Journal of Computational Physics最新文献

{"title":"Annealed adaptive importance sampling method in PINNs for solving high dimensional partial differential equations","authors":"Zhengqi Zhang ,&nbsp;Jing Li ,&nbsp;Bin Liu","doi":"10.1016/j.jcp.2024.113561","DOIUrl":"10.1016/j.jcp.2024.113561","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in accurately resolving PDEs, especially when dealing with singular cases that may lead to unsatisfactory local minima. To address these challenges and improve solution accuracy, we propose an innovative approach called Annealed Adaptive Importance Sampling (AAIS) for computing the discretized PDE residuals of the cost functions, inspired by the Expectation Maximization (EM) algorithm used in finite mixtures to mimic target density. Our objective is to approximate discretized PDE residuals by strategically sampling additional points in regions with elevated residuals, thus enhancing the effectiveness and accuracy of PINNs. Implemented together with a straightforward resampling strategy within PINNs, our AAIS algorithm demonstrates significant improvements in efficiency across a range of tested PDEs, even with limited training datasets. Moreover, our proposed AAIS-PINNs method shows promising capabilities in solving high-dimensional singular PDEs. The adaptive sampling framework introduced here can be integrated into various PINN frameworks.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113561"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method","authors":"Ricardo H. Deucher ,&nbsp;Jacques Franc ,&nbsp;Olav Møyner ,&nbsp;Hamdi A. Tchelepi","doi":"10.1016/j.jcp.2024.113558","DOIUrl":"10.1016/j.jcp.2024.113558","url":null,"abstract":"<div><div>The adaptive implicit method (AIM) is the mainstream approach for compositional reservoir simulation. In its standard form, AIM uses single-point upwinding to reconstruct the fluxes across the interfaces of the control volume, and this leads to substantial numerical diffusion and loss of accuracy. Previous efforts to improve the accuracy of AIM focused on using high-order schemes to reconstruct the interfacial fluxes; those schemes introduced additional numerical nonlinearities to the system of equations or compromised the accuracy of the Jacobian by neglecting the high-order terms in its construction. In this work, we describe a high-resolution compact-stencil (HRCS) AIM. The new scheme is applied to compositional reservoir simulation. In addition to the mixed implicit/explicit time discretization of standard AIM, the HRCS AIM scheme uses a mixed time and space discretization. Specifically, we blend low- and high-order fluxes according to a well defined rule that uses a high-order reconstruction in the explicit regions of the domain and a low-order reconstruction in the implicit regions. This strategy ensures that additional nonlinearities introduced by the high-order reconstruction do not impact the Jacobian matrix, thus preserving the algebraic structure of standard AIM. The HRCS AIM method is demonstrated using several compositional problems. The results indicate substantial gains in accuracy with a small additional computational cost compared with standard AIM. Additionally, HRCS AIM is more robust and has a smaller computational cost compared with its full high-resolution counterpart.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113558"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs","authors":"Zengyan Zhang,&nbsp;Jia Zhao","doi":"10.1016/j.jcp.2024.113562","DOIUrl":"10.1016/j.jcp.2024.113562","url":null,"abstract":"<div><div>In this paper, we propose a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws, offering a unique approach from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent form, where the free energy of the transformed system takes a quadratic form in terms of the state variables. This transformation is known as energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced-order models that continue to respect the energy dissipation law. Secondly, our proposed numerical approach automatically provides algorithms to discretize these reduced-order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. Our platform offers a distinctive approach for deriving structure-preserving reduced-order models for a wide range of PDE systems with underlying thermodynamic principles.