Communications in Computational Physics (CiCP) is an international, peer reviewed, interdisciplinary journal edited by a group of world-wide leading scientists in diverse fields to guarantee its highest standard. It publishes original research and survey papers of high scientific value in computational modeling of physical problems. Results in multi-physics and multi-scale innovative computational methods and modeling in all physical sciences will be featured. The journal is published by Global Science Press in printed and electronic forms.
The First- and Second-Order Energy Stable, Mass Conservative and Bounds-Preserving Schemes for Two-Phase Incompressible Flow with Rock Compressibility
In this paper, we construct a unified framework of first- and second-order schemes in time for thermodynamically consistent modeling of two-phase incompressible and immiscible flow in porous media with rock compressibility. We rigorously prove that the proposed schemes can preserve the energy dissipation law, conserve the mass of each phase as well as pore volumes and are bounds-preserving for both phases without any restrictions on the time step size. Moreover, the proposed schemes can achieve local mass conservation for both phases and preserve the pore volumes when the saturation of the whole region is within the bounds, and we only need to solve one linear system and several linear algebraic equations, whereas for points where saturation is outside the bounds, we just need to solve an additional nonlinear algebraic equation and hence the proposed schemes can achieve global mass conservation. The developed schemes exhibit high efficiency owing to a substantial reduction in the scale of nonlinear computations and their straightforward implementation. The key point is that we propose a new Lagrange multiplier method that is based on the judgment of saturation bound. This addresses the limitation of the classical Lagrange multiplier method, which is only capable of ensuring global mass conservation. Finally, a variety of illustrative numerical examples including several benchmark problems are provided to validate the accuracy and efficiency of the proposed schemes.
Copyright (c) 2026 Communications in Computational Physics
Local Discontinuous Galerkin Method for Nonlinear BSPDEs of Neumann Boundary Conditions with Deep Backward Dynamic Programming Time-Marching
This paper aims to present a local discontinuous Galerkin (LDG) method for solving nonlinear backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.
backward stochastic partial differential equations
Scale-Decomposed Physics-Informed Neural Networks for Singular Perturbation Problems
A novel physics-informed neural network (PINN), scale-decomposed PINN (SD-PINN), is proposed for singular perturbation problems, such as boundary-layer flows. Singular perturbation problems exhibit sharp changes in solutions at different scales, owing to large gradients in equations, which presents a challenge to a conventional PINN. In SD-PINN, the solutions at different scales are represented by different neural networks, and their matching is achieved by imposed matching conditions. In comparison with the conventional procedure for singular perturbation problems, no domain decomposition is used. A nonlinear stretching transformation is introduced to prevent the occurrence of semi-infinite regions in the neural networks. Six benchmark singular perturbation problems are used to evaluate SD-PINN, including linear and nonlinear second-order ordinary differential equations, two-dimensional hyperbolic problems, and a flat-plate boundary-layer flow. It is demonstrated that SD-PINN can reproduce asymptotic and non-asymptotic solutions for small and finite perturbation parameters, respectively. Application of SD-PINN to the inference of perturbation parameters from available data is also discussed. Codes are available at https://github.com/LeiZhang-code/SD-PINN.
Solving Multi-Group Neutron Diffusion Eigenvalue Problem with Decoupling Residual Loss Function
In the midst of the neural network’s success in solving partial differential equations, tackling eigenvalue problems using neural networks remains a challenging task. However, the Physics Constrained-General Inverse Power Method Neural Network (PC-GIPMNN) approach was proposed and successfully applied to solve the single-group critical problems in reactor physics. This paper aims to solve critical problems in multi-group scenarios and in more complex geometries. Hence, inspired by the merits of traditional source iterative method, which can overcome the ill-condition of the right side of the equations effectively and solve the multi-group problem effectively, we propose two residual loss function called Decoupling Residual loss function and Direct Iterative loss function. Our loss function can deal with multi-group eigenvalue problem, and also single-group eigenvalue problem. Using the new residual loss functions, our study solves one-dimensional, two-dimensional, and three-dimensional multi-group problems in nuclear reactor physics without prior data. In numerical experiments, our approach demonstrates superior generalization capabilities compared to previous work.
