Pub Date : 2024-10-29DOI: 10.1016/j.jcp.2024.113538
Chiu-Yen Kao , Junshan Lin , Braxton Osting
Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.
{"title":"A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals","authors":"Chiu-Yen Kao , Junshan Lin , Braxton Osting","doi":"10.1016/j.jcp.2024.113538","DOIUrl":"10.1016/j.jcp.2024.113538","url":null,"abstract":"<div><div>Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113538"},"PeriodicalIF":3.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554356","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 : 2024-10-29DOI: 10.1016/j.jcp.2024.113535
François Vilar
This article aims at presenting a new local subcell monolithic Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme solving system of conservation laws on 2D unstructured grids. This is known that DG method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, a convex blending of high-order DG and first-order FV schemes will be locally performed, at the subcell scale, where it is needed. To this end, by means of the theory developed in [58], [59], we first recall that it is possible to rewrite DG scheme as a subcell FV method, defined on a subgrid, provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the subcell monolithic DG/FV method will be defined as follows: to each face of each subcell we will assign two fluxes, a 1st-order FV one and a high-order reconstructed one, that in the end will be blended in a convex way. The goal is then to determine, through analysis, optimal blending coefficients to achieve the desired properties. Numerical results on various type problems will be presented to assess the very good performance of the design method.
A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: is this possible through this monolithic framework to ensure any entropy stability? What do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed while aiming for high-order accuracy?
{"title":"Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration","authors":"François Vilar","doi":"10.1016/j.jcp.2024.113535","DOIUrl":"10.1016/j.jcp.2024.113535","url":null,"abstract":"<div><div>This article aims at presenting a new local subcell monolithic Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme solving system of conservation laws on 2D unstructured grids. This is known that DG method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, a convex blending of high-order DG and first-order FV schemes will be locally performed, at the subcell scale, where it is needed. To this end, by means of the theory developed in <span><span>[58]</span></span>, <span><span>[59]</span></span>, we first recall that it is possible to rewrite DG scheme as a subcell FV method, defined on a subgrid, provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the subcell monolithic DG/FV method will be defined as follows: to each face of each subcell we will assign two fluxes, a 1st-order FV one and a high-order reconstructed one, that in the end will be blended in a convex way. The goal is then to determine, through analysis, optimal blending coefficients to achieve the desired properties. Numerical results on various type problems will be presented to assess the very good performance of the design method.</div><div>A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: is this possible through this monolithic framework to ensure any entropy stability? What do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed while aiming for high-order accuracy?</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113535"},"PeriodicalIF":3.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554353","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 : 2024-10-29DOI: 10.1016/j.jcp.2024.113533
Zakari Eckert , Jeremiah J. Boerner , Taylor H. Hall , Russell Hooper , Anne M. Grillet , Jose L. Pacheco
We examine a number of common verification and benchmark problems for Particle-in-Cell and Direct Simulation Monte Carlo codes. Since results, including convergence rates, comparison to analytic solutions, and code-to-code comparisons, for these problems are often used as evidence of correctness for simulation codes, it is necessary to understand what successful verification using one or more of these problems implies about the correctness of the simulation code. To that end, a series of benchmark problems is performed in Aleph, a PIC-DSMC code developed at Sandia National Laboratories, including both at the canonical numerical parameters and others where verification should fail. The results presented suggest that improvements and extensions to current benchmark problems and additional problem specifications would benefit existing and future codes thereby providing greater confidence in predictive results.
