Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1169-1195, September 2024. Abstract. In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred to as leaders, whose influence acts on their nearest neighbors, in the following referred to as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behavior of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a [math]-nearest neighbors search algorithm with a binary tree. Finally, we conduct various numerical experiments for different scenarios to validate the algorithm’s effectiveness and investigate collective dynamics in both two and three dimensions.
{"title":"Kinetic Description of Swarming Dynamics with Topological Interaction and Transient Leaders","authors":"Giacomo Albi, Federica Ferrarese","doi":"10.1137/23m1588615","DOIUrl":"https://doi.org/10.1137/23m1588615","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1169-1195, September 2024. <br/> Abstract. In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred to as leaders, whose influence acts on their nearest neighbors, in the following referred to as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behavior of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a [math]-nearest neighbors search algorithm with a binary tree. Finally, we conduct various numerical experiments for different scenarios to validate the algorithm’s effectiveness and investigate collective dynamics in both two and three dimensions.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marie Touboul, Benjamin Vial, Raphaël Assier, Sébastien Guenneau, Richard V. Craster
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1136-1168, September 2024. Abstract. High-frequency homogenization is used to study dispersive media, containing inclusions placed periodically, for which the properties of the material depend on the frequency (Lorentz or Drude model with damping, for example). Effective properties are obtained near a given point of the dispersion diagram in frequency-wavenumber space. The asymptotic approximations of the dispersion diagrams and the wavefields so obtained are then cross-validated via detailed comparison with finite element method simulations in both one and two dimensions.
{"title":"High-Frequency Homogenization for Periodic Dispersive Media","authors":"Marie Touboul, Benjamin Vial, Raphaël Assier, Sébastien Guenneau, Richard V. Craster","doi":"10.1137/23m159648x","DOIUrl":"https://doi.org/10.1137/23m159648x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1136-1168, September 2024. <br/> Abstract. High-frequency homogenization is used to study dispersive media, containing inclusions placed periodically, for which the properties of the material depend on the frequency (Lorentz or Drude model with damping, for example). Effective properties are obtained near a given point of the dispersion diagram in frequency-wavenumber space. The asymptotic approximations of the dispersion diagrams and the wavefields so obtained are then cross-validated via detailed comparison with finite element method simulations in both one and two dimensions.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1097-1135, September 2024. Abstract. Image registration matches the features of two images by minimizing the intensity difference, so that useful and complementary information can be extracted from the mapping. However, in real life problems, images may be affected by the imaging environment, such as varying illumination and noise during the process of imaging acquisition. This may lead to the local intensity distortion, which makes it meaningless to minimize the intensity difference in the traditional registration framework. To address this problem, we propose a variational model for joint image registration and intensity correction. Based on this model, a related greedy matching problem is solved by introducing a multiscale approach for joint image registration and intensity correction. An alternating direction method (ADM) is proposed to solve each multiscale step, and the convergence of the ADM method is proved. For the numerical implementation, a coarse-to-fine strategy is further proposed to accelerate the numerical algorithm, and the convergence of the proposed coarse-to-fine strategy is also established. Some numerical tests are performed to validate the efficiency of the proposed algorithm.
{"title":"Multiscale Approach for Variational Problem Joint Diffeomorphic Image Registration and Intensity Correction: Theory and Application","authors":"Peng Chen, Ke Chen, Huan Han, Daoping Zhang","doi":"10.1137/23m155952x","DOIUrl":"https://doi.org/10.1137/23m155952x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1097-1135, September 2024. <br/> Abstract. Image registration matches the features of two images by minimizing the intensity difference, so that useful and complementary information can be extracted from the mapping. However, in real life problems, images may be affected by the imaging environment, such as varying illumination and noise during the process of imaging acquisition. This may lead to the local intensity distortion, which makes it meaningless to minimize the intensity difference in the traditional registration framework. To address this problem, we propose a variational model for joint image registration and intensity correction. Based on this model, a related greedy matching problem is solved by introducing a multiscale approach for joint image registration and intensity correction. An alternating direction method (ADM) is proposed to solve each multiscale step, and the convergence of the ADM method is proved. For the numerical implementation, a coarse-to-fine strategy is further proposed to accelerate the numerical algorithm, and the convergence of the proposed coarse-to-fine strategy is also established. Some numerical tests are performed to validate the efficiency of the proposed algorithm.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1068-1096, September 2024. Abstract. In this work, we derive a homogenized mathematical model for a porous intercalation electrode with a phase separating active material. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann–boundary condition modeling the lithium intercalation reaction [math]. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn–Hilliard equation, whereas the limit model consists of a diffusion and an Allen–Cahn equation. Thus, we observe a Cahn–Hilliard to Allen–Cahn transition during the upscaling process. In the sense of gradient flows, the transition coincides with a change in the underlying metric structure of the PDE system.