A Locally Mass-Conservative Enriched Petrov-Galerkin Method Without Penalty for the Darcy Flow in Porous Media
In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed method.
Maximum-Principle-Preserving and Positivity-Preserving Central WENO Schemes on Overlapping Meshes
In this paper, we propose a class of maximum-principle-preserving central WENO schemes for scalar conservation laws, and positivity-preserving central WENO schemes for compressible Euler equations. Formulated in a finite volume framework on overlapping meshes, the central schemes require neither flux splitting nor numerical fluxes that are often exact or approximate Riemann solvers. A new fifth-order WENO reconstruction is applied for the spatial discretization, and the linear weights of such reconstruction can be any positive number as long as their sum equals one, which leads to much simpler implementation. The sufficient conditions are provided for the cell average values to preserve maximum principle or positivity property with Euler forward time discretization. The method can be generalized to high-order strong stability preserving Runge-Kutta method without technical difficulties. Extensive numerical examples are presented to illustrate the accuracy and performance of the proposed methods.
Long-Time Integration of Nonlinear Wave Equations with Neural Operators
Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.
Steady-State Simulation of Euler Equations by the Discontinuous Galerkin Method with the Jump Filter
In this study, we introduce a jump filter for discontinuous Galerkin (DG) methods on both rectangular and triangular meshes, specifically designed for steady-state solutions of the Euler equations. Traditional limiters have occasionally been observed to produce slight post-shock oscillations, but the integration of the jump filter effectively reduces these fluctuations. This improvement allows the residual to stabilize at a level indistinguishable from machine zero in steady-state simulations. The jump filter operates by applying distinct damping factors to the different moments of the solution, based on the jump information at cell interfaces. This strategy creates transition zones from smooth regions to shock waves, which is critical for achieving steady-state solutions with residuals approaching the threshold of machine zero. The jump filter on unstructured meshes using Dubiner polynomials associated with the Sturm-Liouville operator is also investigated. Furthermore, the DG method with the jump filter retains many of the key advantages of the classical DG approach, including compactness, conservation, stability, and high-order accuracy. Extensive numerical experiments, conducted on both rectangular and triangular meshes, confirm the effectiveness of the jump filter in not only suppressing numerical oscillations but also significantly driving the residual towards machine zero.
High-Order Structure-Preserving Spectral Difference Methods for Hyperbolic Conservation Laws
This paper presents a study focused on solving hyperbolic conservation laws using arbitrary-order spectral difference (SD) methods. The study is structured around several crucial aspects. Firstly, in order to ensure the maximum principle for scalar equations and positivity preservation for Euler systems, we adopt the concept of flux limiters. This adoption leads to the development of the structure-preserving SD scheme with a flux limiter (SDFL), which is proven to preserve the original high-order accuracy. However, the SDFL scheme with lower order might lack conservational properties, despite its strong performance in short-term simulations. Consequently, we have developed a specific variant of the SDFL scheme with conservational properties, referred to as the CSDFL scheme. Secondly, we introduce a modified WENO-ZQ (MWENO-ZQ) reconstruction to suppress spurious oscillations when simulating problems with strong discontinuities. Finally, we conduct extensive numerical experiments to validate the effectiveness of the proposed high-order SD (SDFL, CSDFL) methods with MWENO-ZQ reconstruction. The results demonstrate the robustness and efficiency of these techniques in solving problems involving strong discontinuities, low pressure, and low density.
Do We Need Decay-Preserving Error Estimate for Solving Parabolic Equations with Initial Singularity?
Solutions exhibiting weak initial singularities arise in various equations, including diffusion and subdiffusion equations. When employing the well-known L1 scheme to solve subdiffusion equations with weak singularities, numerical simulations reveal that this scheme exhibits varying convergence rates for different choices of model parameters (i.e., domain size, final time $T$, and reaction coefficient $\kappa$). This elusive phenomenon is not unique to the L1 scheme but is also observed in other numerical methods for reaction-diffusion equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme, and two-step backward differentiation formula (BDF2) scheme. The existing literature lacks an explanation for the existence of two different convergence regimes, which has puzzled us for a long while and motivated us to study this inconsistency between the standard convergence theory and numerical experiences. In this paper, we provide a general methodology to systematically obtain error estimates that incorporate the exponential decaying feature of the solution. We term this novel error estimate the ‘decay-preserving error estimate’ and apply it to the aforementioned IE, C-N, and BDF2 schemes. Our decay-preserving error estimate consists of a low-order term with an exponential coefficient and a high-order term with an algebraic coefficient, both of which depend on the model parameters. Our estimates reveal that the varying convergence rates are caused by a trade-off between these two components in different model parameter regimes. By considering the model parameters, we capture different states of the convergence rate that traditional error estimates fail to explain. This approach retains more properties of the continuous solution. We validate our analysis with numerical results.
