Journal of Computational Physics最新文献_第7页

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations 针对不可压缩纳维-斯托克斯方程的带能量消耗的动态正则化拉格朗日乘法器方案

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-30 DOI: 10.1016/j.jcp.2024.113550

Cao-Kha Doan , Thi-Thao-Phuong Hoang , Lili Ju , Rihui Lan

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

本文提出了基于拉格朗日乘法的纳维-斯托克斯方程高效数值计算方案。通过引入动态方程（涉及动能、拉格朗日乘数和正则化参数），我们形成了一个包含能量演化过程但仍等效于原始方程的新系统。然后，根据反向微分公式对这种非线性系统进行时间离散化，从而形成动态正则化拉格朗日乘法器（DRLM）方法。推导出了一阶和二阶 DRLM 方案，并证明这些方案相对于原始变量具有无条件的能量稳定性。所提出的方案在每个时间步仅需要两个线性斯托克斯系统和一个标量二次方程的解。此外，由于引入了正则化参数，拉格朗日乘数可以从二次方程中唯一确定，即使时间步长较大，也不会影响数值解的精度和稳定性。利用空间标记和单元（MAC）离散法也证明了完全离散的能量稳定性。二维和三维的各种数值实验验证了收敛性和能量耗散，并证明了所提出的 DRLM 方案的准确性和稳健性。

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引用次数: 0

IGA-Graph-Net: Isogeometric analysis-reuse method based on graph neural networks for topology-consistent models IGA-Graph-Net：基于拓扑一致性模型图神经网络的等时分析-重复使用方法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-30 DOI: 10.1016/j.jcp.2024.113544

Gang Xu , Jin Xie , Weizhen Zhong , Masahiro Toyoura , Ran Ling , Jinlan Xu , Renshu Gu , Charlie C.L. Wang , Timon Rabczuk

This paper introduces a novel isogeometric analysis-reuse framework called IGA-Graph-Net, which combines Graph Neural Networks with Isogeometric Analysis to overcome the limitations of Convolutional Neural Networks when dealing with B-spline data. Our network architecture incorporates ResNetV2 and PointTransformer for enhanced performance. We transformed the dataset creation process from using Convolutional Neural Networks to Graph Neural Networks. Additionally, we proposed a new loss function tailored for Dirichlet boundary conditions and enriched the input features. Several examples are presented to demonstrate the effectiveness of the proposed framework. In terms of accuracy when tested on the same set of partial differential equation data, our framework demonstrates significant improvements compared to the reuse method based on Convolutional Neural Networks for Isogeometric Analysis on topology-consistent geometries with complex boundaries.

本文介绍了一种名为 IGA-Graph-Net 的新颖等距分析复用框架，它将图神经网络与等距分析相结合，克服了卷积神经网络在处理 B-样条数据时的局限性。我们的网络架构结合了 ResNetV2 和 PointTransformer，以提高性能。我们将数据集创建过程从使用卷积神经网络转变为图形神经网络。此外，我们还针对 Dirichlet 边界条件提出了新的损失函数，并丰富了输入特征。我们列举了几个例子来证明所提框架的有效性。在对同一组偏微分方程数据进行测试时，与基于卷积神经网络的等几何分析重用方法相比，我们的框架在具有复杂边界的拓扑一致性几何图形上的准确性有了显著提高。

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引用次数: 0

Particulate transport in porous media at pore-scale. Part 1: Unresolved-resolved four-way coupling CFD-DEM 多孔介质中孔隙尺度的颗粒传输。第 1 部分：非解析-解析四向耦合 CFD-DEM