{"title":"Homogenization of a Porous Intercalation Electrode with Phase Separation","authors":"Martin Heida, Manuel Landstorfer, Matthias Liero","doi":"10.1137/21m1466189","DOIUrl":"https://doi.org/10.1137/21m1466189","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1068-1096, September 2024. <br/> Abstract. In this work, we derive a homogenized mathematical model for a porous intercalation electrode with a phase separating active material. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann–boundary condition modeling the lithium intercalation reaction [math]. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn–Hilliard equation, whereas the limit model consists of a diffusion and an Allen–Cahn equation. Thus, we observe a Cahn–Hilliard to Allen–Cahn transition during the upscaling process. In the sense of gradient flows, the transition coincides with a change in the underlying metric structure of the PDE system.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1030-1067, September 2024. Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.
{"title":"Quantum Algorithms for Multiscale Partial Differential Equations","authors":"Junpeng Hu, Shi Jin, Lei Zhang","doi":"10.1137/23m1566340","DOIUrl":"https://doi.org/10.1137/23m1566340","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 1030-1067, September 2024. <br/> Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"361 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 973-1029, September 2024. Abstract. We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale [math] in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter [math] around the support of the charge. We show that the algorithm in [J. Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv:2109.01616, 2021], suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion [P. Bella, A. Giunti, and F. Otto, Comm. Partial Differential Equations, 45 (2020), pp. 561–640], still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of [math], [math], and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multiscale logarithmic Sobolev inequality, where our main tool is an extension of the semigroup estimates in [N. Clozeau, Stoch. Partial Differ. Equ. Anal. Comput., 11 (2023), pp. 1254–1378]. As part of our strategy, we construct sublinear second-order correctors in this correlated setting, which is of independent interest.
多尺度建模与仿真》，第 22 卷第 3 期，第 973-1029 页，2024 年 9 月。 摘要我们对计算无限异质相关随机介质中电荷分布局部尺度[math]产生的电场的数值算法很感兴趣。我们证明了[J. Lu, F. Otto, and J. M.Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv:2109.01616, 2021]中的算法，提出了以多极扩展为动机的最优 Dirichlet 边界条件[P. Bella, A. Giunti, and L. Wang.Bella, A. Giunti, and F. Otto, Comm.Partial Differential Equations, 45 (2020), pp.我们以压倒性的概率获得了[math]、[math]和相关性大小的收敛速率，数值模拟支持了这些收敛速率的最优性。这些估计值是为满足多尺度对数索博列夫不等式的集合提供的，我们的主要工具是[N. Clozeau, Stoch.Clozeau, Stoch.Partial Differ.Equ.Anal.Comput., 11 (2023), pp.］作为我们策略的一部分，我们在这种相关设置中构建了亚线性二阶修正器，这也是我们的兴趣所在。
{"title":"Artificial Boundary Conditions for Random Elliptic Systems with Correlated Coefficient Field","authors":"Nicolas Clozeau, Lihan Wang","doi":"10.1137/23m1603819","DOIUrl":"https://doi.org/10.1137/23m1603819","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 973-1029, September 2024. <br/> Abstract. We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale [math] in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter [math] around the support of the charge. We show that the algorithm in [J. Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv:2109.01616, 2021], suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion [P. Bella, A. Giunti, and F. Otto, Comm. Partial Differential Equations, 45 (2020), pp. 561–640], still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of [math], [math], and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multiscale logarithmic Sobolev inequality, where our main tool is an extension of the semigroup estimates in [N. Clozeau, Stoch. Partial Differ. Equ. Anal. Comput., 11 (2023), pp. 1254–1378]. As part of our strategy, we construct sublinear second-order correctors in this correlated setting, which is of independent interest.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zecheng Zhang, Christian Moya, Wing Tat Leung, Guang Lin, Hayden Schaeffer
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 956-972, September 2024. Abstract. We present a new framework for computing fine-scale solutions of multiscale partial differential equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using observations of a limited number of (possible noisy) fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.
