advection-dominated linear and nonlinear PDE problems where we consistently outperform state-of-the-art ROMs.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113957"},"PeriodicalIF":3.8,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-26DOI: 10.1016/j.jcp.2025.113954
Jamie M. Taylor , David Pardo , Judit Muñoz-Matute
Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions –the natural function spaces for PDEs– by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor approximations in practice. For example, classical fully-connected feed-forward NNs fail to approximate continuous functions whose gradient is discontinuous when employing strong formulations like in Physics Informed Neural Networks (PINNs). In this article, we propose the use of regularity-conforming neural networks, where a priori information on the regularity of solutions to PDEs can be employed to construct proper architectures. We illustrate the potential of such architectures via a two-dimensional (2D) transmission problem, where the solution may admit discontinuities in the gradient across interfaces, as well as power-like singularities at certain points. In particular, we formulate the weak transmission problem in a PINNs-like strong formulation with interface and continuity conditions. Such architectures are partially explainable; discontinuities are explicitly described, allowing the introduction of novel terms into the loss function. We demonstrate via several model problems in one and two dimensions the advantages of using regularity-conforming architectures in contrast to classical architectures. The ideas presented in this article easily extend to problems in higher dimensions.
{"title":"Regularity-conforming neural networks (ReCoNNs) for solving partial differential equations","authors":"Jamie M. Taylor , David Pardo , Judit Muñoz-Matute","doi":"10.1016/j.jcp.2025.113954","DOIUrl":"10.1016/j.jcp.2025.113954","url":null,"abstract":"<div><div>Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions –the natural function spaces for PDEs– by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor approximations in practice. For example, classical fully-connected feed-forward NNs fail to approximate continuous functions whose gradient is discontinuous when employing strong formulations like in Physics Informed Neural Networks (PINNs). In this article, we propose the use of regularity-conforming neural networks, where <em>a priori</em> information on the regularity of solutions to PDEs can be employed to construct proper architectures. We illustrate the potential of such architectures via a two-dimensional (2D) transmission problem, where the solution may admit discontinuities in the gradient across interfaces, as well as power-like singularities at certain points. In particular, we formulate the weak transmission problem in a PINNs-like strong formulation with interface and continuity conditions. Such architectures are partially explainable; discontinuities are explicitly described, allowing the introduction of novel terms into the loss function. We demonstrate via several model problems in one and two dimensions the advantages of using regularity-conforming architectures in contrast to classical architectures. The ideas presented in this article easily extend to problems in higher dimensions.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113954"},"PeriodicalIF":3.8,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143745789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores the application of hybrid of the finite element method and multiscale finite element method (FEM-MsFEM) to address the steady-state Stokes-Darcy problems with Beavers-Joseph-Saffman (BJS) interface conditions in highly heterogeneous porous media. We propose an algorithm for this multiscale Stokes-Darcy model. The FEM-MsFEM hybrid method adapts to the characteristics of different regions. MsFEM basis functions are applied to the Darcy region, whereas standard finite element basis functions are utilized for the Stokes region. Afterward, the FEM-MsFEM basis functions are used for computations with the Robin-Robin domain decomposition algorithm. Furthermore, this special domain decomposition algorithm preserves a convergence rate independent of the mesh size. Subsequently, we conduct error analysis based on and norms for this FEM-MsFEM hybrid method. Finally, we present extensive numerical tests, illustrating the results of error and convergence analysis.
