Pub Date : 2025-02-18DOI: 10.1016/j.jcp.2025.113850
Wei Liu , Zhifeng Wang , Pengshan Wang
This work presents a development in the context of dimensionally reduced multi-layer models in three-dimensional fractured reservoirs, in which the fractures and damage zones are reduced to be surfaces. The single phase flow is governed by fully coupled models with Neumann boundary involving Brinkman equation in two-dimensional hydraulic fractures, Darcy equation in surrounding two-dimensional damage zones and three-dimensional rock matrix. A hybrid-dimensional finite volume method is proposed to demonstrate efficient handling of various multi-layer configurations with Brinkman-type and Darcy-type transmission interface conditions. Additionally, the convergence analysis of proposed numerical method yields fully space-time second-order on staggered nonuniform grids. The sensitivity analysis of effective viscosity is tested by numerical results for three-dimensional problems with intersecting fractures and damage zones to show the performance of the proposed method.
{"title":"Finite volume method for reduced multi-layer model of compressible Brinkman flow in high-dimensional fractured reservoirs with damage zones","authors":"Wei Liu , Zhifeng Wang , Pengshan Wang","doi":"10.1016/j.jcp.2025.113850","DOIUrl":"10.1016/j.jcp.2025.113850","url":null,"abstract":"<div><div>This work presents a development in the context of dimensionally reduced multi-layer models in three-dimensional fractured reservoirs, in which the fractures and damage zones are reduced to be surfaces. The single phase flow is governed by fully coupled models with Neumann boundary involving Brinkman equation in two-dimensional hydraulic fractures, Darcy equation in surrounding two-dimensional damage zones and three-dimensional rock matrix. A hybrid-dimensional finite volume method is proposed to demonstrate efficient handling of various multi-layer configurations with Brinkman-type and Darcy-type transmission interface conditions. Additionally, the convergence analysis of proposed numerical method yields fully space-time second-order on staggered nonuniform grids. The sensitivity analysis of effective viscosity is tested by numerical results for three-dimensional problems with intersecting fractures and damage zones to show the performance of the proposed method.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113850"},"PeriodicalIF":3.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437832","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 : 2025-02-18DOI: 10.1016/j.jcp.2025.113860
Seungchan Ko, Sanghyeon Park
Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it remains unclear in many aspects how to effectively train PINNs if the solutions of PDEs exhibit stiff behaviors or high frequencies. In this paper, we propose a new method for training PINNs using variable-scaling techniques. This method is simple and it can be applied to a wide range of problems including PDEs with rapidly-varying solutions. Throughout various numerical experiments, we will demonstrate the effectiveness of the proposed method for these problems and confirm that it can significantly improve the training efficiency and performance of PINNs. Furthermore, based on the analysis of the neural tangent kernel (NTK), we will provide theoretical evidence for this phenomenon and show that our methods can indeed improve the performance of PINNs.
{"title":"VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior","authors":"Seungchan Ko, Sanghyeon Park","doi":"10.1016/j.jcp.2025.113860","DOIUrl":"10.1016/j.jcp.2025.113860","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it remains unclear in many aspects how to effectively train PINNs if the solutions of PDEs exhibit stiff behaviors or high frequencies. In this paper, we propose a new method for training PINNs using variable-scaling techniques. This method is simple and it can be applied to a wide range of problems including PDEs with rapidly-varying solutions. Throughout various numerical experiments, we will demonstrate the effectiveness of the proposed method for these problems and confirm that it can significantly improve the training efficiency and performance of PINNs. Furthermore, based on the analysis of the neural tangent kernel (NTK), we will provide theoretical evidence for this phenomenon and show that our methods can indeed improve the performance of PINNs.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113860"},"PeriodicalIF":3.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453715","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 : 2025-02-18DOI: 10.1016/j.jcp.2025.113851
Duc Thach Son Vu, Tan M. Nguyen, Weiqing Ren
In this paper, we investigate the time evolution of compositional multiphase flows in porous media using Graph Neural Networks (GNN). A recent approach to this problem is the unified formulation introduced by Lauser et al. (2011) [2], which incorporates the complementarity conditions. The advantage of this formulation is its ability to automatically handle the appearance and disappearance of phases. To solve the system of equations numerically, Ben Gharbia and Flauraud (2019) [13] employed the Newton-min method. More recently, Vu et al. (2021) [14] proposed a new strategy called NPIPM (NonParametric Interior-Point Method). However, these existing methods still face challenges, particularly with convergence when using large time steps during iterations. Inspired by the relationships between a cell and its neighborhood cells in the mesh when applying the finite volume method (FVM) to solve the problem, we recognize that these connections can be represented as a graph of nodes and edges in a Graph Neural Network. This GNN approach provides a promising framework for predicting long-term phenomena in porous media flows, especially when integrated into a hybrid algorithm along with other numerical solvers.
