Pub Date : 2025-04-04DOI: 10.1016/j.jcp.2025.113982
A. Vergnaud , A. Lemoine , J. Breil
In this paper we investigate cell-based Compatible Discrete Operator (CDO) scheme for elliptic problems. These mimetic schemes rely on mixed potential degrees of freedom at the cells and flux degrees of freedom at the faces. In this paper, we propose a compact rewriting of this scheme by eliminating a large part of the degrees of freedom associated with the faces, which greatly reduces the size of the linear systems to be inverted and therefore the computational cost. For this, we take advantage of Cartesian AMR (Adaptive Mesh Refinement) meshes, for which the discrete CDO Hodge operator can be made diagonal almost everywhere. We also propose a formulation of general Robin-type boundary conditions for these cell-based CDO schemes, also valid in the presence of Immersed Boundaries with a cut-cell approach. The proposed scheme and its performances is validated through various test cases.
{"title":"Compact cell-based Compatible Discrete Operator diffusion scheme on Cartesian AMR mesh","authors":"A. Vergnaud , A. Lemoine , J. Breil","doi":"10.1016/j.jcp.2025.113982","DOIUrl":"10.1016/j.jcp.2025.113982","url":null,"abstract":"<div><div>In this paper we investigate cell-based Compatible Discrete Operator (CDO) scheme for elliptic problems. These mimetic schemes rely on mixed potential degrees of freedom at the cells and flux degrees of freedom at the faces. In this paper, we propose a compact rewriting of this scheme by eliminating a large part of the degrees of freedom associated with the faces, which greatly reduces the size of the linear systems to be inverted and therefore the computational cost. For this, we take advantage of Cartesian AMR (Adaptive Mesh Refinement) meshes, for which the discrete CDO Hodge operator can be made diagonal almost everywhere. We also propose a formulation of general Robin-type boundary conditions for these cell-based CDO schemes, also valid in the presence of Immersed Boundaries with a cut-cell approach. The proposed scheme and its performances is validated through various test cases.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113982"},"PeriodicalIF":3.8,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807378","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 : 2025-04-03DOI: 10.1016/j.jcp.2025.113981
Lukas Lundgren , Christian Helanow , Jonathan Wiskandt , Inga Monika Koszalka , Josefin Ahlkrona
We introduce a continuous Galerkin finite element discretization of the non-hydrostatic Boussinesq approximation of the Navier-Stokes equations, suitable for various applications such as coastal ocean dynamics and ice-ocean interactions, among others. In particular, we introduce a consistent modification of the gravity force term which enhances conservation properties for Galerkin methods without strictly enforcing the divergence-free condition. We show that this modification results in a sharp energy estimate, including both kinetic and potential energy. Additionally, we propose a new, symmetric, tensor-based viscosity operator that is especially suitable for modeling turbulence in stratified flow. The viscosity coefficients are constructed using a residual-based shock-capturing method and the method conserves angular momentum and dissipates kinetic energy. We validate our proposed method through numerical tests and use it to model the ocean circulation and basal melting beneath the ice tongue of the Ryder Glacier and the adjacent Sherard Osborn Fjord in two dimensions on a fully unstructured mesh. Our results compare favorably with a standard numerical ocean model, showing better resolved turbulent flow features and reduced artificial diffusion.