{"title":"Benchmark verification of PIC-DSMC programs","authors":"Zakari Eckert , Jeremiah J. Boerner , Taylor H. Hall , Russell Hooper , Anne M. Grillet , Jose L. Pacheco","doi":"10.1016/j.jcp.2024.113533","DOIUrl":"10.1016/j.jcp.2024.113533","url":null,"abstract":"<div><div>We examine a number of common verification and benchmark problems for Particle-in-Cell and Direct Simulation Monte Carlo codes. Since results, including convergence rates, comparison to analytic solutions, and code-to-code comparisons, for these problems are often used as evidence of correctness for simulation codes, it is necessary to understand what successful verification using one or more of these problems implies about the correctness of the simulation code. To that end, a series of benchmark problems is performed in Aleph, a PIC-DSMC code developed at Sandia National Laboratories, including both at the canonical numerical parameters and others where verification should fail. The results presented suggest that improvements and extensions to current benchmark problems and additional problem specifications would benefit existing and future codes thereby providing greater confidence in predictive results.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113533"},"PeriodicalIF":3.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560646","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 : 2024-10-29DOI: 10.1016/j.jcp.2024.113541
Yashar Mehmani, Kangan Li
Fluid flow through porous media is central to many subsurface (e.g., CO2 storage) and industrial (e.g., fuel cell) applications. The optimization of design and operational protocols, and quantifying the associated uncertainties, requires fluid-dynamics simulations inside the microscale void space of porous samples. This often results in large and ill-conditioned linear(ized) systems that require iterative solvers, for which preconditioning is key to ensure rapid convergence. We present robust and efficient preconditioners for the accelerated solution of saddle-point systems arising from the discretization of the Stokes equation on geometrically complex porous microstructures. They are based on the recently proposed pore-level multiscale method (PLMM) and the more established reduced-order method called the pore network model (PNM). The four preconditioners presented are the monolithic PLMM, monolithic PNM, block PLMM, and block PNM. Compared to existing block preconditioners, accelerated by the algebraic multigrid method, we show our preconditioners are far more robust and efficient. The monolithic PLMM is an algebraic reformulation of the original PLMM, which renders it portable and amenable to non-intrusive implementation in existing software. Similarly, the monolithic PNM is an algebraization of PNM, allowing it to be used as an accelerator of direct numerical simulations (DNS). This bestows PNM with the, heretofore absent, ability to estimate and control prediction errors. The monolithic PLMM/PNM can also be used as approximate solvers that yield globally flux-conservative solutions, usable in many practical settings. We systematically test all preconditioners on 2D/3D geometries and show the monolithic PLMM outperforms all others. All preconditioners can be built and applied on parallel machines.
{"title":"Multiscale preconditioning of Stokes flow in complex porous geometries","authors":"Yashar Mehmani, Kangan Li","doi":"10.1016/j.jcp.2024.113541","DOIUrl":"10.1016/j.jcp.2024.113541","url":null,"abstract":"<div><div>Fluid flow through porous media is central to many subsurface (e.g., CO<sub>2</sub> storage) and industrial (e.g., fuel cell) applications. The optimization of design and operational protocols, and quantifying the associated uncertainties, requires fluid-dynamics simulations inside the microscale void space of porous samples. This often results in large and ill-conditioned linear(ized) systems that require iterative solvers, for which preconditioning is key to ensure rapid convergence. We present robust and efficient preconditioners for the accelerated solution of saddle-point systems arising from the discretization of the Stokes equation on geometrically complex porous microstructures. They are based on the recently proposed pore-level multiscale method (PLMM) and the more established reduced-order method called the pore network model (PNM). The four preconditioners presented are the monolithic PLMM, monolithic PNM, block PLMM, and block PNM. Compared to existing block preconditioners, accelerated by the algebraic multigrid method, we show our preconditioners are far more robust and efficient. The monolithic PLMM is an algebraic reformulation of the original PLMM, which renders it portable and amenable to non-intrusive implementation in existing software. Similarly, the monolithic PNM is an algebraization of PNM, allowing it to be used as an accelerator of direct numerical simulations (DNS). This bestows PNM with the, heretofore absent, ability to estimate and control prediction errors. The monolithic PLMM/PNM can also be used as approximate solvers that yield globally flux-conservative solutions, usable in many practical settings. We systematically test all preconditioners on 2D/3D geometries and show the monolithic PLMM outperforms all others. All preconditioners can be built and applied on parallel machines.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113541"},"PeriodicalIF":3.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593480","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 : 2024-10-29DOI: 10.1016/j.jcp.2024.113539
Yingxia Xi , Xia Ji
The paper considers the computation of scattering resonances of the fluid-solid interaction problem. Scattering resonances are the replacement of discrete spectral data for problems on non-compact domains which are very important in many areas of science and engineering. For the special disk case, we get the analytical solution which can be used as reference solutions. For the general case, we truncate the unbounded domain using the Dirichlet-to-Neumann (DtN) mapping. Standard linear Lagrange element is used to do the discretization which leads to nonlinear algebraic eigenvalue problems. We then solve the nonlinear algebraic eigenvalue problems by the parallel spectral indicator methods. Finally, numerical examples are presented.