{"title":"Learning and predicting dynamics of compositional multiphase mixtures using Graph Neural Networks","authors":"Duc Thach Son Vu, Tan M. Nguyen, Weiqing Ren","doi":"10.1016/j.jcp.2025.113851","DOIUrl":"10.1016/j.jcp.2025.113851","url":null,"abstract":"<div><div>In this paper, we investigate the time evolution of compositional multiphase flows in porous media using Graph Neural Networks (GNN). A recent approach to this problem is the unified formulation introduced by Lauser et al. (2011) <span><span>[2]</span></span>, which incorporates the complementarity conditions. The advantage of this formulation is its ability to automatically handle the appearance and disappearance of phases. To solve the system of equations numerically, Ben Gharbia and Flauraud (2019) <span><span>[13]</span></span> employed the Newton-min method. More recently, Vu et al. (2021) <span><span>[14]</span></span> proposed a new strategy called NPIPM (NonParametric Interior-Point Method). However, these existing methods still face challenges, particularly with convergence when using large time steps during iterations. Inspired by the relationships between a cell and its neighborhood cells in the mesh when applying the finite volume method (FVM) to solve the problem, we recognize that these connections can be represented as a graph of nodes and edges in a Graph Neural Network. This GNN approach provides a promising framework for predicting long-term phenomena in porous media flows, especially when integrated into a hybrid algorithm along with other numerical solvers.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113851"},"PeriodicalIF":3.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437794","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 : 2025-02-17DOI: 10.1016/j.jcp.2025.113858
Hongtao Liu , Chang Lu , Guangqing Xia , Rony Keppens , Giovanni Lapenta
In this paper, we develop an efficient energy conserving semi-Lagrangian (ECSL) kinetic scheme for the Vlasov-Maxwell (VM) system. The proposed ECSL scheme is the first semi-Lagrangian VM solver to achieve unconditional stability with respect to the plasma period while conserving total energy, without requiring nonlinear iterations. The new method is built on three key components: an efficient field solver, an exact splitting (ES) method, and a conservative Semi-Lagrangian (CSL) scheme. Specifically, the efficient field solver is developed by semi-implicitly coupling the Ampère and Vlasov moment equations, removing the numerical constraints imposed by the plasma period. Furthermore, the ES method is applied to the Vlasov equation and integrated with an implicit Maxwell solver, ensuring energy conservation in rotational dynamics. The ES method reduces the multidimensional Vlasov equation to one-dimensional advections exactly in time, which are then solved with the CSL scheme to ensure mass conservation and remove CFL restrictions. These synergistic components enable the ECSL scheme to conserve both total energy and mass at the fully discrete level, regardless of spatial and temporal resolution. Finally, several numerical experiments are presented to demonstrate the accuracy, efficiency, and conservation properties of the proposed method.
{"title":"An efficient energy conserving semi-Lagrangian kinetic scheme for the Vlasov-Maxwell system","authors":"Hongtao Liu , Chang Lu , Guangqing Xia , Rony Keppens , Giovanni Lapenta","doi":"10.1016/j.jcp.2025.113858","DOIUrl":"10.1016/j.jcp.2025.113858","url":null,"abstract":"<div><div>In this paper, we develop an efficient energy conserving semi-Lagrangian (ECSL) kinetic scheme for the Vlasov-Maxwell (VM) system. The proposed ECSL scheme is the first semi-Lagrangian VM solver to achieve unconditional stability with respect to the plasma period while conserving total energy, without requiring nonlinear iterations. The new method is built on three key components: an efficient field solver, an exact splitting (ES) method, and a conservative Semi-Lagrangian (CSL) scheme. Specifically, the efficient field solver is developed by semi-implicitly coupling the Ampère and Vlasov moment equations, removing the numerical constraints imposed by the plasma period. Furthermore, the ES method is applied to the Vlasov equation and integrated with an implicit Maxwell solver, ensuring energy conservation in rotational dynamics. The ES method reduces the multidimensional Vlasov equation to one-dimensional advections exactly in time, which are then solved with the CSL scheme to ensure mass conservation and remove CFL restrictions. These synergistic components enable the ECSL scheme to conserve both total energy and mass at the fully discrete level, regardless of spatial and temporal resolution. Finally, several numerical experiments are presented to demonstrate the accuracy, efficiency, and conservation properties of the proposed method.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113858"},"PeriodicalIF":3.8,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445277","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 : 2025-02-15DOI: 10.1016/j.jcp.2025.113841
Jan Glaubitz , Hendrik Ranocha , Andrew R. Winters , Michael Schlottke-Lakemper , Philipp Öffner , Gregor Gassner
High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously—sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre–Gauss–Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin–Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes, as FD schemes typically have more unresolved modes than nodal DG methods.