{"title":"A potential energy conserving finite element method for turbulent variable density flow: Application to glacier-fjord circulation","authors":"Lukas Lundgren , Christian Helanow , Jonathan Wiskandt , Inga Monika Koszalka , Josefin Ahlkrona","doi":"10.1016/j.jcp.2025.113981","DOIUrl":"10.1016/j.jcp.2025.113981","url":null,"abstract":"<div><div>We introduce a continuous Galerkin finite element discretization of the non-hydrostatic Boussinesq approximation of the Navier-Stokes equations, suitable for various applications such as coastal ocean dynamics and ice-ocean interactions, among others. In particular, we introduce a consistent modification of the gravity force term which enhances conservation properties for Galerkin methods without strictly enforcing the divergence-free condition. We show that this modification results in a sharp energy estimate, including both kinetic and potential energy. Additionally, we propose a new, symmetric, tensor-based viscosity operator that is especially suitable for modeling turbulence in stratified flow. The viscosity coefficients are constructed using a residual-based shock-capturing method and the method conserves angular momentum and dissipates kinetic energy. We validate our proposed method through numerical tests and use it to model the ocean circulation and basal melting beneath the ice tongue of the Ryder Glacier and the adjacent Sherard Osborn Fjord in two dimensions on a fully unstructured mesh. Our results compare favorably with a standard numerical ocean model, showing better resolved turbulent flow features and reduced artificial diffusion.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113981"},"PeriodicalIF":3.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143806852","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 : 2025-04-03DOI: 10.1016/j.jcp.2025.113980
Li Luo , Qian Zhang , Haochen Liu , Jinpeng Zhang , Xiao-Ping Wang
A phase-field model consisting of the Cahn-Hilliard and Navier-Stokes equations with a generalized Navier slip and penetration boundary condition is proposed to simulate the behavior of two-phase flows through permeable surfaces. The proposed boundary condition is a generalization of the generalized Navier boundary condition to penetrable boundaries, enabling the simulation of significant scientific problems such as gas penetration through polymer films and oil filtration through porous materials. To address the challenges imposed by the new boundary condition to conventional numerical schemes, we develop a new numerical algorithm by using a finite element method that naturally incorporates the boundary condition into the weak formulation. The algorithm solves a semi-implicit system for the Cahn-Hilliard equation and a fully implicit system for the Navier-Stokes equations. Complex geometries required in the applications are handled by using body-conforming unstructured meshes. Furthermore, an adaptive mesh refinement strategy based on a gradient-jump error indicator is devised to accelerate the simulation process while obtaining a reliable solution on an optimally refined mesh. Extensive numerical experiments, including two practical applications, are conducted to validate the effectiveness and efficiency of the proposed approach. The first application involves bubble penetration through a polymer film, encompassing processes such as bouncing, spreading, pinning, slipping, and penetrating. The numerical results show qualitative agreement with experimental results. In the second application, we examine the robustness of the algorithm by testing different physical parameters with high contrast for the displacement and infiltration of two-phase flows in a complex pore structure.
{"title":"A numerical study of two-phase flows in complex domain with a generalized Navier slip and penetration boundary condition on permeable boundaries","authors":"Li Luo , Qian Zhang , Haochen Liu , Jinpeng Zhang , Xiao-Ping Wang","doi":"10.1016/j.jcp.2025.113980","DOIUrl":"10.1016/j.jcp.2025.113980","url":null,"abstract":"<div><div>A phase-field model consisting of the Cahn-Hilliard and Navier-Stokes equations with a generalized Navier slip and penetration boundary condition is proposed to simulate the behavior of two-phase flows through permeable surfaces. The proposed boundary condition is a generalization of the generalized Navier boundary condition to penetrable boundaries, enabling the simulation of significant scientific problems such as gas penetration through polymer films and oil filtration through porous materials. To address the challenges imposed by the new boundary condition to conventional numerical schemes, we develop a new numerical algorithm by using a finite element method that naturally incorporates the boundary condition into the weak formulation. The algorithm solves a semi-implicit system for the Cahn-Hilliard equation and a fully implicit system for the Navier-Stokes equations. Complex geometries required in the applications are handled by using body-conforming unstructured meshes. Furthermore, an adaptive mesh refinement strategy based on a gradient-jump error indicator is devised to accelerate the simulation process while obtaining a reliable solution on an optimally refined mesh. Extensive numerical experiments, including two practical applications, are conducted to validate the effectiveness and efficiency of the proposed approach. The first application involves bubble penetration through a polymer film, encompassing processes such as bouncing, spreading, pinning, slipping, and penetrating. The numerical results show qualitative agreement with experimental results. In the second application, we examine the robustness of the algorithm by testing different physical parameters with high contrast for the displacement and infiltration of two-phase flows in a complex pore structure.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113980"},"PeriodicalIF":3.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143785551","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 : 2025-04-02DOI: 10.1016/j.jcp.2025.113979
Zhenming Wang , Jun Zhu , Yan Tan , Linlin Tian , Ning Zhao
The high-order weighted essentially non-oscillatory (WENO) schemes are widely used in practical engineering problems due to their excellent shock-capturing features, especially for unstructured meshes. However, its characteristic decomposition and nonlinear weights calculation process bring a lot of computational overhead. Therefore, a hybrid unequal-sized WENO (US-WENO) scheme is developed for hyperbolic conservation laws on tetrahedral meshes. Firstly, an extremum properties (EP)-based discontinuous sensor was designed according to the highest degree polynomial. This proposed discontinuous sensor does not depend on the specific problems and is well adapted to the tetrahedral unstructured meshes in this paper. Secondly, based on the developed EP-based sensor, a hybrid US-WENO scheme was proposed for the first time on three-dimensional unstructured meshes. This method can inherit the excellent features of the US-WENO scheme while improving computational efficiency by about 30% on the same mesh level. Finally, several classical examples are provided to verify the numerical accuracy, shock capture characteristics, and computational efficiency of the proposed method. Numerical results show that the presented method performs well and has a good engineering application prospect.