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113562"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A time-relaxation reduced order model for the turbulent channel flow","authors":"Ping-Hsuan Tsai ,&nbsp;Paul Fischer ,&nbsp;Traian Iliescu","doi":"10.1016/j.jcp.2024.113563","DOIUrl":"10.1016/j.jcp.2024.113563","url":null,"abstract":"<div><div>Regularized reduced order models (Reg-ROMs) are stabilization strategies that leverage spatial filtering to alleviate the spurious numerical oscillations generally displayed by the classical Galerkin ROM (G-ROM) in under-resolved numerical simulations of turbulent flows. In this paper, we propose a new Reg-ROM, the time-relaxation ROM (TR-ROM), which filters the marginally resolved scales. We compare the new TR-ROM with the two other Reg-ROMs in current use, i.e., the Leray ROM (L-ROM) and the evolve-filter-relax ROM (EFR-ROM) and one eddy viscosity model, the mixing-length model, in the numerical simulation of the turbulent channel flow at <span><math><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>=</mo><mn>180</mn></math></span> and <span><math><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>=</mo><mn>395</mn></math></span> in both the reproduction and the predictive regimes. For each Reg-ROM, we investigate two different filters: (i) the differential filter (DF), and (ii) the higher-order algebraic filter (HOAF). In our numerical investigation, we monitor the Reg-ROM performance with respect to the ROM dimension, <em>N</em>, and the filter order. We also perform sensitivity studies of the three Reg-ROMs with respect to the time interval, relaxation parameter, and filter radius. The numerical results yield the following conclusions: (i) All three Reg-ROMs are significantly more accurate than the G-ROM. (ii) All three Reg-ROMs are more accurate than the ROM projection in terms of Reynolds stresses. (iii) With the optimal parameter values, the new TR-ROM yields more accurate results than the L-ROM and the EFR-ROM in all tests. (iv) The new TR-ROM is more accurate than the mixing-length ROM. (v) For most <em>N</em> values, DF yields the most accurate results for all three Reg-ROMs. (vi) The optimal parameters trained in the reproduction regime are also optimal for the predictive regime for most <em>N</em> values, demonstrating the Reg-ROM predictive capabilities. (vii) All three Reg-ROMs are sensitive to the filter order and the filter radius, and the EFR-ROM and the TR-ROM are sensitive to the relaxation parameter. (viii) The optimal range for the filter radius and the effect of relaxation parameter are similar for the two <span><math><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> values.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113563"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids","authors":"Alexander Cicchino ,&nbsp;Siva Nadarajah","doi":"10.1016/j.jcp.2024.113532","DOIUrl":"10.1016/j.jcp.2024.113532","url":null,"abstract":"<div><div>To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113532"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A dynamic linearized wall model for turbulent flow simulation: Towards grid convergence in wall-modeled simulations","authors":"Marc Terracol,&nbsp;Lucas Manueco","doi":"10.1016/j.jcp.2024.113555","DOIUrl":"10.1016/j.jcp.2024.113555","url":null,"abstract":"<div><div>This paper addresses the common issue of (no-)grid convergence in wall-modeled numerical simulations and proposes a dynamic linearization technique applied to the Spalart-Allmaras wall model to achieve a proper behavior on fine grids and low-friction areas. A theoretical analysis of the numerical error committed on the shear stress balance close to the walls is performed. It shows that the error is due to the inappropriate imposition of too steep wall-normal velocity gradients that cannot be properly accounted for on the typical grids used for wall-modeled simulations. Based on this error quantification, a dedicated wall model linearization technique is proposed, following the approach developed by Tamaki, Harada and Imamura in 2017. In the proposed modified linearization method, the linearization distance is modified and adjusted dynamically. This is done according to the theoretical shear stress error estimate, in order to keep the numerical error below a user-defined threshold. The method is applied to well-referenced test cases of increasing complexity from the Turbulence Modeling Resource. Overall, the proposed wall model clearly exhibits appropriate grid convergence properties and is also able to predict accurately non-equilibrium boundary layers and flow separation using proper grid refinement.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113555"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Small-sample Bayesian error estimation for ergodic, chaotic systems of ordinary differential equations","authors":"Cory Frontin,&nbsp;David L. Darmofal","doi":"10.1016/j.jcp.2024.113559","DOIUrl":"10.1016/j.jcp.2024.113559","url":null,"abstract":"<div><div>The discretization of chaotic systems introduces statistical errors in addition to discretization errors into the estimation of quantities of interest. In order to efficiently arrive at estimates of quantities of interest, these two forms of error should be balanced; however, simulations are run without knowledge of the true/asymptotic outputs of interest or their error behaviors. In this work, we develop a framework for error modeling and identification using small-sample Bayesian inference that allows approximation of the optimal balance between sampling time and discretization precision without the computation of high-cost libraries of reference solutions. The result enables the possibility of running chaotic and turbulent simulations in a way that minimizes the total error between sampling and discretization without prior knowledge of the error behavior of the system.