{"title":"Bayesian Deep Operator Learning for Homogenized to Fine-Scale Maps for Multiscale PDE","authors":"Zecheng Zhang, Christian Moya, Wing Tat Leung, Guang Lin, Hayden Schaeffer","doi":"10.1137/23m160342x","DOIUrl":"https://doi.org/10.1137/23m160342x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 956-972, September 2024. <br/> Abstract. We present a new framework for computing fine-scale solutions of multiscale partial differential equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using observations of a limited number of (possible noisy) fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jeremy Kazimer, Manlio De Domenico, Peter J. Mucha, Dane Taylor
Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 925-955, September 2024. Abstract. Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge’s removal would change a system’s von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks, since the possible removal of [math] edges requires that one compute all the eigenvalues of [math] distinct matrices. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and has a computational complexity that scales as [math] for networks with [math] nodes. Focusing on a form of VNE that is associated with a transport operator [math], where [math] is a graph Laplacian matrix and [math] is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate, and long timescales [math] for diffusion.
{"title":"Ranking Edges by Their Impact on the Spectral Complexity of Information Diffusion over Networks","authors":"Jeremy Kazimer, Manlio De Domenico, Peter J. Mucha, Dane Taylor","doi":"10.1137/22m153135x","DOIUrl":"https://doi.org/10.1137/22m153135x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 3, Page 925-955, September 2024. <br/> Abstract. Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge’s removal would change a system’s von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks, since the possible removal of [math] edges requires that one compute all the eigenvalues of [math] distinct matrices. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and has a computational complexity that scales as [math] for networks with [math] nodes. Focusing on a form of VNE that is associated with a transport operator [math], where [math] is a graph Laplacian matrix and [math] is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate, and long timescales [math] for diffusion.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 891-924, June 2024. Abstract. In this article, we introduce semi-implicit particle-in-cell (PIC) methods based on a discretization of the Vlasov–Maxwell system in the electrostatic regime and embedding sparse grid reconstructions: the semi-implict sparse-PIC (SISPIC-sg) scheme, its standard extension (SISPIC-std), and the energy-conserving sparse-PIC (ECSPIC) scheme. These schemes are inspired by the energy-conserving semi-implicit method introduced in [G. Lapenta, J. Comput. Phys., 334 (2017), pp. 349–366]. The particle equations are linearized so that the particle response to the field can be computed by solving a linear system with a stiffness matrix. The methods feature the three following properties: the scheme is unconditionally stable with respect to the plasma period; the finite grid instability is eliminated, allowing the user to use any desired grid discretization; the statistical error is significantly reduced compared to semi-implicit and explicit schemes with standard grid for the same number of particles. The ECSPIC scheme conserves exactly the discrete total energy of the system but we have experienced numerical instability related to the loss of the field energy nonnegativity genuine to the sparse grid combination technique. The SISPIC methods are exempted from this instability and are unconditionally stable with respect to the time and spatial discretization, but do not conserve exactly the discrete total energy. The methods have been investigated on a series of two-dimensional test cases, and gains in terms of memory storage and computational time compared to explicit and existing semi-implicit methods have been observed. These gains are expected to be larger for three-dimensional computations for which the full potential of sparse grid reconstructions can be achieved.