Least-Squares Recovery of Dual Variables in ROMs for Saddle-Point Problems
In this work, we introduce a methodology for recovering the reduced Lagrange multiplier in Reduced Order Models (ROMs) of general saddle-point problems. Specifically, the multiplier is determined as the least-squares minimizer of the residual of the reduced primal variable in a dual norm. We prove that this procedure yields a unique solution under the condition that the full-order primal and dual spaces satisfy the inf-sup stability condition. This is an extension to general saddle-point problems of the method recently introduced in [6] to recover the reduced pressure in ROMs of incompressible flows. We further show that the proposed approach is equivalent to solving the reduced mixed problem with an enriched reduced primal basis, augmented by the supremizers of the reduced multiplier. We establish optimal error estimates for the reduced multiplier applicable to general saddle-point problems. Numerical experiments for heat diffusion in composite media validate our theoretical findings and demonstrate the suitability of the method to solve engineering problems.
Quantum Simulation of the Fokker-Planck Equation via Schrödingerization
This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method – it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.
Reconstruction of Dynamical Systems Without Time Label
In this paper, we study the method to reconstruct dynamical systems from data without time labels. Data without time labels appear in many applications, such as molecular dynamics, single-cell RNA sequencing, etc. Reconstruction of dynamical system from time sequence data has been studied extensively. However, these methods do not apply if time labels are unknown. Without time labels, sequence data become distribution data. Based on this observation, we propose to treat the data as samples from a probability distribution and try to reconstruct the underlying dynamical system by minimizing the distribution loss, sliced Wasserstein distance more specifically. Extensive experiment results demonstrate the effectiveness of the proposed method.
Derivation of Asymptotic Limiting Equations for Multigroup Anisotropic-Scattering Transport Processes in Optically Thick Regimes
Asymptotic limit methods play a crucial role in deriving macroscopic fluid equations from equations governing mesoscopic particle transport mechanics, constituting a significant research area within kinetic theory. In this paper, we present asymptotic expansions applied to the multigroup linear transport equation with a general anisotropic scattering kernel. We derive the diffusion limit for such transport equations under three different scalings. For the conventional scaling, the standard multigroup diffusion equations are derived as an asymptotic approximation to the multigroup transport equations. For the other two scalings where the scattering effect between groups is strong, new group-collapsed (monoenergetic) diffusion approximations with different coefficients depending on the form of the scaling are derived. Numerical experiments validate the prediction of the theoretical models. This paper provides a deeper understanding of the diffusion physics for the multigroup anisotropic particle transport in the optically thick regime.
Towards the Efficient Calculation of Quantity of Interest from Steady Euler Equations II: A CNNs-Based Automatic Implementation
In [J. Wang, G. Hu, arxiv: 2302.14262], a dual-consistent dual-weighted residual-based $h$-adaptive method has been proposed based on a Newton-GMG framework, toward the accurate calculation of a given quantity of interest from Euler equations. The performance of such a numerical method is satisfactory, i.e., the stable convergence of the quantity of interest can be observed. In this paper, we will focus on the efficiency issue to further develop this method. Three approaches are studied for addressing the efficiency issue, i.e., i). using convolutional neural networks as a solver for dual equations, ii). designing an automatic adjustment strategy for the tolerance in the $h$-adaptive process to conduct the local refinement and/or coarsening of mesh grids, and iii). introducing OpenMP, a shared memory parallelization technique, to accelerate the module such as the solution reconstruction in the method. The feasibility of each approach and numerical issues are discussed in depth, and significant acceleration from those approaches in simulations can be observed clearly from a number of numerical experiments. In convolutional neural networks, it is worth mentioning that the dual consistency plays an important role in guaranteeing the efficiency of the whole method and that unstructured meshes are employed in all simulations.
