Pub Date : 2025-01-15DOI: 10.1016/j.jcp.2025.113745
Andreas Granath, Siyang Wang
We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible discontinuous Galerkin methods on unstructured grids by a penalty based technique. At the interface between the two methods, we construct projection operators for the pointwise finite difference solutions and discontinuous Galerkin solutions based on piecewise polynomials. In addition, we optimize the projection operators for both accuracy and spectrum. We prove that the overall discretization conserves a discrete energy, and verify optimal convergence in numerical experiments.
{"title":"A hybrid numerical method for elastic wave propagation in discontinuous media with complex geometry","authors":"Andreas Granath, Siyang Wang","doi":"10.1016/j.jcp.2025.113745","DOIUrl":"10.1016/j.jcp.2025.113745","url":null,"abstract":"<div><div>We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible discontinuous Galerkin methods on unstructured grids by a penalty based technique. At the interface between the two methods, we construct projection operators for the pointwise finite difference solutions and discontinuous Galerkin solutions based on piecewise polynomials. In addition, we optimize the projection operators for both accuracy and spectrum. We prove that the overall discretization conserves a discrete energy, and verify optimal convergence in numerical experiments.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113745"},"PeriodicalIF":3.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131966","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 : 2025-01-15DOI: 10.1016/j.jcp.2025.113746
Eric J. Ching , Ryan F. Johnson , Sarah Burrows , Jacklyn Higgs , Andrew D. Kercher
This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang (2017) [16], which was adapted to multicomponent flows by Du and Yang (2022) [17] in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also detail how to account for boundary conditions and incorporate previously developed techniques to reduce spurious pressure oscillations into the positivity-preserving framework. Furthermore, potential issues associated with the Lax-Friedrichs-type viscous flux function in the case of zero species concentrations are discussed and addressed. Comparisons between the Lax-Friedrichs-type viscous flux function and more conventional flux functions are provided, the results of which motivate an adaptive solution procedure that employs the former only when the element-local solution average has negative species concentrations, nonpositive density, or nonpositive pressure. The resulting formulation is compatible with curved, multidimensional elements and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction.
{"title":"Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations","authors":"Eric J. Ching , Ryan F. Johnson , Sarah Burrows , Jacklyn Higgs , Andrew D. Kercher","doi":"10.1016/j.jcp.2025.113746","DOIUrl":"10.1016/j.jcp.2025.113746","url":null,"abstract":"<div><div>This article concerns the development of a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for the multicomponent, chemically reacting, compressible Navier-Stokes equations with complex thermodynamics. In particular, we extend to viscous flows the fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin method for the chemically reacting Euler equations that we previously introduced. An important component of the formulation is the positivity-preserving Lax-Friedrichs-type viscous flux function devised by Zhang (2017) <span><span>[16]</span></span>, which was adapted to multicomponent flows by Du and Yang (2022) <span><span>[17]</span></span> in a manner that treats the inviscid and viscous fluxes as a single flux. Here, we similarly extend the aforementioned flux function to multicomponent flows but separate the inviscid and viscous fluxes, resulting in a different dissipation coefficient. This separation of the fluxes allows for use of other inviscid flux functions, as well as enforcement of entropy boundedness on only the convective contribution to the evolved state, as motivated by physical and mathematical principles. We also detail how to account for boundary conditions and incorporate previously developed techniques to reduce spurious pressure oscillations into the positivity-preserving framework. Furthermore, potential issues associated with the Lax-Friedrichs-type viscous flux function in the case of zero species concentrations are discussed and addressed. Comparisons between the Lax-Friedrichs-type viscous flux function and more conventional flux functions are provided, the results of which motivate an adaptive solution procedure that employs the former only when the element-local solution average has negative species concentrations, nonpositive density, or nonpositive pressure. The resulting formulation is compatible with curved, multidimensional elements and general quadrature rules with positive weights. A variety of multicomponent, viscous flows is computed, ranging from a one-dimensional shock tube problem to multidimensional detonation waves and shock/mixing-layer interaction.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"525 ","pages":"Article 113746"},"PeriodicalIF":3.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175038","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 : 2025-01-14DOI: 10.1016/j.jcp.2025.113737