{"title":"Semi-Implicit Particle-in-Cell Methods Embedding Sparse Grid Reconstructions","authors":"C. Guillet","doi":"10.1137/23m1579340","DOIUrl":"https://doi.org/10.1137/23m1579340","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 891-924, June 2024. <br/> Abstract. In this article, we introduce semi-implicit particle-in-cell (PIC) methods based on a discretization of the Vlasov–Maxwell system in the electrostatic regime and embedding sparse grid reconstructions: the semi-implict sparse-PIC (SISPIC-sg) scheme, its standard extension (SISPIC-std), and the energy-conserving sparse-PIC (ECSPIC) scheme. These schemes are inspired by the energy-conserving semi-implicit method introduced in [G. Lapenta, J. Comput. Phys., 334 (2017), pp. 349–366]. The particle equations are linearized so that the particle response to the field can be computed by solving a linear system with a stiffness matrix. The methods feature the three following properties: the scheme is unconditionally stable with respect to the plasma period; the finite grid instability is eliminated, allowing the user to use any desired grid discretization; the statistical error is significantly reduced compared to semi-implicit and explicit schemes with standard grid for the same number of particles. The ECSPIC scheme conserves exactly the discrete total energy of the system but we have experienced numerical instability related to the loss of the field energy nonnegativity genuine to the sparse grid combination technique. The SISPIC methods are exempted from this instability and are unconditionally stable with respect to the time and spatial discretization, but do not conserve exactly the discrete total energy. The methods have been investigated on a series of two-dimensional test cases, and gains in terms of memory storage and computational time compared to explicit and existing semi-implicit methods have been observed. These gains are expected to be larger for three-dimensional computations for which the full potential of sparse grid reconstructions can be achieved.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"184 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 869-890, June 2024. Abstract. We consider the fluid-structure interaction of viscoelastic solids and Stokesian multiphase fluid flows with moving capillary interfaces and investigate the impact of moving contact lines. Thermodynamic consistency of Lagrangian diffuse and sharp-interface models is ensured even on the discrete level by providing a monolithic incremental time discretization and a finite element space discretization. We numerically analyze how phase-field models converge to sharp-interface limits when the interface thickness tends to zero, [math], and investigate scalings of the Cahn–Hilliard mobility [math] for [math]. In the presence of interfaces, certain sharp-interface limits are only valid for an interval [math], i.e., there is an upper and lower bound on the range of valid scaling exponents [math]. We show that with moving contact lines scaling is more restrictive since [math] causes significant errors due to excess diffusion. Similarly, we demonstrate that [math] leads to nonconvergence to the sharp-interface limit. We propose [math] as a range of exponents that ensure optimal convergence of the phase field dynamics towards the sharp interface dynamics as [math].
{"title":"Sharp-Interface Limits of Cahn–Hilliard Models and Mechanics with Moving Contact Lines","authors":"Leonie Schmeller, Dirk Peschka","doi":"10.1137/23m1546592","DOIUrl":"https://doi.org/10.1137/23m1546592","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 869-890, June 2024. <br/> Abstract. We consider the fluid-structure interaction of viscoelastic solids and Stokesian multiphase fluid flows with moving capillary interfaces and investigate the impact of moving contact lines. Thermodynamic consistency of Lagrangian diffuse and sharp-interface models is ensured even on the discrete level by providing a monolithic incremental time discretization and a finite element space discretization. We numerically analyze how phase-field models converge to sharp-interface limits when the interface thickness tends to zero, [math], and investigate scalings of the Cahn–Hilliard mobility [math] for [math]. In the presence of interfaces, certain sharp-interface limits are only valid for an interval [math], i.e., there is an upper and lower bound on the range of valid scaling exponents [math]. We show that with moving contact lines scaling is more restrictive since [math] causes significant errors due to excess diffusion. Similarly, we demonstrate that [math] leads to nonconvergence to the sharp-interface limit. We propose [math] as a range of exponents that ensure optimal convergence of the phase field dynamics towards the sharp interface dynamics as [math].","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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