Neural-Network-Augmented Empirical Interpolation Method for Field Reconstruction with Noise and Vibration Tolerance
Field reconstruction methods based on machine learning often face challenges when handling unstructured data and designing complex model structures. On the other hand, Reduced Order Models (ROM) struggle with robustness in the presence of observation noise and sensor vibrations. Striking a balance between lightweight design, the ability to handle unstructured data, and robustness presents a significant challenge. In this paper, we introduce the EIM-NN algorithm, which leverages neural networks to determine the coefficients of the subspace found by the EIM algorithm. Furthermore, we present the EIM-TNN algorithm, which enhances robustness by designing a loss function incorporating Tikhonov regularization. Our neural network consists of only two fully connected layers, allowing it to handle unstructured data while maintaining a lightweight profile. Experimental results demonstrate the algorithm’s ability to significantly enhance robustness against noise and sensor vibrations without compromising its accuracy in fitting the original data. Additionally, the algorithm’s lightweight nature ensures that the added training time and memory requirements over EIM remain acceptable, making it adaptable to a range of industrial applications.
Thermodynamically Consistent Modeling and Energy Stable Numerical Simulation of Multicomponent Compressible Flow in Poroelastic Media
Modeling and numerical simulation of coupled poromechanical problems with multicomponent compressible flow are of particular importance in many fields including shale and natural gas engineering, carbon dioxide sequestration and geotechnical engineering. In this paper, using the second law of thermodynamics, we rigorously derive a thermodynamically consistent model for multicomponent compressible flow in deformable porous media coupled with poroelasticity. The model herein takes molar densities as the primary unknowns rather than pressure and molar fractions as well as introduces fluid and solid free energies, so that it naturally follows an energy dissipation law. Additionally, the Maxwell-Stefan model of multicomponent diffusion is generalized as a multicomponent fluid-solid coupling model accounting for the solid deformation, which not only satisfies Onsager’s reciprocal principle but also yields a thermodynamically consistent poro-visco-elastic equation. For numerical simulation, we propose a novel energy stable and mass conservative numerical scheme for the model. We first design the semi-discrete time scheme using the stabilized energy factorization approach to deal with the multicomponent Helmholtz free energy as well as subtle semi-implicit treatments for the coupling between multicomponent fluids and solids. A nontrivial treatment is the use of the discrete Gibbs-Duhem equation substituting for the pressure gradient contributed by the multicomponent fluids in the solid mechanical balance equation, which establishes the thermodynamically consistent relation between fluids and solids at the discrete level. Based on the cell-centered finite volume method on staggered grids, the fully discrete scheme is constructed using the upwind strategy for both molar densities and porosity. The scheme is proved to preserve the discrete energy dissipation law and Onsager’s reciprocal principle as well as to conserve the mass of fluid components and solids. Numerical experiments are performed to confirm our theories, especially to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theoretical analysis.
Enhanced Second-Order Gauss-Seidel Projection Methods for the Landau-Lifshitz Equation
The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.
A Highly Efficient CN-Like Scheme for the Transient Wigner Equation
The transient Wigner equation (TWE) plays an important role in modeling quantum effects of nano-scale semiconductor devices. The pseudo-differential term contained within the TWE provides quantum descriptions for the model, however, it is costly in numerical simulations. We develop a CN-like scheme for time integration of the TWE, based on the two-step method. Additionally, the spatial discretization is the hybrid finite-difference/sinc-Galerkin scheme [H. Jiang, T. Lu, W. Yao and W. Zhang, SIAM J. Sci. Comput. 45 (2023)]. Rigorous proofs are provided to show that the CN-like scheme is unconditionally $L^2$-stable and has second-order accuracy in time. More importantly, the computational efficiency of the new CN-like scheme could be considered to be higher than that of any explicit multi-stage one-step time integration scheme. Numerical experiments are also carried out to verify the accuracy, stability and efficiency of the new CN-like scheme. In addition to the verification experiments, resonant tunneling diodes (RTDs) with various sets of parameters are simulated. I-V characteristics and associated curves are obtained to demonstrate the resonant tunneling effect and the variations in I-V characteristic curves in relation to changes in structure parameters.