Jun Zhang , Kejia Pan , Xiaofeng Yang
The development of effective numerical methods for simulating the dendritic solidification process using the flow-coupled, melt-convective phase-field model has consistently encountered challenges due to the model's complex nonlinear and coupled structures. The major concern of algorithm design is to ensure a numerical scheme that achieves second-order accuracy in time, maintains linearity and attains a fully decoupled structure. All these objectives are accomplished within the framework of guaranteed unconditional energy stability, which is addressed by the fully discrete finite element scheme proposed in this paper. The developed scheme uses the modified projection-type method to deal with the “weighted” Navier-Stokes equations, and the complete decoupling feature is achieved by using the explicit-SAV (Scalar Auxiliary Variable) method, which also helps to linearize the nonlinear potentials. The scheme simplifies its procedure by only requiring the solution of several completely decoupled and linear elliptic equations at every time step, which facilitates its easy implementation. The solvability and energy stability are further rigorously validated. Comprehensive details of the procedural steps for implementation are also provided, accompanied by plenty of numerical tests conducted in both 2D and 3D, serving to numerically ascertain the accuracy and robustness of the scheme.
{"title":"Fully discrete finite element method with full decoupling structure and second-order temporal accuracy for a flow-coupled dendritic solidification phase-field model","authors":"Jun Zhang , Kejia Pan , Xiaofeng Yang","doi":"10.1016/j.jcp.2025.113737","DOIUrl":"10.1016/j.jcp.2025.113737","url":null,"abstract":"<div><div>The development of effective numerical methods for simulating the dendritic solidification process using the flow-coupled, melt-convective phase-field model has consistently encountered challenges due to the model's complex nonlinear and coupled structures. The major concern of algorithm design is to ensure a numerical scheme that achieves second-order accuracy in time, maintains linearity and attains a fully decoupled structure. All these objectives are accomplished within the framework of guaranteed unconditional energy stability, which is addressed by the fully discrete finite element scheme proposed in this paper. The developed scheme uses the modified projection-type method to deal with the “weighted” Navier-Stokes equations, and the complete decoupling feature is achieved by using the explicit-SAV (Scalar Auxiliary Variable) method, which also helps to linearize the nonlinear potentials. The scheme simplifies its procedure by only requiring the solution of several completely decoupled and linear elliptic equations at every time step, which facilitates its easy implementation. The solvability and energy stability are further rigorously validated. Comprehensive details of the procedural steps for implementation are also provided, accompanied by plenty of numerical tests conducted in both 2D and 3D, serving to numerically ascertain the accuracy and robustness of the scheme.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113737"},"PeriodicalIF":3.8,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131963","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 : 2025-01-14DOI: 10.1016/j.jcp.2025.113743
Lei Wang , Robert Krasny
To account for hydrodynamic interactions among solvated molecules, Brownian dynamics simulations require correlated random displacements , where D is the Rotne-Prager-Yamakawa diffusion tensor for a system of N particles and z is a standard normal random vector. The Spectral Lanczos Decomposition Method (SLDM) computes a sequence of Krylov subspace approximations , but each step requires a dense matrix-vector product with a Lanczos vector q, and the cost of computing the product by direct summation (DS) is an obstacle for large-scale simulations. This work employs the barycentric Lagrange treecode (BLTC) to reduce the cost of the matrix-vector product to while introducing a controllable approximation error. Numerical experiments compare the performance of SLDM-DS and SLDM-BLTC in serial and parallel (32 core, GPU) calculations.