{"title":"A finite element contour integral method for computing the scattering resonances of fluid-solid interaction problem","authors":"Yingxia Xi , Xia Ji","doi":"10.1016/j.jcp.2024.113539","DOIUrl":"10.1016/j.jcp.2024.113539","url":null,"abstract":"<div><div>The paper considers the computation of scattering resonances of the fluid-solid interaction problem. Scattering resonances are the replacement of discrete spectral data for problems on non-compact domains which are very important in many areas of science and engineering. For the special disk case, we get the analytical solution which can be used as reference solutions. For the general case, we truncate the unbounded domain using the Dirichlet-to-Neumann (DtN) mapping. Standard linear Lagrange element is used to do the discretization which leads to nonlinear algebraic eigenvalue problems. We then solve the nonlinear algebraic eigenvalue problems by the parallel spectral indicator methods. Finally, numerical examples are presented.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113539"},"PeriodicalIF":3.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561369","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 : 2024-10-28DOI: 10.1016/j.jcp.2024.113534
Yaqing Yang , Liang Pan , Kun Xu
For the simulations of unsteady flow, the global time step becomes really small with a large variation of local cell size. In this paper, an implicit high-order gas-kinetic scheme (HGKS) is developed to alleviate the restrictions on the time step for unsteady simulations. In order to improve the efficiency and keep the high-order accuracy, a two-stage third-order implicit time-accurate discretization is proposed. In each stage, an artificial steady solution is obtained for the implicit system with the pseudo-time iteration. In the iteration, the classical implicit methods are adopted to solve the nonlinear system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and generalized minimum residual (GMRES) methods. To achieve the spatial accuracy, the HGKSs with both non-compact and compact reconstructions are constructed. For the non-compact scheme, the weighted essentially non-oscillatory (WENO) reconstruction is used. For the compact one, the Hermite WENO (HWENO) reconstruction is adopted due to the updates of both cell-averaged flow variables and their derivatives. The expected third-order temporal accuracy is achieved with the two-stage temporal discretization. For the smooth flow, only a single artificial iteration is needed. For uniform meshes, the efficiency of the current implicit method improves significantly in comparison with the explicit one. For the flow with discontinuities, compared with the well-known Crank-Nicholson method, the spurious oscillations in the current schemes are well suppressed. The increase of the artificial iteration steps introduces extra reconstructions associating with a reduction of the computational efficiency. Overall, the current implicit method leads to an improvement in efficiency over the explicit one in the cases with a large variation of mesh size. Meanwhile, for the cases with strong discontinuities on a uniform mesh, the efficiency of the current method is comparable with that of the explicit scheme.
{"title":"Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes II: Unsteady flows","authors":"Yaqing Yang , Liang Pan , Kun Xu","doi":"10.1016/j.jcp.2024.113534","DOIUrl":"10.1016/j.jcp.2024.113534","url":null,"abstract":"<div><div>For the simulations of unsteady flow, the global time step becomes really small with a large variation of local cell size. In this paper, an implicit high-order gas-kinetic scheme (HGKS) is developed to alleviate the restrictions on the time step for unsteady simulations. In order to improve the efficiency and keep the high-order accuracy, a two-stage third-order implicit time-accurate discretization is proposed. In each stage, an artificial steady solution is obtained for the implicit system with the pseudo-time iteration. In the iteration, the classical implicit methods are adopted to solve the nonlinear system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and generalized minimum residual (GMRES) methods. To achieve the spatial accuracy, the HGKSs with both non-compact and compact reconstructions are constructed. For the non-compact scheme, the weighted essentially non-oscillatory (WENO) reconstruction is used. For the compact one, the Hermite WENO (HWENO) reconstruction is adopted due to the updates of both cell-averaged flow variables and their derivatives. The expected third-order temporal accuracy is achieved with the two-stage temporal discretization. For the smooth flow, only a single artificial iteration is needed. For uniform meshes, the efficiency of the current implicit method improves significantly in comparison with the explicit one. For the flow with discontinuities, compared with the well-known Crank-Nicholson method, the spurious oscillations in the current schemes are well suppressed. The increase of the artificial iteration steps introduces extra reconstructions associating with a reduction of the computational efficiency. Overall, the current implicit method leads to an improvement in efficiency over the explicit one in the cases with a large variation of mesh size. Meanwhile, for the cases with strong discontinuities on a uniform mesh, the efficiency of the current method is comparable with that of the explicit scheme.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113534"},"PeriodicalIF":3.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560647","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 : 2024-10-28DOI: 10.1016/j.jcp.2024.113528
Christina G. Taylor , Lucas C. Wilcox , Jesse Chan
Cut meshes are a type of mesh that is formed by allowing embedded boundaries to “cut” a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani in [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is energy stable under arbitrary quadrature. We prove that state redistribution can be added to a provably energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.