{"title":"FEM-MsFEM hybrid method for the Stokes-Darcy model","authors":"Yachen Hong , Wenhan Zhang , Lina Zhao , Haibiao Zheng","doi":"10.1016/j.jcp.2025.113952","DOIUrl":"10.1016/j.jcp.2025.113952","url":null,"abstract":"<div><div>This paper explores the application of hybrid of the finite element method and multiscale finite element method (FEM-MsFEM) to address the steady-state Stokes-Darcy problems with Beavers-Joseph-Saffman (BJS) interface conditions in highly heterogeneous porous media. We propose an algorithm for this multiscale Stokes-Darcy model. The FEM-MsFEM hybrid method adapts to the characteristics of different regions. MsFEM basis functions are applied to the Darcy region, whereas standard finite element basis functions are utilized for the Stokes region. Afterward, the FEM-MsFEM basis functions are used for computations with the Robin-Robin domain decomposition algorithm. Furthermore, this special domain decomposition algorithm preserves a convergence rate independent of the mesh size. Subsequently, we conduct error analysis based on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norms for this FEM-MsFEM hybrid method. Finally, we present extensive numerical tests, illustrating the results of error and convergence analysis.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113952"},"PeriodicalIF":3.8,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-25DOI: 10.1016/j.jcp.2025.113953
Xinyu Wu , Hui Guo , Ziyao Xu , Lulu Tian , Yang Yang
Wormholes are high-permeability, deep-penetrating, narrow channels formed during the acidizing process, which serves as a popular stimulation treatment. For the study of wormhole formation in naturally fractured porous media, we develop a novel hybrid-dimensional two-scale continuum wormhole model, with fractures represented as Dirac-δ functions. As an extension of the reinterpreted discrete fracture model (RDFM) [50], the model is applicable to nonconforming meshes and adaptive to intersecting fractures in reservoirs without introducing additional computational complexity. A numerical scheme based on the local discontinuous Galerkin (LDG) method is constructed for the corresponding dimensionless model to accommodate the presence of Dirac-δ functions and the property of flux discontinuity. Moreover, a bound-preserving technique is introduced to theoretically ensure the boundedness of acid concentration and porosity between 0 and 1, as well as the monotone increase in porosity during simulation. The performance of the model and algorithms is validated, and the effects of various parameters on wormhole propagation are analyzed through several numerical experiments, contributing to the acidizing design in fractured reservoirs.
{"title":"A reinterpreted discrete fracture model for wormhole propagation in fractured porous media","authors":"Xinyu Wu , Hui Guo , Ziyao Xu , Lulu Tian , Yang Yang","doi":"10.1016/j.jcp.2025.113953","DOIUrl":"10.1016/j.jcp.2025.113953","url":null,"abstract":"<div><div>Wormholes are high-permeability, deep-penetrating, narrow channels formed during the acidizing process, which serves as a popular stimulation treatment. For the study of wormhole formation in naturally fractured porous media, we develop a novel hybrid-dimensional two-scale continuum wormhole model, with fractures represented as Dirac-<em>δ</em> functions. As an extension of the reinterpreted discrete fracture model (RDFM) <span><span>[50]</span></span>, the model is applicable to nonconforming meshes and adaptive to intersecting fractures in reservoirs without introducing additional computational complexity. A numerical scheme based on the local discontinuous Galerkin (LDG) method is constructed for the corresponding dimensionless model to accommodate the presence of Dirac-<em>δ</em> functions and the property of flux discontinuity. Moreover, a bound-preserving technique is introduced to theoretically ensure the boundedness of acid concentration and porosity between 0 and 1, as well as the monotone increase in porosity during simulation. The performance of the model and algorithms is validated, and the effects of various parameters on wormhole propagation are analyzed through several numerical experiments, contributing to the acidizing design in fractured reservoirs.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113953"},"PeriodicalIF":3.8,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143725289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Deep learning is widely used to predict complex dynamical systems in many scientific and engineering areas. However, the black-box nature of these deep learning models presents significant challenges for carrying out simultaneous data assimilation (DA), which is a crucial technique for state estimation, model identification, and reconstructing missing data. Integrating ensemble-based DA methods with nonlinear deep learning models is computationally expensive and may suffer from large sampling errors. To address these challenges, we introduce a deep learning framework designed to simultaneously provide accurate forecasts and efficient DA. It is named Conditional Gaussian Koopman Network (CGKN), which transforms general nonlinear systems into nonlinear neural differential equations with conditional Gaussian structures. CGKN aims to retain essential nonlinear components while applying systematic and minimal simplifications to facilitate the development of analytic formulae for nonlinear DA. This allows for seamless integration of DA performance into the deep learning training process, eliminating the need for empirical tuning as required in ensemble methods. CGKN compensates for structural simplifications by lifting the dimension of the system, which is motivated by Koopman theory. Nevertheless, CGKN exploits special nonlinear dynamics within the lifted space. This enables the model to capture extreme events and strong non-Gaussian features in joint and marginal distributions with appropriate uncertainty quantification. We demonstrate the effectiveness of CGKN for both prediction and DA on three strongly nonlinear and non-Gaussian turbulent systems: the projected stochastic Burgers–Sivashinsky equation, the Lorenz 96 system, and the El Niño-Southern Oscillation. The results justify the robustness and computational efficiency of CGKN.