{"title":"Numerical experiments using the barycentric Lagrange treecode to compute correlated random displacements for Brownian dynamics simulations","authors":"Lei Wang , Robert Krasny","doi":"10.1016/j.jcp.2025.113743","DOIUrl":"10.1016/j.jcp.2025.113743","url":null,"abstract":"<div><div>To account for hydrodynamic interactions among solvated molecules, Brownian dynamics simulations require correlated random displacements <span><math><mi>g</mi><mo>=</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>z</mi></math></span>, where <em>D</em> is the <span><math><mn>3</mn><mi>N</mi><mo>×</mo><mn>3</mn><mi>N</mi></math></span> Rotne-Prager-Yamakawa diffusion tensor for a system of <em>N</em> particles and <strong>z</strong> is a standard normal random vector. The Spectral Lanczos Decomposition Method (SLDM) computes a sequence of Krylov subspace approximations <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>→</mo><mi>g</mi></math></span>, but each step requires a dense matrix-vector product <span><math><mi>D</mi><mi>q</mi></math></span> with a Lanczos vector <strong>q</strong>, and the <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> cost of computing the product by direct summation (DS) is an obstacle for large-scale simulations. This work employs the barycentric Lagrange treecode (BLTC) to reduce the cost of the matrix-vector product to <span><math><mi>O</mi><mo>(</mo><mi>N</mi><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></math></span> while introducing a controllable approximation error. Numerical experiments compare the performance of SLDM-DS and SLDM-BLTC in serial and parallel (32 core, GPU) calculations.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"525 ","pages":"Article 113743"},"PeriodicalIF":3.8,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175031","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 : 2025-01-13DOI: 10.1016/j.jcp.2025.113729
Francesco Romor , Giovanni Stabile , Gianluigi Rozza
A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to perform accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modeling that employs neural networks for the solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov-Galerkin method, opportunely hyper-reduced in order to be independent of the number of degrees of freedom. New adaptive hyper-reduction strategies are developed along with the employment of local nonlinear approximants. We test our methodology on two nonlinear time dependent parametric benchmarks involving a supersonic flow past a NACA airfoil with changing Mach number and an incompressible turbulent flow around the Ahmed body with changing slant angle.
{"title":"Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations","authors":"Francesco Romor , Giovanni Stabile , Gianluigi Rozza","doi":"10.1016/j.jcp.2025.113729","DOIUrl":"10.1016/j.jcp.2025.113729","url":null,"abstract":"<div><div>A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to perform accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modeling that employs neural networks for the solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov-Galerkin method, opportunely hyper-reduced in order to be independent of the number of degrees of freedom. New adaptive hyper-reduction strategies are developed along with the employment of local nonlinear approximants. We test our methodology on two nonlinear time dependent parametric benchmarks involving a supersonic flow past a NACA airfoil with changing Mach number and an incompressible turbulent flow around the Ahmed body with changing slant angle.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113729"},"PeriodicalIF":3.8,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131970","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 : 2025-01-13DOI: 10.1016/j.jcp.2025.113738
Jian Meng , Xu Qian , Jiali Qiu , Jingmin Xia
The singularly perturbed theory mainly arises in the system of differential equations with the small enough perturbed parameters acting on the highest-order derivatives. In this paper, we introduce the adaptive mixed virtual element method for the fourth-order singularly perturbed problem and the associated eigenvalue problem. It allows to apply the -conforming virtual elements to discrete the continuous spaces and reduces the total number of required degrees of freedom of the -conforming virtual element method. Basically, the great flexibility of virtual element method becomes appealing in mesh refinement because the locally mesh post-processing to remove hanging nodes is never needed. This naturally motivates us to develop an a posteriori error estimate for the model problem. Based on the numerical solutions, the interior and edge residual terms, and the error terms related to the inconsistency of the virtual element scheme, the error estimators applied to adaptively refine meshes are constructed and then proved to be equivalent to numerical errors under the balanced energy norms. Moreover, we also consider the approximation method for the fourth-order singularly perturbed eigenvalue problem in two-dimensional space. Analogous with the source problem, we not only discuss the boundedness of the eigenfunctions, but also present the upper bound for the error of the approximated eigenvalues by these error estimators. Necessitated by supporting the theoretical analysis, representative numerical examples are reported. We show that the current numerical method converges at the optimal rate uniformly with respect to the singularly perturbed parameters when using the adaptive polygonal meshes.