{"title":"Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods","authors":"Jan Glaubitz , Hendrik Ranocha , Andrew R. Winters , Michael Schlottke-Lakemper , Philipp Öffner , Gregor Gassner","doi":"10.1016/j.jcp.2025.113841","DOIUrl":"10.1016/j.jcp.2025.113841","url":null,"abstract":"<div><div>High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously—sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre–Gauss–Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin–Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes, as FD schemes typically have more unresolved modes than nodal DG methods.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113841"},"PeriodicalIF":3.8,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429041","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 : 2025-02-14DOI: 10.1016/j.jcp.2025.113843
Nanxi Chen , Sergio Lucarini , Rujin Ma , Airong Chen , Chuanjie Cui
Physics-informed neural networks (PINNs) have emerged as a promising tool for effectively resolving diverse partial differential equations. Despite the numerous recent advances, PINNs often encounter significant challenges when dealing with complex nonlinear systems, such as the coupling Allen-Cahn (AC) and Cahn-Hilliard (CH) equations for phase field interfacial problems. In this work, we present an enhanced PINN framework, termed PF-PINNs, for the robust and efficient resolution of AC-CH coupled PDEs. Key features of the PF-PINNs framework include: (1) a normalisation and de-normalisation method to bridge the disparity in temporal and spatial scales in real-world physical problems, (2) an advanced sampling strategy designed to efficiently diffuse the initial interface and dynamically monitor its evolution throughout the training process, and (3) an NTK-based adaptive weighting strategy with random-batch method to balance the complex loss terms associated with phase field governing equations. We conduct extensive benchmarks on electrochemical corrosion, to showcase the accuracy and efficiency of the proposed PF-PINNs framework. The comparison of our results with reference solutions from FEniCS demonstrates that our PF-PINNs framework is a versatile and powerful tool for a wide range of AC-CH phase field applications.
{"title":"PF-PINNs: Physics-informed neural networks for solving coupled Allen-Cahn and Cahn-Hilliard phase field equations","authors":"Nanxi Chen , Sergio Lucarini , Rujin Ma , Airong Chen , Chuanjie Cui","doi":"10.1016/j.jcp.2025.113843","DOIUrl":"10.1016/j.jcp.2025.113843","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) have emerged as a promising tool for effectively resolving diverse partial differential equations. Despite the numerous recent advances, PINNs often encounter significant challenges when dealing with complex nonlinear systems, such as the coupling Allen-Cahn (AC) and Cahn-Hilliard (CH) equations for phase field interfacial problems. In this work, we present an enhanced PINN framework, termed PF-PINNs, for the robust and efficient resolution of AC-CH coupled PDEs. Key features of the PF-PINNs framework include: (1) a normalisation and de-normalisation method to bridge the disparity in temporal and spatial scales in real-world physical problems, (2) an advanced sampling strategy designed to efficiently diffuse the initial interface and dynamically monitor its evolution throughout the training process, and (3) an NTK-based adaptive weighting strategy with random-batch method to balance the complex loss terms associated with phase field governing equations. We conduct extensive benchmarks on electrochemical corrosion, to showcase the accuracy and efficiency of the proposed PF-PINNs framework. The comparison of our results with reference solutions from <span>FEniCS</span> demonstrates that our PF-PINNs framework is a versatile and powerful tool for a wide range of AC-CH phase field applications.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113843"},"PeriodicalIF":3.8,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429042","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 : 2025-02-14DOI: 10.1016/j.jcp.2025.113848
Andreas H. Akselsen
A two-dimensional water wave model based on conformal mapping is presented. The model is exact in the sense that it does not rely on truncated series expansions, nor suffer any numerical diffusion. Additionally, it is computationally highly efficient as it numerically evaluates only the surface line while using a fixed number of FFT operations per time step. A double layered mapping enforces prescribed outer boundaries without iteration. The model also supports transient boundaries, including walls. Mapping models are presented that support smooth bathymetries and angled overhanging geometries. An exact piston-type wavemaker model demonstrates the method's potential as a numerical wave tank. The model is tested and validated through a number of examples covering shallow water waves, wavemaker generation, rising bathymetry shelves, and wave reflection from slanting structures. A paddle-type wavemaker model, developed from the present theory, will be detailed in a forthcoming paper.
