Journal of Computational Physics最新文献

{"title":"Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations","authors":"Dmitri Kuzmin ,&nbsp;Sanghyun Lee ,&nbsp;Yi-Yung Yang","doi":"10.1016/j.jcp.2025.114323","DOIUrl":"10.1016/j.jcp.2025.114323","url":null,"abstract":"<div><div>In this paper, we develop monolithic limiting techniques for enforcing nonlinear stability constraints in enriched Galerkin (EG) discretizations of nonlinear scalar hyperbolic equations. To achieve local mass conservation and gain control over the cell averages, the space of continuous (multi-)linear finite element approximations is enriched with piecewise-constant functions. The resulting spatial semi-discretization has the structure of a variational multiscale method. For linear advection equations, it is inherently stable but generally not bound preserving. To satisfy discrete maximum principles and ensure entropy stability in the nonlinear case, we use limiters adapted to the structure of our locally conservative EG method. The cell averages are constrained using a flux limiter, while the nodal values of the continuous component are constrained using a clip-and-scale limiting strategy for antidiffusive element contributions. The design and analysis of our new algorithms build on recent advances in the fields of convex limiting and algebraic entropy fixes for finite element methods. In addition to proving the claimed properties of the proposed approach, we conduct numerical studies for two-dimensional nonlinear hyperbolic problems. The numerical results demonstrate the ability of our limiters to prevent violations of the imposed constraints, while preserving the optimal order of accuracy in experiments with smooth solutions.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114323"},"PeriodicalIF":3.8,"publicationDate":"2025-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997086","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}

{"title":"Deep learning for high-dimensional PDEs with fat-tailed Lévy measure","authors":"Kamran Arif ,&nbsp;Guojiang Xi ,&nbsp;Heng Wang ,&nbsp;Weihua Deng","doi":"10.1016/j.jcp.2025.114327","DOIUrl":"10.1016/j.jcp.2025.114327","url":null,"abstract":"<div><div>The partial differential equations (PDEs) for jump process with Lévy measure have wide applications. When the measure has fat tails, it will bring big challenges for both computational cost and accuracy. In this work, we develop a deep learning method for high-dimensional PDEs related to fat-tailed Lévy measure, which can be naturally extended to the general case. Building on the theory of backward stochastic differential equations for Lévy processes, our deep learning method avoids the need for neural network differentiation and introduces a novel technique to address the singularity of fat-tailed Lévy measures. The developed method is used to solve four kinds of high-dimensional PDEs: the diffusion equation with fractional Laplacian; the advective diffusion equation with fractional Laplacian; the advective diffusion reaction equation with fractional Laplacian; and the nonlinear reaction diffusion equation with fractional Laplacian. The parameter <span><math><mi>β</mi></math></span> in fractional Laplacian is an indicator of the strength of the singularity of Lévy measure. Specifically, for <span><math><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, the model describes super-ballistic diffusion; while for <span><math><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>, it characterizes super-diffusion. In addition, we experimentally verify that the developed algorithm can be easily extended to solve fractional PDEs with finite general Lévy measures. Our method achieves a relative error of <span><math><mrow><mi>O</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></math></span> for low-dimensional problems and <span><math><mrow><mi>O</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math></span> for high-dimensional ones. We also investigate three factors that influence the algorithm’s performance: the number of hidden layers; the number of Monte Carlo samples; and the choice of activation functions. Furthermore, we test the efficiency of the algorithm in solving problems in 3D, 10D, 20D, 50D, and 100D. Our numerical results demonstrate that the algorithm achieves excellent performance with deeper hidden layers, a larger number of Monte Carlo samples, and the Softsign activation function.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114327"},"PeriodicalIF":3.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997088","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}

{"title":"Discontinuous extreme learning machine for interface and free boundary problems","authors":"Anci Lin ,&nbsp;Zhiwen Zhang ,&nbsp;Weidong Zhao ,&nbsp;Wenju Zhao","doi":"10.1016/j.jcp.2025.114329","DOIUrl":"10.1016/j.jcp.2025.114329","url":null,"abstract":"<div><div>We present a machine-learning framework for interface and free-boundary problems, focusing on physics-informed neural networks (PINNs). Two major challenges are addressed: (i) interface-induced discontinuities and (ii) moving boundaries inherent to free-boundary problems. To meet these challenges, we introduce the discontinuous extreme learning machine (DELM), a mesh-free method that leverages an “artificial discontinuity” mechanism, and the local extreme learning machine (locELM) architecture. Our first innovation augments the input of a single-layer neural network with two additional variables: a piecewise-constant indicator that enforces discontinuities in the solution itself, and the absolute value of a signed-distance level-set function that produces the correct gradient jump across the interface. This design captures discontinuities without splitting the network into multiple pieces or inflating the parameter count. For problems with evolving interfaces (e.g., the Stefan problem), we devise a decoupled discrete-DELM strategy that integrates seamlessly with the classical front-tracking and time-discretization technique. At each time step, the front-tracking module updates the interface geometry, and DELM subsequently solves the governing PDE in the updated domain. To further reduce complexity while maintaining accuracy, the computational domain is partitioned, and an independent single-layer ELM is trained within each subdomain. Various numerical experiments validate the proposed framework, demonstrating high accuracy and fast computational speed across a wide range of benchmark problems.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114329"},"PeriodicalIF":3.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997085","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}