{"title":"Adaptive mixed virtual element method for the fourth-order singularly perturbed problem","authors":"Jian Meng , Xu Qian , Jiali Qiu , Jingmin Xia","doi":"10.1016/j.jcp.2025.113738","DOIUrl":"10.1016/j.jcp.2025.113738","url":null,"abstract":"<div><div>The singularly perturbed theory mainly arises in the system of differential equations with the small enough perturbed parameters acting on the highest-order derivatives. In this paper, we introduce the adaptive mixed virtual element method for the fourth-order singularly perturbed problem and the associated eigenvalue problem. It allows to apply the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-conforming virtual elements to discrete the continuous spaces and reduces the total number of required degrees of freedom of the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-conforming virtual element method. Basically, the great flexibility of virtual element method becomes appealing in mesh refinement because the locally mesh post-processing to remove hanging nodes is never needed. This naturally motivates us to develop an <em>a posteriori</em> error estimate for the model problem. Based on the numerical solutions, the interior and edge residual terms, and the error terms related to the inconsistency of the virtual element scheme, the error estimators applied to adaptively refine meshes are constructed and then proved to be equivalent to numerical errors under the balanced energy norms. Moreover, we also consider the approximation method for the fourth-order singularly perturbed eigenvalue problem in two-dimensional space. Analogous with the source problem, we not only discuss the boundedness of the eigenfunctions, but also present the upper bound for the error of the approximated eigenvalues by these error estimators. Necessitated by supporting the theoretical analysis, representative numerical examples are reported. We show that the current numerical method converges at the optimal rate uniformly with respect to the singularly perturbed parameters when using the adaptive polygonal meshes.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113738"},"PeriodicalIF":3.8,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131961","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 : 2025-01-13DOI: 10.1016/j.jcp.2025.113741
Opal Issan , Oleksandr Koshkarov , Federico D. Halpern , Boris Kramer , Gian Luca Delzanno
We derive conservative closures for the Vlasov-Poisson equations discretized in velocity via the symmetrically weighted Hermite spectral expansion. We demonstrate that no closure can simultaneously restore the conservation of mass, momentum, and energy in this formulation. The properties of the analytically derived conservative closures of each conserved quantity are validated numerically by simulating an electrostatic benchmark problem: the Langmuir wave. Both the numerical results and analytical analysis indicate that closure by truncation (i.e. setting the last Hermite moment to zero) is the most suitable conservative closure for the symmetrically weighted Hermite formulation.