{"title":"A precise conformally mapped method for water waves in complex transient environments","authors":"Andreas H. Akselsen","doi":"10.1016/j.jcp.2025.113848","DOIUrl":"10.1016/j.jcp.2025.113848","url":null,"abstract":"<div><div>A two-dimensional water wave model based on conformal mapping is presented. The model is exact in the sense that it does not rely on truncated series expansions, nor suffer any numerical diffusion. Additionally, it is computationally highly efficient as it numerically evaluates only the surface line while using a fixed number of FFT operations per time step. A double layered mapping enforces prescribed outer boundaries without iteration. The model also supports transient boundaries, including walls. Mapping models are presented that support smooth bathymetries and angled overhanging geometries. An exact piston-type wavemaker model demonstrates the method's potential as a numerical wave tank. The model is tested and validated through a number of examples covering shallow water waves, wavemaker generation, rising bathymetry shelves, and wave reflection from slanting structures. A paddle-type wavemaker model, developed from the present theory, will be detailed in a forthcoming paper.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"528 ","pages":"Article 113848"},"PeriodicalIF":3.8,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419231","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 : 2025-02-14DOI: 10.1016/j.jcp.2025.113849
G. Gros , B. Faugeras , C. Boulbe , J.-F. Artaud , R. Nouailletas , F. Rapetti
This paper focuses on the numerical methods recently developed in order to simulate the time evolution of a tokamak plasma equilibrium at the resistive diffusion time scale. Starting from the method proposed by Heumann in 2021 for the coupling of magnetic equilibrium and current diffusion, we introduce a new space discretization for the poloidal flux using coupled and finite elements. This, together with the use of cubic spline functions to represent the poloidal current function in the resistive diffusion equation, enables to restrain numerical oscillations which can occur with the original method. In order to compute consistently the plasma resistivity and the non-inductive bootstrap current terms needed in the resistive diffusion equation we add to the model an evolution equation for electron temperature in the plasma. It is also used to evolve the pressure term in the simulation. These numerical methods are implemented in the plasma equilibrium code NICE. A free plasma displacement is simulated and comparison with experimental results from the WEST tokamak are used to validate the simulation. The code is also coupled to a magnetic feedback controller making it possible to simulate a prescribed plasma scenario. The results for an X-point formation scenario in the WEST tokamak are presented as an illustration of the efficiency of the developed numerical methods.
{"title":"Numerical simulation of tokamak plasma equilibrium evolution","authors":"G. Gros , B. Faugeras , C. Boulbe , J.-F. Artaud , R. Nouailletas , F. Rapetti","doi":"10.1016/j.jcp.2025.113849","DOIUrl":"10.1016/j.jcp.2025.113849","url":null,"abstract":"<div><div>This paper focuses on the numerical methods recently developed in order to simulate the time evolution of a tokamak plasma equilibrium at the resistive diffusion time scale. Starting from the method proposed by Heumann in 2021 for the coupling of magnetic equilibrium and current diffusion, we introduce a new space discretization for the poloidal flux using coupled <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> finite elements. This, together with the use of cubic spline functions to represent the poloidal current function in the resistive diffusion equation, enables to restrain numerical oscillations which can occur with the original method. In order to compute consistently the plasma resistivity and the non-inductive bootstrap current terms needed in the resistive diffusion equation we add to the model an evolution equation for electron temperature in the plasma. It is also used to evolve the pressure term in the simulation. These numerical methods are implemented in the plasma equilibrium code NICE. A free plasma displacement is simulated and comparison with experimental results from the WEST tokamak are used to validate the simulation. The code is also coupled to a magnetic feedback controller making it possible to simulate a prescribed plasma scenario. The results for an X-point formation scenario in the WEST tokamak are presented as an illustration of the efficiency of the developed numerical methods.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113849"},"PeriodicalIF":3.8,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478734","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 : 2025-02-13DOI: 10.1016/j.jcp.2025.113847
Yunlong Li , Fei Wang