{"title":"An energy stable high-order cut cell discontinuous Galerkin method with state redistribution for wave propagation","authors":"Christina G. Taylor , Lucas C. Wilcox , Jesse Chan","doi":"10.1016/j.jcp.2024.113528","DOIUrl":"10.1016/j.jcp.2024.113528","url":null,"abstract":"<div><div>Cut meshes are a type of mesh that is formed by allowing embedded boundaries to “cut” a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the <em>small cell problem</em>, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani in <span><span>[1]</span></span>, can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> energy stable under arbitrary quadrature. We prove that state redistribution can be added to a provably <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113528"},"PeriodicalIF":3.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554355","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 : 2024-10-28DOI: 10.1016/j.jcp.2024.113536
Philipp Horn , Veronica Saz Ulibarrena , Barry Koren , Simon Portegies Zwart
When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approximate the trajectories with extreme precision. So, integrating Hamilton's equations to a level of scientific reliability such that the answer can be used for scientific interpretation, may be computationally expensive. However, a neural network can be a viable alternative to numerical integrators, offering high-fidelity solutions orders of magnitudes faster.
To understand whether it is also important to preserve the symplecticity when neural networks are used, we analyze three well-known neural network architectures that are including the symplectic structure inside the neural network's topology. Between these neural network architectures many similarities can be found. This allows us to formulate a new, generalized framework for these architectures. In the generalized framework Symplectic Recurrent Neural Networks, SympNets and HénonNets are included as special cases. Additionally, this new framework enables us to find novel neural network topologies by transitioning between the established ones.
We compare new Generalized Hamiltonian Neural Networks (GHNNs) against the already established SympNets, HénonNets and physics-unaware multilayer perceptrons. This comparison is performed with data for a pendulum, a double pendulum and a gravitational 3-body problem. In order to achieve a fair comparison, the hyperparameters of the different neural networks are chosen such that the prediction speeds of all four architectures are the same during inference. A special focus lies on the capability of the neural networks to generalize outside the training data. The GHNNs outperform all other neural network architectures for the problems considered.
{"title":"A generalized framework of neural networks for Hamiltonian systems","authors":"Philipp Horn , Veronica Saz Ulibarrena , Barry Koren , Simon Portegies Zwart","doi":"10.1016/j.jcp.2024.113536","DOIUrl":"10.1016/j.jcp.2024.113536","url":null,"abstract":"<div><div>When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approximate the trajectories with extreme precision. So, integrating Hamilton's equations to a level of scientific reliability such that the answer can be used for scientific interpretation, may be computationally expensive. However, a neural network can be a viable alternative to numerical integrators, offering high-fidelity solutions orders of magnitudes faster.</div><div>To understand whether it is also important to preserve the symplecticity when neural networks are used, we analyze three well-known neural network architectures that are including the symplectic structure inside the neural network's topology. Between these neural network architectures many similarities can be found. This allows us to formulate a new, generalized framework for these architectures. In the generalized framework Symplectic Recurrent Neural Networks, SympNets and HénonNets are included as special cases. Additionally, this new framework enables us to find novel neural network topologies by transitioning between the established ones.</div><div>We compare new Generalized Hamiltonian Neural Networks (GHNNs) against the already established SympNets, HénonNets and physics-unaware multilayer perceptrons. This comparison is performed with data for a pendulum, a double pendulum and a gravitational 3-body problem. In order to achieve a fair comparison, the hyperparameters of the different neural networks are chosen such that the prediction speeds of all four architectures are the same during inference. A special focus lies on the capability of the neural networks to generalize outside the training data. The GHNNs outperform all other neural network architectures for the problems considered.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113536"},"PeriodicalIF":3.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578701","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 : 2024-10-28DOI: 10.1016/j.jcp.2024.113531
Jean-Luc Guermond , Zuodong Wang
A limiting technique for scalar transport equations is presented. The originality of the method is that it does not require solving nonlinear optimization problems nor does it rely on the construction of a low-order approximation. The method has minimal complexity and is numerically demonstrated to maintain high-order accuracy. The performance of the method is illustrated on the radiation transport equation.