United Filter for Jointly Estimating State and Parameters of Stochastic Dynamical Systems
This paper tackles the intricate task of jointly estimating state and parameters in data assimilation for stochastic dynamical systems that are affected by noise and observed only partially. While the concept of “optimal filtering” serves as the customary approach to estimate the state of the target dynamical system, traditional methods such as Kalman filters and particle filters encounter significant challenges when dealing with high-dimensional and nonlinear problems. When we also consider the scenario where the model parameters are unknown, the problem transforms into a joint state-parameter estimation problem. Presently, the leading-edge technique known as the Augmented Ensemble Kalman Filter (AugEnKF) addresses this issue by treating unknown parameters as additional state variables and employing the Ensemble Kalman Filter to estimate the augmented state-parameter vector. Despite its considerable progress, AugEnKF does exhibit certain limitations in terms of accuracy and stability. To address these challenges, we introduce an innovative approach, referred to as the United Filter. This method combines a remarkably stable and efficient ensemble score filter (EnSF) for state estimation with a precise direct filter dedicated to online parameter estimation. Utilizing the EnSF’s generative capabilities grounded in diffusion models, the United Filter iteratively fine-tunes both state and parameter estimates within a single temporal data assimilation step. Thanks to the robustness of the EnSF, the proposed United Filter method offers a promising solution for enhancing our understanding and modeling of dynamical systems, as demonstrated by results from numerical experiments.
An Improved Ghost-Point Technique for the Electrostatic Equation with Curved Boundaries and Discontinuous Permittivity in ECR Ion Source PIC Simulations
In this paper, we propose an efficient Poisson solver for a two-dimensional axisymmetric domain to simulate plasma in a 2.45 GHz Electron Cyclotron Resonance (ECR) ion source. The solver employs a uniform Cartesian mesh for a curved domain, utilizing an unfitted boundary method. The presence of two Boron Nitride disks inside the plasma chamber creates a discontinuity in permittivity, leading to a Poisson problem with discontinuous coefficients. We discretize the Laplace operator by standard central difference formulas, and apply a ghost-point technique at nodes outside the domain and near the interfaces between the disks and the chamber, enforcing appropriate boundary and interface conditions for the ghost values. This method significantly simplifies and accelerates computations compared to boundary-conforming mesh approaches. The primary objective of this work is to upgrade our Poisson solver (employed in the Particle-In-Cell (PIC) code) from a fitted boundary method, such as COMSOL, to a custom implementation based on an unfitted boundary approach. The performance improvements are substantial, reducing computation time from a minimum of 5 seconds per solution with COMSOL to just 0.062 seconds with our custom solver.
numerical treatment of material discontinuities
Weak Minimization-Sampling Network for Solving PDEs
In many engineering applications, people concentrate on deriving the weak solution to a given partial differential equation (PDE) model. Traditionally, both the trial space and the test space are constructed as the span of a set of basis functions with compact supports and the approximate solution is achieved by solving a (linear) system of equations. A recent advancement in this field is the Weak Adversarial Network (WAN), a machine learning-based method that employs deep neural networks (DNNs) to approximate trial and test spaces. WAN addresses PDEs through a min-max optimization framework, showcasing its effectiveness in solving high dimensional PDEs and surpassing the limitations of traditional numerical methods. In this paper, motivated by WAN, we propose a Weak Minimization-Sampling Network (WMSN) for solving PDEs which also leverages DNNs to approximate the trial and test spaces. Instead of the min-max loss function, our key idea is to reformulate the weak formulation into a loss function defined as an expectation over all possible DNNs in the test space with only the network parameters in the trial space to be trained. This leads to a minimization problem in the trial space and a sampling problem over the test space, which can be solved by stochastic gradient descent type methods with neat implementation. Through a series of numerical tests, we find that WMSN exhibits comparable or slightly better results with smaller computational cost when compared to WAN.