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 10.1016/j.jcp.2024.113540

Laurez Maya Fogouang , Laurent André , Cyprien Soulaine

Computational Fluid Dynamics - Discrete Element Method (CFD-DEM) is a powerful approach to simulate particulate flow in porous media at the pore-scale, and hence decipher the complex interplay between particle transport and retention. Two separate CFD-DEM approaches are commonly used in the literature: the unresolved (particle smaller than the grid cell size) and the resolved (particle bigger than the grid cell size) approach. In this paper, we propose a novel CFD-DEM coupling approach that combines both unresolved and resolved coupling. Our new modeling technique allows for the simulation of particulate flows in complex pore morphology characteristic of porous materials. It relies on an efficient searching strategy to find grid cells covered by the particles and on an appropriate calculation of the fluid-solid momentum exchange term. The robustness and efficiency of the computational model are demonstrated using cases for which reference solutions – analytical or experimental – exist. The new unresolved-resolved four-way coupling CFD-DEM is used to investigate pore-clogging and permeability reduction due to the sieving and bridging of particles.

计算流体动力学--离散元法（CFD-DEM）是一种强大的方法，可在孔隙尺度上模拟颗粒在多孔介质中的流动，从而破解颗粒传输和滞留之间复杂的相互作用。文献中通常使用两种不同的 CFD-DEM 方法：非解析法（颗粒小于网格单元尺寸）和解析法（颗粒大于网格单元尺寸）。在本文中，我们提出了一种新颖的 CFD-DEM 耦合方法，它结合了非解析耦合和解析耦合两种方法。我们的新建模技术可以模拟多孔材料特有的复杂孔隙形态中的颗粒流动。它依靠高效的搜索策略来寻找颗粒覆盖的网格单元，并对流固动量交换项进行适当计算。计算模型的稳健性和高效性通过已有参考解（分析或实验解）的案例得以证明。新的非解析-解析四向耦合 CFD-DEM 被用于研究颗粒的筛分和架桥导致的孔隙堵塞和渗透率降低。

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引用次数: 0

Quantifying the checkerboard problem to reduce numerical dissipation 量化棋盘问题以减少数值耗散

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 10.1016/j.jcp.2024.113537

J.A. Hopman , D. Santos , À. Alsalti-Baldellou , J. Rigola , F.X. Trias

This work provides a comprehensive exploration of various methods in solving incompressible flows using a projection method, and their relation to the occurrence and management of checkerboard oscillations. It employs an algebraic symmetry-preserving framework, clarifying the derivation and implementation of discrete operators while also addressing the associated numerical errors. The lack of a proper definition for the checkerboard problem is addressed by proposing a physics-based coefficient. This coefficient, rooted in the disparity between the compact- and wide-stencil Laplacian operators, is able to quantify oscillatory solution fields with a physics-based, global, normalised, non-dimensional value. The influence of mesh and time-step refinement on the occurrence of checkerboarding is highlighted. Therefore, single measurements using this coefficient should be considered with caution, as the value presents little use without any context and can either suggest mesh refinement or use of a different solver. In addition, an example is given on how to employ this coefficient, by establishing a negative feedback between the level of checkerboarding and the inclusion of a pressure predictor, to dynamically balance the checkerboarding and numerical dissipation. This method is tested for laminar and turbulent flows, demonstrating its capabilities in obtaining this dynamical balance, without requiring user input. The method is able to achieve low numerical dissipation in absence of oscillations or diminish oscillation on skew meshes, while it shows minimal loss in accuracy for a turbulent test case. Despite its advantages, the method exhibits a slight decrease in the second-order relation between time-step size and pressure error, suggesting that other feedback mechanisms could be of interest.