{"title":"Conservative closures of the Vlasov-Poisson equations based on symmetrically weighted Hermite spectral expansion","authors":"Opal Issan , Oleksandr Koshkarov , Federico D. Halpern , Boris Kramer , Gian Luca Delzanno","doi":"10.1016/j.jcp.2025.113741","DOIUrl":"10.1016/j.jcp.2025.113741","url":null,"abstract":"<div><div>We derive conservative closures for the Vlasov-Poisson equations discretized in velocity via the symmetrically weighted Hermite spectral expansion. We demonstrate that no closure can simultaneously restore the conservation of mass, momentum, and energy in this formulation. The properties of the analytically derived conservative closures of each conserved quantity are validated numerically by simulating an electrostatic benchmark problem: the Langmuir wave. Both the numerical results and analytical analysis indicate that closure by truncation (i.e. setting the last Hermite moment to zero) is the most suitable conservative closure for the symmetrically weighted Hermite formulation.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113741"},"PeriodicalIF":3.8,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131960","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 : 2025-01-13DOI: 10.1016/j.jcp.2025.113739
C. Guillet
Sparse grid reconstructions have recently been applied to Particle-In-Cell (PIC) methods with a semi-implicit formulation, as demonstrated in [28], to reduce computational costs. By linearizing the particle equations and using a finite difference discretization of the field's equations, along with incorporating sparse grid reconstructions through the combination technique, an exactly energy-conserving scheme was proposed. However, this scheme exhibited numerical instability due to the loss of non-negativity in electric energy, inherent to the combination technique. This paper introduces a novel PIC method with a semi-implicit formulation that embeds sparse grid techniques to exactly conserve discrete total energy, defined as the sum of non-negative kinetic and field energies, ensuring nonlinear stability. The method utilizes a Galerkin approach for the field equations, employing a hierarchical sparse grid representation in the approximation space. This distinguishes it from previous sparse grid PIC methods, which typically use the combination technique and nodal representation. The enhancement of hierarchical subspaces, serving as the truncated combination technique counterpart of the newly introduced method, is proposed to address the limitations of sparse-PIC methods— notably the difficulty in capturing non-smooth and non-axis-aligned solutions. Key features of the method include: unconditional stability with respect to the plasma period; mitigation of grid heating, allowing flexible grid discretization irrespective of the Debye length; exact conservation of discrete total energy; significant reduction in statistical error compared to standard grid schemes for the same number of particles; and decreased computational complexity, particularly in the size of the linear system to be solved. We validate the method through a series of two-dimensional test cases, demonstrating its numerical stability and robust performance.
{"title":"Energy-conserving Particle-In-Cell scheme based on Galerkin methods with sparse grids","authors":"C. Guillet","doi":"10.1016/j.jcp.2025.113739","DOIUrl":"10.1016/j.jcp.2025.113739","url":null,"abstract":"<div><div>Sparse grid reconstructions have recently been applied to Particle-In-Cell (PIC) methods with a semi-implicit formulation, as demonstrated in <span><span>[28]</span></span>, to reduce computational costs. By linearizing the particle equations and using a finite difference discretization of the field's equations, along with incorporating sparse grid reconstructions through the combination technique, an exactly energy-conserving scheme was proposed. However, this scheme exhibited numerical instability due to the loss of non-negativity in electric energy, inherent to the combination technique. This paper introduces a novel PIC method with a semi-implicit formulation that embeds sparse grid techniques to exactly conserve discrete total energy, defined as the sum of non-negative kinetic and field energies, ensuring nonlinear stability. The method utilizes a Galerkin approach for the field equations, employing a hierarchical sparse grid representation in the approximation space. This distinguishes it from previous sparse grid PIC methods, which typically use the combination technique and nodal representation. The enhancement of hierarchical subspaces, serving as the truncated combination technique counterpart of the newly introduced method, is proposed to address the limitations of sparse-PIC methods— notably the difficulty in capturing non-smooth and non-axis-aligned solutions. Key features of the method include: unconditional stability with respect to the plasma period; mitigation of grid heating, allowing flexible grid discretization irrespective of the Debye length; exact conservation of discrete total energy; significant reduction in statistical error compared to standard grid schemes for the same number of particles; and decreased computational complexity, particularly in the size of the linear system to be solved. We validate the method through a series of two-dimensional test cases, demonstrating its numerical stability and robust performance.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113739"},"PeriodicalIF":3.8,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131964","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 : 2025-01-13DOI: 10.1016/j.jcp.2025.113740
Charles Bienvenue , Ahmed Naceur , Alain Hébert , Jean-François Carrier
For many contemporary applications, ionizing radiation transport plays a pivotal role, requiring an accurate assessment of its impact on the exposed environment. While Monte Carlo simulations are widely considered the gold standard for accurate general-purpose coupled transport of photons, electrons and positrons in matter, discrete ordinates algorithms provide a viable alternative. This work consolidates cross-section models for coupled photon-electron-positron transport and provides the methodology to generate the data required by the multigroup Boltzmann Fokker-Planck transport equation and by energy and charge deposition formulas. It includes elastic, collisional and radiative inelastic interactions of leptons, annihilation of positrons, Compton scattering, Rayleigh scattering, photoelectric effect, pair production as well as fluorescence and Auger electron production from relaxation cascades following ionization. Comparative analyses of energy deposition in water, aluminum, and gold are conducted for incident beams of 1 MeV, 10 MeV, and 100 MeV electrons and photons, and juxtaposed against Monte Carlo reference calculations. While disparities of a few percent are typical, higher deviation can be observed due to discretization or physical model limitations. Energy spectrums per particle type at varying depths in the medium are also contrasted with Monte Carlo calculations to discern limitations in the current implementation and to propose potential avenues for enhancing the presented models. Energy and charge deposition calculations are also compared to experimental measurements. The cross-section production and transport algorithms are implemented in an open-source Julia package, Radiant.jl.