{"title":"An extremum properties (EP)-based discontinuous sensor and hybrid weighted essentially non-oscillatory scheme on tetrahedral meshes","authors":"Zhenming Wang , Jun Zhu , Yan Tan , Linlin Tian , Ning Zhao","doi":"10.1016/j.jcp.2025.113979","DOIUrl":"10.1016/j.jcp.2025.113979","url":null,"abstract":"<div><div>The high-order weighted essentially non-oscillatory (WENO) schemes are widely used in practical engineering problems due to their excellent shock-capturing features, especially for unstructured meshes. However, its characteristic decomposition and nonlinear weights calculation process bring a lot of computational overhead. Therefore, a hybrid unequal-sized WENO (US-WENO) scheme is developed for hyperbolic conservation laws on tetrahedral meshes. Firstly, an extremum properties (EP)-based discontinuous sensor was designed according to the highest degree polynomial. This proposed discontinuous sensor does not depend on the specific problems and is well adapted to the tetrahedral unstructured meshes in this paper. Secondly, based on the developed EP-based sensor, a hybrid US-WENO scheme was proposed for the first time on three-dimensional unstructured meshes. This method can inherit the excellent features of the US-WENO scheme while improving computational efficiency by about 30% on the same mesh level. Finally, several classical examples are provided to verify the numerical accuracy, shock capture characteristics, and computational efficiency of the proposed method. Numerical results show that the presented method performs well and has a good engineering application prospect.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113979"},"PeriodicalIF":3.8,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807382","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 : 2025-04-02DOI: 10.1016/j.jcp.2025.113976
Jinwoo Go, Peng Chen
We develop a new computational framework to solve sequential Bayesian optimal experimental design (SBOED) problems constrained by large-scale partial differential equations with infinite-dimensional random parameters. We propose an adaptive terminal formulation of the optimality criterion for SBOED to achieve adaptive global optimality. We also establish an equivalent optimization formulation to achieve computational simplicity enabled by Laplace and low-rank approximations of the posterior. To accelerate the solution of the SBOED problem, we develop a derivative-informed latent attention neural operator (LANO), a new neural network surrogate model that leverages (1) derivative-informed dimension reduction for latent encoding, (2) an attention mechanism to capture the dynamics in the latent space, (3) an efficient training in the latent space augmented by projected Jacobian, which collectively leads to an efficient, accurate, and scalable surrogate in computing not only the parameter-to-observable (PtO) maps but also their Jacobians. We further develop the formulation for the computation of the MAP points, the eigenpairs, and the sampling from the posterior by LANO in the reduced spaces and use these computations to solve the SBOED problem. We demonstrate the superior accuracy of LANO compared to two other neural architectures and the high accuracy of LANO compared to the finite element method (FEM) for the computation of MAP points and eigenvalues in solving the SBOED problem with application to the experimental design of the time to take MRI images in monitoring tumor growth. We show that the proposed computational framework achieves an amortized 180× speed-up.