{"title":"Mass conservative limiting and applications to the approximation of the steady-state radiation transport equations","authors":"Jean-Luc Guermond , Zuodong Wang","doi":"10.1016/j.jcp.2024.113531","DOIUrl":"10.1016/j.jcp.2024.113531","url":null,"abstract":"<div><div>A limiting technique for scalar transport equations is presented. The originality of the method is that it does not require solving nonlinear optimization problems nor does it rely on the construction of a low-order approximation. The method has minimal complexity and is numerically demonstrated to maintain high-order accuracy. The performance of the method is illustrated on the radiation transport equation.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113531"},"PeriodicalIF":3.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587082","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 : 2024-10-28DOI: 10.1016/j.jcp.2024.113525
Jinghua Wang
This paper presents a new numerical model based on the highly nonlinear potential flow theory for simulating the propagation of water waves in variable depth. A new set of equations for estimating the surface vertical velocity is derived based on the boundary integral equation considering the water depth variability. A successive approximation scheme is also proposed in this study for calculating the surface vertical velocity. With the usage of Fast Fourier Transform, the model can be efficiently used for simulating highly nonlinear water waves on large spatiotemporal scale in a phase-resolving approach. The new model is comprehensively verified and validated through simulating a variety of nonlinear wave phenomenon including free propagating solitary wave, wave transformations over submerged bar, Bragg reflection over undulating bars, nonlinear evolution of Peregrine breather, obliquely propagating uniform waves and extreme waves in crossing random seas. Good agreements are achieved between the numerical simulations and laboratory measurements, indicating that the new model is sufficiently accurate. A discussion is presented on the accuracy and efficiency of the present model, which is compared with the Higher-Order Spectral method. The results show that the present model can be significantly more efficient at the same level of accuracy. It is suggested that the new model developed in the paper can be reliably used to simulate the nonlinear evolution of ocean waves in phase-resolving approach to shed light on the dynamics of nonlinear wave phenomenon taking place on a large spatiotemporal scale, which may be computationally expensive by using other existing methods.
{"title":"An enhanced spectral boundary integral method for modeling highly nonlinear water waves in variable depth","authors":"Jinghua Wang","doi":"10.1016/j.jcp.2024.113525","DOIUrl":"10.1016/j.jcp.2024.113525","url":null,"abstract":"<div><div>This paper presents a new numerical model based on the highly nonlinear potential flow theory for simulating the propagation of water waves in variable depth. A new set of equations for estimating the surface vertical velocity is derived based on the boundary integral equation considering the water depth variability. A successive approximation scheme is also proposed in this study for calculating the surface vertical velocity. With the usage of Fast Fourier Transform, the model can be efficiently used for simulating highly nonlinear water waves on large spatiotemporal scale in a phase-resolving approach. The new model is comprehensively verified and validated through simulating a variety of nonlinear wave phenomenon including free propagating solitary wave, wave transformations over submerged bar, Bragg reflection over undulating bars, nonlinear evolution of Peregrine breather, obliquely propagating uniform waves and extreme waves in crossing random seas. Good agreements are achieved between the numerical simulations and laboratory measurements, indicating that the new model is sufficiently accurate. A discussion is presented on the accuracy and efficiency of the present model, which is compared with the Higher-Order Spectral method. The results show that the present model can be significantly more efficient at the same level of accuracy. It is suggested that the new model developed in the paper can be reliably used to simulate the nonlinear evolution of ocean waves in phase-resolving approach to shed light on the dynamics of nonlinear wave phenomenon taking place on a large spatiotemporal scale, which may be computationally expensive by using other existing methods.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113525"},"PeriodicalIF":3.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560644","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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