{"title":"Solving cluster moment relaxation with hierarchical matrix","authors":"Yi Wang ,&nbsp;Rizheng Huang ,&nbsp;Yuehaw Khoo","doi":"10.1016/j.jcp.2025.114331","DOIUrl":"10.1016/j.jcp.2025.114331","url":null,"abstract":"<div><div>Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114331"},"PeriodicalIF":3.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997089","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}

{"title":"P-adaptation of successive correction k-exact finite volume schemes for compressible flows","authors":"M. Salihoglu ,&nbsp;A. Liapi ,&nbsp;A. Belme ,&nbsp;P. Brenner ,&nbsp;G. Pont ,&nbsp;P. Cinnella","doi":"10.1016/j.jcp.2025.114330","DOIUrl":"10.1016/j.jcp.2025.114330","url":null,"abstract":"<div><div>A <span><math><mi>p</mi></math></span>-adaptation strategy is developed in the framework of successive correction <span><math><mi>k</mi></math></span>-exact finite volume schemes. A new adaptation indicator based on the decay of the successive correction terms used to reconstruct the solution within one cell is introduced to drive the adaptation process. The criterion relies on low-order derivatives, is efficiently estimated as part of the successive correction process, and identifies well flow regions characterized by steep gradients. Unlike other strategies in the literature, <span><math><mi>p</mi></math></span>-adaptation is only used to increase solution accuracy, while robust slope limiters are used to control the appearance of spurious oscillations. The performance of the proposed adaptive method is evaluated for a variety of 2D steady and unsteady, inviscid and viscous compressible flow configurations, as well as for a 3D transonic wing. The results show the effectiveness of <span><math><mi>p</mi></math></span>-adaptivity in achieving high-order solution quality while maintaining the computational effort close to that of a second-order (one-exact) scheme in terms of memory load and computation time.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"542 ","pages":"Article 114330"},"PeriodicalIF":3.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097584","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}

{"title":"Time-splitting methods for the cold-plasma model using Finite Element Exterior Calculus","authors":"Elena Moral Sánchez ,&nbsp;Martin Campos Pinto ,&nbsp;Yaman [Güşlü] ,&nbsp;Omar Maj","doi":"10.1016/j.jcp.2025.114305","DOIUrl":"10.1016/j.jcp.2025.114305","url":null,"abstract":"<div><div>In this work we propose a high-order structure-preserving discretization of the cold plasma model which describes the propagation of electromagnetic waves in magnetized plasmas. By utilizing B-Splines Finite Elements Exterior Calculus, we derive a space discretization that preserves the underlying Hamiltonian structure of the model, and we study two stable time-splitting geometrical integrators. We approximate an incoming wave boundary condition in such a way that the resulting schemes are compatible with a time-harmonic / transient decomposition of the solution, which allows us to establish their long-time stability. This approach readily applies to curvilinear and complex domains. We perform a numerical study of these schemes which compares their cost and accuracy against a standard Crank-Nicolson time integrator, and we run realistic simulations where the long-term behaviour is assessed using frequency-domain solutions. Our solvers are three-dimensional and parallel. They are implemented in the Python library PSYDAC, which makes them memory-efficient.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114305"},"PeriodicalIF":3.8,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010100","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}

{"title":"Discontinuous Galerkin methods for the complete stochastic Euler equations","authors":"Dominic Breit ,&nbsp;Thamsanqa Castern Moyo ,&nbsp;Philipp Öffner","doi":"10.1016/j.jcp.2025.114324","DOIUrl":"10.1016/j.jcp.2025.114324","url":null,"abstract":"<div><div>In recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids in computational fluid dynamics are the Euler equations of gas dynamics which we focus on in this paper. To take stochastic effects into account, we incorporate a stochastic forcing term in the momentum equation of the Euler system. To solve the extended system, we apply an entropy dissipative discontinuous Galerkin spectral element method including the Finite Volume setting, adjust it to the stochastic Euler equations and analyse its convergence properties. Our analysis is based on the concept of <em>dissipative martingale solutions</em>, as recently introduced by Moyo (J. Diff. Equ. 365, 408–464, 2023). Assuming no vacuum formation and bounded total energy, we prove that the scheme converges in law to a dissipative martingale solution. During the lifespan of a pathwise strong solution, we achieve convergence of at least order 1/2, measured by the expected <span><math><msup><mi>L</mi><mn>1</mn></msup></math></span> norm of the relative entropy. The results build a counterpart of those obtained in the deterministic case. In numerical simulations, we show the robustness of our scheme, visualise different stochastic realisations and support/extend the theoretical findings.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114324"},"PeriodicalIF":3.8,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144933157","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}