{"title":"Toward highly accurate multigroup coupled photon-electron-positron cross-sections for the Boltzmann Fokker-Planck equation","authors":"Charles Bienvenue , Ahmed Naceur , Alain Hébert , Jean-François Carrier","doi":"10.1016/j.jcp.2025.113740","DOIUrl":"10.1016/j.jcp.2025.113740","url":null,"abstract":"<div><div>For many contemporary applications, ionizing radiation transport plays a pivotal role, requiring an accurate assessment of its impact on the exposed environment. While Monte Carlo simulations are widely considered the gold standard for accurate general-purpose coupled transport of photons, electrons and positrons in matter, discrete ordinates algorithms provide a viable alternative. This work consolidates cross-section models for coupled photon-electron-positron transport and provides the methodology to generate the data required by the multigroup Boltzmann Fokker-Planck transport equation and by energy and charge deposition formulas. It includes elastic, collisional and radiative inelastic interactions of leptons, annihilation of positrons, Compton scattering, Rayleigh scattering, photoelectric effect, pair production as well as fluorescence and Auger electron production from relaxation cascades following ionization. Comparative analyses of energy deposition in water, aluminum, and gold are conducted for incident beams of 1 MeV, 10 MeV, and 100 MeV electrons and photons, and juxtaposed against Monte Carlo reference calculations. While disparities of a few percent are typical, higher deviation can be observed due to discretization or physical model limitations. Energy spectrums per particle type at varying depths in the medium are also contrasted with Monte Carlo calculations to discern limitations in the current implementation and to propose potential avenues for enhancing the presented models. Energy and charge deposition calculations are also compared to experimental measurements. The cross-section production and transport algorithms are implemented in an open-source Julia package, <em>Radiant.jl</em>.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"524 ","pages":"Article 113740"},"PeriodicalIF":3.8,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143131959","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 : 2025-01-10DOI: 10.1016/j.jcp.2025.113728
Maxim A. Olshanskii , Leo G. Rebholz
This paper extends a low-rank tensor decomposition (LRTD) reduced order model (ROM) methodology to simulate viscous flows and in particular to predict a smooth branch of solutions for the incompressible Navier-Stokes equations (by branch we refer to the continuation of the solution over a range of viscosities). Additionally, it enhances the LRTD-ROM methodology by introducing a non-interpolatory variant, which demonstrates improved accuracy compared to the interpolatory method utilized in previous LRTD-ROM studies. After presenting both the interpolatory and non-interpolatory LRTD-ROM, we demonstrate that with snapshots from a few different viscosities, the proposed method is able to accurately predict the statistics of a 2D flow passing a cylinder in the Reynolds number range . This is a significantly wider and higher range than state of the art (and similar size) ROMs built for use on varying Reynolds number have been successful on. The paper also discusses how LRTD may offer new insights into the properties of parametric solutions.
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