{"title":"CGKN: A deep learning framework for modeling complex dynamical systems and efficient data assimilation","authors":"Chuanqi Chen , Nan Chen , Yinling Zhang , Jin-Long Wu","doi":"10.1016/j.jcp.2025.113950","DOIUrl":"10.1016/j.jcp.2025.113950","url":null,"abstract":"<div><div>Deep learning is widely used to predict complex dynamical systems in many scientific and engineering areas. However, the black-box nature of these deep learning models presents significant challenges for carrying out simultaneous data assimilation (DA), which is a crucial technique for state estimation, model identification, and reconstructing missing data. Integrating ensemble-based DA methods with nonlinear deep learning models is computationally expensive and may suffer from large sampling errors. To address these challenges, we introduce a deep learning framework designed to simultaneously provide accurate forecasts and efficient DA. It is named Conditional Gaussian Koopman Network (CGKN), which transforms general nonlinear systems into nonlinear neural differential equations with conditional Gaussian structures. CGKN aims to retain essential nonlinear components while applying systematic and minimal simplifications to facilitate the development of analytic formulae for nonlinear DA. This allows for seamless integration of DA performance into the deep learning training process, eliminating the need for empirical tuning as required in ensemble methods. CGKN compensates for structural simplifications by lifting the dimension of the system, which is motivated by Koopman theory. Nevertheless, CGKN exploits special nonlinear dynamics within the lifted space. This enables the model to capture extreme events and strong non-Gaussian features in joint and marginal distributions with appropriate uncertainty quantification. We demonstrate the effectiveness of CGKN for both prediction and DA on three strongly nonlinear and non-Gaussian turbulent systems: the projected stochastic Burgers–Sivashinsky equation, the Lorenz 96 system, and the El Niño-Southern Oscillation. The results justify the robustness and computational efficiency of CGKN.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113950"},"PeriodicalIF":3.8,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, higher-order spatial models including the gradient model and Laplacian model based on Taylor's series and using their coordinates as coefficients are evaluated by calculating some simple functions and a diffusion problem. The numerical convergence is achieved by the models as the smaller particle distance can calculate more accurate results. By using the models, when the particle distribution is significantly irregular, increasing the search radius can involve more neighboring particles which consequently improves the accuracy. Based on the proposed higher-order models, the higher-order Explicit Incompressible version of the Moving Particle Semi-implicit method (EI-MPS) is developed. The numerical scheme is validated by simulating various free surface flows including the rotation of a fluid square patch, the impact of two identical rectangular fluid patches, oscillating drop under a central force field, a hydrostatic problem, and dam-break flow. The parameters of the particle distance, search radius, and repeated time in the pressure calculation are all examined in the free surface flows. The proposed method can reproduce the free surface variations, kinetic energy, and total energy variation in the violent flows. It can also obtain the hydrostatic pressure achieving numerical convergence. Increasing the search radius can result in larger errors in simulating the hydrostatic pressure. The impacting pressure caused by the dam-break flow is reflected by the method in good agreement with the experimental measurements.
{"title":"Higher-order MPS models and higher-order Explicit Incompressible MPS (EI-MPS) method to simulate free-surface flows","authors":"Tibing Xu , Seiichi Koshizuka , Tsuyoshi Koyama , Toshihide Saka , Osamu Imazeki","doi":"10.1016/j.jcp.2025.113951","DOIUrl":"10.1016/j.jcp.2025.113951","url":null,"abstract":"<div><div>In this study, higher-order spatial models including the gradient model and Laplacian model based on Taylor's series and using their coordinates as coefficients are evaluated by calculating some simple functions and a diffusion problem. The numerical convergence is achieved by the models as the smaller particle distance can calculate more accurate results. By using the models, when the particle distribution is significantly irregular, increasing the search radius can involve more neighboring particles which consequently improves the accuracy. Based on the proposed higher-order models, the higher-order Explicit Incompressible version of the Moving Particle Semi-implicit method (EI-MPS) is developed. The numerical scheme is validated by simulating various free surface flows including the rotation of a fluid square patch, the impact of two identical rectangular fluid patches, oscillating drop under a central force field, a hydrostatic problem, and dam-break flow. The parameters of the particle distance, search radius, and repeated time in the pressure calculation are all examined in the free surface flows. The proposed method can reproduce the free surface variations, kinetic energy, and total energy variation in the violent flows. It can also obtain the hydrostatic pressure achieving numerical convergence. Increasing the search radius can result in larger errors in simulating the hydrostatic pressure. The impacting pressure caused by the dam-break flow is reflected by the method in good agreement with the experimental measurements.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113951"},"PeriodicalIF":3.8,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143706232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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