{"title":"Sequential infinite-dimensional Bayesian optimal experimental design with derivative-informed latent attention neural operator","authors":"Jinwoo Go, Peng Chen","doi":"10.1016/j.jcp.2025.113976","DOIUrl":"10.1016/j.jcp.2025.113976","url":null,"abstract":"<div><div>We develop a new computational framework to solve sequential Bayesian optimal experimental design (SBOED) problems constrained by large-scale partial differential equations with infinite-dimensional random parameters. We propose an adaptive terminal formulation of the optimality criterion for SBOED to achieve adaptive global optimality. We also establish an equivalent optimization formulation to achieve computational simplicity enabled by Laplace and low-rank approximations of the posterior. To accelerate the solution of the SBOED problem, we develop a derivative-informed latent attention neural operator (LANO), a new neural network surrogate model that leverages (1) derivative-informed dimension reduction for latent encoding, (2) an attention mechanism to capture the dynamics in the latent space, (3) an efficient training in the latent space augmented by projected Jacobian, which collectively leads to an efficient, accurate, and scalable surrogate in computing not only the parameter-to-observable (PtO) maps but also their Jacobians. We further develop the formulation for the computation of the MAP points, the eigenpairs, and the sampling from the posterior by LANO in the reduced spaces and use these computations to solve the SBOED problem. We demonstrate the superior accuracy of LANO compared to two other neural architectures and the high accuracy of LANO compared to the finite element method (FEM) for the computation of MAP points and eigenvalues in solving the SBOED problem with application to the experimental design of the time to take MRI images in monitoring tumor growth. We show that the proposed computational framework achieves an amortized 180× speed-up.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113976"},"PeriodicalIF":3.8,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760397","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 : 2025-04-02DOI: 10.1016/j.jcp.2025.113977
Cuiling Ma, Xufeng Xiao, Xinlong Feng
In this paper, we propose and analyze an energy-stable approximation for axisymmetric Willmore flow of closed surfaces. This approach extends the original work of Bao and Li [4] for the planar Willmore flow of curves. Through relations among various geometric quantities, we derive a system of equivalent geometric equations for the axisymmetric Willmore flow, including the evolution equations for the parameterization and mean curvature. The proposed method consists of the linear parametric finite element method in space and the backward Euler method in time. Furthermore, we prove that the fully discrete scheme is unconditionally energy-stable. The Newton-Raphson iteration method is adopted to solve the nonlinear system. Finally, numerical examples are presented to illustrate the efficiency and energy stability of the proposed method for Willmore flow in an axisymmetric setting.
{"title":"An energy-stable parametric finite element approximation for axisymmetric Willmore flow of closed surfaces","authors":"Cuiling Ma, Xufeng Xiao, Xinlong Feng","doi":"10.1016/j.jcp.2025.113977","DOIUrl":"10.1016/j.jcp.2025.113977","url":null,"abstract":"<div><div>In this paper, we propose and analyze an energy-stable approximation for axisymmetric Willmore flow of closed surfaces. This approach extends the original work of Bao and Li <span><span>[4]</span></span> for the planar Willmore flow of curves. Through relations among various geometric quantities, we derive a system of equivalent geometric equations for the axisymmetric Willmore flow, including the evolution equations for the parameterization and mean curvature. The proposed method consists of the linear parametric finite element method in space and the backward Euler method in time. Furthermore, we prove that the fully discrete scheme is unconditionally energy-stable. The Newton-Raphson iteration method is adopted to solve the nonlinear system. Finally, numerical examples are presented to illustrate the efficiency and energy stability of the proposed method for Willmore flow in an axisymmetric setting.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113977"},"PeriodicalIF":3.8,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807379","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 : 2025-04-01DOI: 10.1016/j.jcp.2025.113978
Jaeyoung Jung , Manuel Schmid , Jacob Fish , Ensheng Weng , Marco Giometto
This study develops a novel numerical scheme to the volume-averaged Navier–Stokes equations, specifically addressing challenges posed by discontinuous porosity fields. Utilizing the path-conservative approach, we propose tailored paths that ensure the conservation of mass and energy across discontinuities. Introducing these paths into the generalized Rankine–Hugoniot relation, we define a stationary wave that incorporates the effects of discontinuous porosity onto the flow. This stationary wave serves as a fundamental component of the exact solution to the Riemann problem. Analyzing the solution space via the L–M/R–M curve approach, we prove that the existence and uniqueness of the solution are guaranteed if the artificial parameter is sufficiently large. Building on the identified structure of the exact solution, a path-conservative well-balanced high-order finite-volume solver is designed. The WENO reconstruction is implemented to achieve high-order accuracy. Mimicking the exact solution structure, we formulate the stationary wave reconstruction to capture the effect of discontinuous porosity on the flow. Lastly, the source term is handled by a well-balanced high-order approximation. Numerical tests were conducted to verify the well-balanced property, high-order accuracy, and shock-capturing capability of the proposed method, demonstrating excellent agreement with reference solutions.