这项研究全面探讨了使用投影法求解不可压缩流的各种方法，以及这些方法与棋盘式振荡的发生和处理之间的关系。它采用了代数对称保护框架，明确了离散算子的推导和实现，同时也解决了相关的数值误差问题。针对棋盘问题缺乏适当定义的问题，提出了一个基于物理学的系数。该系数根植于紧凑模板和宽模板拉普拉斯算子之间的差异，能够用一个基于物理的、全局的、归一化的非维度值来量化振荡解场。网格和时间步细化对发生棋盘格现象的影响得到了强调。因此，使用该系数进行的单次测量应慎重考虑，因为在没有任何背景的情况下，该值几乎没有任何用处，可能会建议细化网格或使用不同的求解器。此外，还举例说明了如何使用该系数，即在棋盘格水平和压力预测器之间建立负反馈，以动态平衡棋盘格和数值耗散。该方法针对层流和湍流进行了测试，证明了其无需用户输入即可获得动态平衡的能力。该方法能够在没有振荡的情况下实现低数值耗散，或在倾斜网格上减小振荡，同时在湍流测试案例中显示出最小的精度损失。尽管该方法具有诸多优点，但其时间步长与压力误差之间的二阶关系略有减弱，这表明其他反馈机制可能会引起人们的兴趣。

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引用次数: 0

A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals 拓扑光子晶体共享带隙最大化的半有限优化方法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 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.

拓扑光子晶体（PC）可以支持稳健的边缘模式，从而以高效的方式传输电磁能量。这种边缘模式是将具有不同拓扑不变性的两个光子晶体连接在一起而形成的联合光学结构的 PDE 算子的特征模式，相应的特征频率位于两个单独光子晶体的共享带隙中。这项研究关注的是如何最大化两个具有不同拓扑特征的光子晶体的共享带隙，以增加边缘模式的带宽。我们为底层优化设计问题开发了一个半有限优化框架，它能在每个时间步长上高效更新介电函数，同时尊重对称性约束，并在必要时尊重拓扑不变性约束。在每次迭代时，我们都会对带隙函数和拓扑不变性约束函数进行敏感性分析，使优化问题线性化，并高效地解决凸半有限编程（SDP）问题。数值实例表明，所提出的算法在生成具有稳健边缘模式的优化光学结构方面具有优势。

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引用次数: 0

Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration 非结构网格上的局部子单元整体 DG/FV 凸特性保持方案和熵考虑

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 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?

本文旨在介绍一种新的局部子单元整体连续-格勒金/有限体积（DG/FV）凸特性保持方案，用于求解二维非结构网格上的守恒定律系统。众所周知，DG 方法需要某种非线性限制，以避免可能导致代码崩溃的虚假振荡或非线性不稳定性。本研究的主要思路是提高 DG 方案的鲁棒性，同时尽可能保持其高精度和非常精确的子单元分辨率。为此，将在需要的子单元尺度局部执行高阶 DG 和一阶 FV 方案的凸混合。为此，通过文献[58]、[59]中的理论，我们首先回顾一下，可以将 DG 方案重写为子单元 FV 方法，定义在子网格上，并提供一些特定的数值通量，称为 DG 重构通量。然后，子单元整体 DG/FV 方法将定义如下：我们将为每个子单元的每个面分配两个通量，一个一阶 FV 通量和一个高阶重构通量，最后以凸的方式混合。我们的目标是通过分析确定最佳混合系数，以实现所需的特性。我们将展示各种类型问题的数值结果，以评估该设计方法的良好性能。通过这种子单元整体框架，我们将尝试解决以下问题：通过这种整体框架是否有可能确保任何熵的稳定性？我们所说的熵稳定性指的是什么？这种限制的代价是什么？在追求高阶精度的同时，是否绝对需要这样做？

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引用次数: 0

Benchmark verification of PIC-DSMC programs PIC-DSMC 程序的基准验证

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 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.

我们研究了粒子中单元和直接模拟蒙特卡罗代码的一些常见验证和基准问题。由于这些问题的结果（包括收敛率、与解析解的比较以及代码间的比较）经常被用作仿真代码正确性的证据，因此有必要了解使用一个或多个这些问题的成功验证对仿真代码的正确性意味着什么。为此，我们在桑迪亚国家实验室开发的 PIC-DSMC 代码 Aleph 中执行了一系列基准问题，包括典型数值参数和其他验证应该失败的问题。结果表明，对当前基准问题的改进和扩展以及额外的问题规范将有利于现有和未来的代码，从而为预测结果提供更大的信心。

引用次数: 0

Multiscale preconditioning of Stokes flow in complex porous geometries 复杂多孔几何中斯托克斯流的多尺度预处理

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 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.