{"title":"A robust structure-preserving surface reconstruction scheme for two-layer shallow water equations based on a relaxation model and an extension on adaptive moving triangles","authors":"Jian Dong,&nbsp;Xu Qian,&nbsp;Zige Wei","doi":"10.1016/j.jcp.2025.114328","DOIUrl":"10.1016/j.jcp.2025.114328","url":null,"abstract":"<div><div>We present a structure-preserving surface reconstruction scheme for the relaxation two-layer shallow water equation, inspired by the approach in [Computers &amp; Fluids 272 (2024) 106193], along with its two-dimensional extension on adaptive moving triangular meshes. The original two-layer shallow water equation is conditionally hyperbolic, which presents challenges in designing shock-capturing numerical schemes. To address this, we propose a relaxation two-layer shallow water equation that is hyperbolic. However, this relaxation equation still contains nonconservative products associated with layer heights and bottom topography, which cannot be defined in the distributional sense. Utilizing the surface reconstruction method, we define Riemann states linked to the layer heights and bottom topography. This approach smooths the solution, facilitating the discretization of the nonconservative product across cell boundaries. We introduce a structure-preserving parameter crucial for demonstrating convergence, maintaining stationary steady states, and ensuring the positivity-preserving property. We establish a Lax-Wendroff type convergence theorem for the structure-preserving surface reconstruction scheme applied to the relaxation two-layer shallow water equation. To validate our approach, we conduct several classical numerical experiments for both one-dimensional and two-dimensional cases. Notably, we numerically confirm that the structure-preserving surface reconstruction scheme for the one-dimensional relaxation two-layer shallow water equation exhibits <span><math><mi>K</mi></math></span>-convergence based on the Cesàro average, particularly in the context of the well-known Kelvin-Helmholtz instabilities. Finally, we show several numerical results of the two-dimensional two-layer shallow water equations on adaptive moving triangles to verify the theoretical results.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114328"},"PeriodicalIF":3.8,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144917543","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}

{"title":"Energy-dissipative evolutionary Kolmogorov-Arnold networks for complex PDE systems","authors":"Guang Lin ,&nbsp;Changhong Mou ,&nbsp;Jiahao Zhang","doi":"10.1016/j.jcp.2025.114326","DOIUrl":"10.1016/j.jcp.2025.114326","url":null,"abstract":"<div><div>In this work, we introduce an evolutionary Kolmogorov-Arnold Network (EvoKAN), a novel framework for solving complex partial differential equations (PDEs). EvoKAN builds on Kolmogorov-Arnold Networks (KANs), where activation functions are spline-based and trainable on each edge, offering localized flexibility across multiple scales. Rather than retraining the network repeatedly, EvoKAN encodes only the initial state of the PDE during an initial learning phase. EvoKAN models Kolmogorov-Arnold network weights as time-dependent functions and updates them through the evolution of the governing PDEs. By treating EvoKAN weights as continuous functions in the relevant coordinates and updating them over time, EvoKAN can predict system trajectories over arbitrarily long horizons, a notable challenge for many conventional neural network-based methods. In addition, EvoKAN employs the scalar auxiliary variable (SAV) method to guarantee unconditional energy stability and computational efficiency. At individual time step, SAV only needs to solve the decoupled linear systems with constant coefficients, the implementation is significantly simplified. We test the proposed framework in several complex PDEs, including one-dimensional and two-dimensional Allen–Cahn equations and two-dimensional Navier-Stokes equations. The numerical results show that the EvoKAN solutions closely match the analytical references and established numerical benchmarks, effectively capturing the sharp interfaces in predicting the solution of the Allen-Cahn equation and turbulent flow patterns in predicting the solution of the Navier-Stokes equations.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114326"},"PeriodicalIF":3.8,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144917492","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}

{"title":"An adaptive spectral element method for systems of conservation laws","authors":"Manuel Colera ,&nbsp;Vít Dolejší","doi":"10.1016/j.jcp.2025.114322","DOIUrl":"10.1016/j.jcp.2025.114322","url":null,"abstract":"<div><div>We present a novel method for the numerical solution of systems of conservation laws. For the space discretization, the scheme considers high-order continuous finite elements stabilized via subgrid modeling, as well as highly anisotropic adaptive meshes in order to capture efficiently any sharp features in the solution. Time integration is carried out via a time-step adaptive, linearly implicit Runge–Kutta method, which allows large time steps and requires only the solution of a linear system of equations at each internal stage. An important characteristic of the present method is that the mesh is adapted before solving for the next time interval, and not afterwards as in the common procedure. Furthermore, the error arising from the inexact solution of the linear systems is also estimated and controlled, in such a way that the numerical solution is sufficiently accurate and the linear systems are not oversolved. Numerical experiments, including a computationally difficult case of non-convex flux and the Euler equations for compressible flows, were performed with up to eight-degree elements and a third-order time marching formula to demonstrate the capabilities of the method.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"541 ","pages":"Article 114322"},"PeriodicalIF":3.8,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988010","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}