{"title":"Path-conservative well-balanced high-order finite-volume solver for the volume-averaged Navier–Stokes equations with discontinuous porosity","authors":"Jaeyoung Jung , Manuel Schmid , Jacob Fish , Ensheng Weng , Marco Giometto","doi":"10.1016/j.jcp.2025.113978","DOIUrl":"10.1016/j.jcp.2025.113978","url":null,"abstract":"<div><div>This study develops a novel numerical scheme to the volume-averaged Navier–Stokes equations, specifically addressing challenges posed by discontinuous porosity fields. Utilizing the path-conservative approach, we propose tailored paths that ensure the conservation of mass and energy across discontinuities. Introducing these paths into the generalized Rankine–Hugoniot relation, we define a stationary wave that incorporates the effects of discontinuous porosity onto the flow. This stationary wave serves as a fundamental component of the exact solution to the Riemann problem. Analyzing the solution space via the L–M/R–M curve approach, we prove that the existence and uniqueness of the solution are guaranteed if the artificial parameter is sufficiently large. Building on the identified structure of the exact solution, a path-conservative well-balanced high-order finite-volume solver is designed. The WENO reconstruction is implemented to achieve high-order accuracy. Mimicking the exact solution structure, we formulate the stationary wave reconstruction to capture the effect of discontinuous porosity on the flow. Lastly, the source term is handled by a well-balanced high-order approximation. Numerical tests were conducted to verify the well-balanced property, high-order accuracy, and shock-capturing capability of the proposed method, demonstrating excellent agreement with reference solutions.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113978"},"PeriodicalIF":3.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143823508","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 : 2025-04-01DOI: 10.1016/j.jcp.2025.113975
Chong-Sen Huang , Tian-Yang Han , Jie Zhang , Ming-Jiu Ni
A sharp and conservative numerical method is proposed for studying the 2D contact line dynamics along complex geometrical boundaries, while a hybrid volume-of-fluid and embedded boundary method is designed to model the liquid/gas and fluid/solid interfaces, respectively. Unlike diffusive numerical methods that artificially thicken the interface to enhance stability, our method devises a sharp and precise scheme for discretizing the advection term of the VOF equation, with special attention to arbitrary solid boundaries within the same discretized interfacial cell. This scheme conserves the volume fraction exactly and maintains numerical stability even in small and irregular cells cut by the solid boundary. Another novel aspect of our contribution is the precise imposition of the contact angle condition at the contact line. A special height function method is designed and implemented for cells cut by the solid boundary. Furthermore, the contact angle condition, whether static or dynamic during contact line motion, is extended to more general cases by considering hysteresis phenomena. The code is released on the Basilisk website, enabling the first implementation of a sharp geometrical VOF method capable of accurately simulating contact line dynamics on complex solid substrates in 2D.
{"title":"A 2D sharp and conservative VOF method for modeling the contact line dynamics with hysteresis on complex boundary","authors":"Chong-Sen Huang , Tian-Yang Han , Jie Zhang , Ming-Jiu Ni","doi":"10.1016/j.jcp.2025.113975","DOIUrl":"10.1016/j.jcp.2025.113975","url":null,"abstract":"<div><div>A sharp and conservative numerical method is proposed for studying the 2D contact line dynamics along complex geometrical boundaries, while a hybrid volume-of-fluid and embedded boundary method is designed to model the liquid/gas and fluid/solid interfaces, respectively. Unlike diffusive numerical methods that artificially thicken the interface to enhance stability, our method devises a sharp and precise scheme for discretizing the advection term of the VOF equation, with special attention to arbitrary solid boundaries within the same discretized interfacial cell. This scheme conserves the volume fraction exactly and maintains numerical stability even in small and irregular cells cut by the solid boundary. Another novel aspect of our contribution is the precise imposition of the contact angle condition at the contact line. A special height function method is designed and implemented for cells cut by the solid boundary. Furthermore, the contact angle condition, whether static or dynamic during contact line motion, is extended to more general cases by considering hysteresis phenomena. The code is released on the <em>Basilisk</em> website, enabling the first implementation of a sharp geometrical VOF method capable of accurately simulating contact line dynamics on complex solid substrates in 2D.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"533 ","pages":"Article 113975"},"PeriodicalIF":3.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143807381","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 : 2025-03-31DOI: 10.1016/j.jcp.2025.113974
Georgios Akrivis , Buyang Li , Rong Tang , Hui Zhang
This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.
{"title":"High-order mass-, energy- and momentum-conserving methods for the nonlinear Schrödinger equation","authors":"Georgios Akrivis , Buyang Li , Rong Tang , Hui Zhang","doi":"10.1016/j.jcp.2025.113974","DOIUrl":"10.1016/j.jcp.2025.113974","url":null,"abstract":"<div><div>This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"532 ","pages":"Article 113974"},"PeriodicalIF":3.8,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768610","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 : 2025-03-31DOI: 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}
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