流体在多孔介质中的流动是许多地下（如二氧化碳封存）和工业（如燃料电池）应用的核心。要优化设计和操作规程，并量化相关的不确定性，需要对多孔样品的微尺度空隙空间进行流体动力学模拟。这通常会产生需要迭代求解器的大型、条件不充分的线性（化）系统，而预处理是确保快速收敛的关键。我们提出了稳健高效的预处理方法，用于加速求解由几何复杂多孔微结构上斯托克斯方程离散化产生的鞍点系统。它们基于最近提出的孔隙级多尺度方法（PLMM）和更成熟的称为孔隙网络模型（PNM）的降阶方法。介绍的四种预处理器分别是单片 PLMM、单片 PNM、块 PLMM 和块 PNM。与通过代数多网格法加速的现有块预处理相比，我们的预处理更加稳健高效。单片 PLMM 是对原始 PLMM 的代数重构，这使其具有可移植性，并可在现有软件中以非侵入方式实现。同样，单片式 PNM 是 PNM 的代数化，可用作直接数值模拟 (DNS) 的加速器。这使 PNM 具备了前所未有的估计和控制预测误差的能力。单片 PLMM/PNM 还可用作近似求解器，产生全局通量保守解，适用于许多实际环境。我们在 2D/3D 几何图形上对所有预处理器进行了系统测试，结果表明单片 PLMM 优于所有其他预处理器。所有预处理器都可以在并行机器上构建和应用。

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引用次数: 0

A finite element contour integral method for computing the scattering resonances of fluid-solid interaction problem 计算流固耦合问题散射共振的有限元轮廓积分法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-29 DOI: 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.

本文探讨了流固相互作用问题的散射共振计算。散射共振是非紧凑域问题离散谱数据的替代品，在科学和工程的许多领域都非常重要。对于特殊圆盘情况，我们得到了可用作参考解的解析解。对于一般情况，我们使用 Dirichlet 到 Neumann（DtN）映射截断无界域。使用标准线性拉格朗日元素进行离散化，从而产生非线性代数特征值问题。然后，我们用并行谱指标法求解非线性代数特征值问题。最后，我们给出了数值示例。

引用次数: 0

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes II: Unsteady flows 三维非结构网格上可压缩流动的隐式高阶气体动力学方案 II：非稳态流动

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-28 DOI: 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.

对于非稳态流动的模拟，由于局部单元大小变化较大，全局时间步长会变得非常小。本文开发了一种隐式高阶气体动力学方案（HGKS），以减轻非稳态模拟对时间步长的限制。为了提高效率并保持高阶精度，本文提出了一种两阶段三阶隐式时间精确离散法。在每个阶段，通过伪时间迭代获得隐式系统的人工稳定解。在迭代过程中，采用了经典的隐式方法来求解非线性系统，包括下-上对称高斯-赛德尔（LUSGS）和广义最小残差（GMRES）方法。为了实现空间精度，我们构建了非紧凑型和紧凑型重建的 HGKS。在非紧凑型方案中，使用了加权基本非振荡（WENO）重建法。对于紧凑型方案，由于需要更新单元平均流量变量及其导数，因此采用了赫米特 WENO（HWENO）重建。两阶段时间离散化达到了预期的三阶时间精度。对于平滑流动，只需要一次人工迭代。对于均匀网格，与显式方法相比，当前隐式方法的效率显著提高。对于不连续流动，与著名的 Crank-Nicholson 方法相比，当前方案中的假振荡得到了很好的抑制。人工迭代步数的增加会带来额外的重构，从而降低计算效率。总体而言，在网格尺寸变化较大的情况下，当前的隐式方法比显式方法提高了效率。同时，对于均匀网格上的强不连续性情况，当前方法的效率与显式方案相当。

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引用次数: 0