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113559"},"PeriodicalIF":3.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A unified HTC multiphase model of continuum mechanics","authors":"Davide Ferrari ,&nbsp;Ilya Peshkov ,&nbsp;Evgeniy Romenski ,&nbsp;Michael Dumbser","doi":"10.1016/j.jcp.2024.113553","DOIUrl":"10.1016/j.jcp.2024.113553","url":null,"abstract":"<div><div>In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Hyperbolic Thermodynamically Compatible (HTC) equations, can describe an arbitrary number of phases that can be heat conducting <em>inviscid</em> and <em>viscous fluids</em>, as well as <em>elastoplastic solids</em>. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the novel HTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the HTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging we restrict our numerical examples to a simplified version of the model for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications to compressible multiphase problems. Notably, results are presented for multiphase flows in several relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids. In all cases, the numerical results demonstrate good agreement with established models, including the Euler or Navier-Stokes equations for fluids and the classical hypo-elastic model with plasticity for solids. Importantly, however, this approach achieves these results within a unified multiphase framework of continuum mechanics.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113553"},"PeriodicalIF":3.8,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660004","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}

{"title":"Two-phase regularized phase-field density gradient Navier–Stokes based flow model: Tuning for microfluidic and digital core applications","authors":"Vladislav Balashov ,&nbsp;Evgeny Savenkov ,&nbsp;Aleksey Khlyupin ,&nbsp;Kirill M. Gerke","doi":"10.1016/j.jcp.2024.113554","DOIUrl":"10.1016/j.jcp.2024.113554","url":null,"abstract":"<div><div>Here we present a regularized phase-field Navier–Stokes two-phase flow model with density gradient theory for interface treatment. The usage of regularization allows us for faster computations — for some particular simulation cases the time step could be increased x9 times. The computations are stable and spurious layers are suppressed with the help of nonlinear surface energy model. To verify the model we test rigorously against major classic problems: (1) droplet on a flat wall with given contact angle, (2) analytical solution for capillary rise, (3) hydrodynamical focusing within a micro-channel, (4) displacement in a pore doublet. As the application example, we simulate two-phase flow displacement within an X-ray microtomography scan of oil-bearing rock and demonstrate computation of relative permeabilities. We believe that developed model will be useful in numerous research applications: porous media design with desired physical properties, digital core technology applications to obtain flow properties of rocks and enhance hydrocarbon recovery, hydrological applications to study unsaturated flow in soils. Finally, we discuss potential improvements of the model as related to computational efficiency.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113554"},"PeriodicalIF":3.8,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamic adaptive and fully unstructured tetrahedral gridding: Application to CO2 sequestration with consideration of full fluid compressibility","authors":"Jakub Solovský ,&nbsp;Abbas Firoozabadi","doi":"10.1016/j.jcp.2024.113556","DOIUrl":"10.1016/j.jcp.2024.113556","url":null,"abstract":"<div><div>Numerical simulations of complex subsurface flow problems and geomechanics will advance enormously by dynamic adaptive gridding. Only a small part of a large domain where sharp changes occur may require fine gridding. In this work, we introduce a methodology to carry out dynamic adaptive gridding for large-scale flow problems. The algorithm allows 2D and 3D unstructured gridding with consideration of full fluid compressibility in single-phase, and two-phase compositional flow. We divide a triangular element into four and a tetrahedron element into eight which creates hanging nodes at each level of refinement. The handing nodes are eliminated by splitting extra elements. As a result, a transition region exists between the fine and coarse grid regions of the domain. We have applied the method to CO₂ sequestration in subsurface aquifers. The conditions are selected such that gravity fingers develop from density increase by the dissolution of CO₂ in the aqueous phase. The selected examples include large domains where neither systematic studies nor dynamic adaptive gridding have been reported in the past. Results from comparison with uniform gridding reveal a speedup of up to three orders of magnitude in 3D.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"521 ","pages":"Article 113556"},"PeriodicalIF":3.8,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653185","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}