Accurate modeling of complex physical problems, such as fluid-structure interaction, requires multiphysics coupling across the interface, which often has intricate geometry and dynamic boundaries. Conventional numerical methods face challenges in handling interface conditions. Deep neural networks offer a mesh-free and flexible alternative, but they suffer from drawbacks such as time-consuming optimization and local optima. In this paper, we propose a mesh-free approach based on Randomized Neural Networks (RaNNs) and finite difference methods (FDM), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks with finite difference methods (LRaNN-FDM), uses different RaNNs to approximate solutions in different subdomains. We discretize the interface problem into a linear system at randomly sampled points across the domain, boundary, and interface using a finite difference scheme, and then solve it by a least-square method. Unlike automatic differentiation for partial derivative calculations, the finite difference approach offers significantly faster computation. For time-dependent interface problems, we use a space-time approach based on LRaNNs. We show the effectiveness and robustness of the LRaNN-FDM through numerical examples of elliptic and parabolic interface problems. We also demonstrate that our approach can handle high-dimension interface problems. Compared to conventional numerical methods, our approach achieves higher accuracy with fewer degrees of freedom, eliminates the need for complex interface meshing and fitting, and significantly reduces training time, outperforming deep neural networks.
{"title":"Local randomized neural networks with finite difference methods for interface problems","authors":"Yunlong Li , Fei Wang","doi":"10.1016/j.jcp.2025.113847","DOIUrl":"10.1016/j.jcp.2025.113847","url":null,"abstract":"<div><div>Accurate modeling of complex physical problems, such as fluid-structure interaction, requires multiphysics coupling across the interface, which often has intricate geometry and dynamic boundaries. Conventional numerical methods face challenges in handling interface conditions. Deep neural networks offer a mesh-free and flexible alternative, but they suffer from drawbacks such as time-consuming optimization and local optima. In this paper, we propose a mesh-free approach based on Randomized Neural Networks (RaNNs) and finite difference methods (FDM), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks with finite difference methods (LRaNN-FDM), uses different RaNNs to approximate solutions in different subdomains. We discretize the interface problem into a linear system at randomly sampled points across the domain, boundary, and interface using a finite difference scheme, and then solve it by a least-square method. Unlike automatic differentiation for partial derivative calculations, the finite difference approach offers significantly faster computation. For time-dependent interface problems, we use a space-time approach based on LRaNNs. We show the effectiveness and robustness of the LRaNN-FDM through numerical examples of elliptic and parabolic interface problems. We also demonstrate that our approach can handle high-dimension interface problems. Compared to conventional numerical methods, our approach achieves higher accuracy with fewer degrees of freedom, eliminates the need for complex interface meshing and fitting, and significantly reduces training time, outperforming deep neural networks.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113847"},"PeriodicalIF":3.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429040","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 : 2025-02-13DOI: 10.1016/j.jcp.2025.113837
Shengfeng Xu , Yuanjun Dai , Chang Yan , Zhenxu Sun , Renfang Huang , Dilong Guo , Guowei Yang
Physics-Informed Neural Networks (PINNs) serve as a flexible alternative for tackling forward and inverse problems in differential equations, displaying impressive advancements in diverse areas of applied mathematics. Despite integrating both data and underlying physics to enrich the neural network's understanding, concerns regarding the effectiveness and practicality of PINNs persist. Over the past few years, extensive efforts in the current literature have been made to enhance this evolving method, by drawing inspiration from both machine learning algorithms and numerical methods. Despite notable progressions in PINNs algorithms, the important and fundamental field of data preprocessing remain unexplored, limiting the applications of PINNs especially in solving inverse problems. Therefore in this paper, a concise yet potent data preprocessing method focusing on data normalization was proposed. By applying a linear transformation to both the data and corresponding equations concurrently, the normalized PINNs approach was evaluated on the task of reconstructing flow fields in four turbulent cases. The results illustrate that by adhering to the data preprocessing procedure, PINNs can robustly achieve higher prediction accuracy for all flow quantities under different hyperparameter setups, without incurring extra computational cost, distinctly improving the utilization of limited training data. Though mainly verified in Navier-Stokes (NS) equations, this method holds potential for application to various other equations.
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