Structure-Preserving SAV-RK$s$ Spectral Methods for Nonlinear Hamiltonian System
It is well-established that high-order symplectic Runge–Kutta methods cannot preserve polynomial energy of Hamiltonian systems beyond degree two. However, the scalar auxiliary variable (SAV) approach, which was originally proposed for gradient flow systems, offers an effective strategy to overcome this limitation, albeit with a modified energy. In this paper, we study high-order Runge–Kutta methods for solving general Hamiltonian systems without imposing a lower bound on the high-order part of the energy functional. By integrating a symplectic Runge–Kutta method with a new SAV strategy, we develop a class of high-order nonlinear structure-preserving $s$-stage Runge–Kutta SAV (SAV-RK$s$) spectral methods, and prove that the nonlinear semi-discrete scheme can preserve the Hamiltonian energy and other conservative quantities such as mass/momentum (if they exist). In addition, we design a tailored fast Newton iteration algorithm to efficiently solve the resulting nonlinear algebraic system. Finally, we carry out extensive numerical simulations on several benchmark problems, including cases where the energy functional is either bounded or unbounded, to validate the accuracy and robustness of the proposed algorithms.
Deep Learning-Based Reduced-Order Methods for Fast Transient Dynamics
In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are computationally expensive and, often, the time taken for one single computation is large and does not permit parametric studies. The aim of this work is therefore to facilitate real-time and multi-query calculations by employing a non-intrusive Reduced Order Method (ROM). We propose a deep learning-based (DL) ROM scheme able to deal with fast transient dynamics. In the case of blast waves, the parametrised PDEs are time-dependent and non-linear. For such problems, the Proper Orthogonal Decomposition (POD), which relies on a linear superposition of modes, cannot approximate the solutions efficiently. The piecewise POD-DL scheme developed here is a local ROM based on time-domain partitioning and a first dimensionality reduction obtained through the POD. Autoencoders are used as a second and non-linear dimensionality reduction. The latent space obtained is then reconstructed from the time and parameter space through deep forward neural networks. The proposed scheme is applied to an example consisting of a blast wave propagating in air and impacting on the outside of a building. The efficiency of the deep learning-based ROM in approximating the time-dependent pressure field is shown.
A New Approximate Method to Mean Field Stochastic Differential Equation with One-Sided Lipschitz Drift Coefficient
In this paper, we construct a numerical method to approximate a class of mean field stochastic differential equations whose drift coefficient satisfies a one-sided Lipschitz condition. The key idea is to independently approximate the distribution via solving a nonlinear Fokker-Planck equation which governs the density function of the solution. Meanwhile, based on the approximate density function, we construct an approximate stochastic differential equation to approach the mean field one. Then, we also apply a truncated Euler-Maruyama method to achieve discretization and derive error estimates. Finally, we present several numerical experiments to illustrate our theoretical analysis.
Mean field stochastic differential equation
Copyright (c) 2025 Communications in Computational Physics
Uncertainty Quantification of Phase Transition Problems with an Injection Boundary
We develop an enthalpy-based modeling and computational framework to quantify uncertainty in Stefan problems with an injection boundary. Inspired by airfoil icing studies, we consider a system featuring an injection boundary inducing domain changes and a free boundary separating phases, resulting in two types of moving boundaries. Our proposed enthalpy-based formulation seamlessly integrates thermal diffusion across the domain with energy fluxes at the boundaries, addressing a modified injection condition for boundary movement. Uncertainty then stems from random variations in the injection boundary. The primary focus of our Uncertainty Quantification (UQ) centers on investigating the effects of uncertainty on free boundary propagation. Through mapping to a reference domain, we derive an enthalpy-based numerical scheme tailored to the transformed coordinate system, facilitating a simple and efficient simulation. Numerical and UQ studies in one and two dimensions validate the proposed model and the extended enthalpy method. They offer intriguing insights into ice accretion and other multiphysics processes involving phase transitions.
Adaptive Neural Network Basis Methods for Partial Differential Equations with Low-Regular Solutions
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semi-linear partial differential equation with low-regular solutions in two/three dimensions. The method is obtained by combining basis functions from a class of shallow neural networks and the resulting multi-scale analogues, a residual strategy in adaptive methods and the non-overlapping domain decomposition method. Firstly, based on the solution residual, the domain $\Omega$ is iteratively decomposed and eventually partitioned into $K+1$ non-overlapping subdomains, denoted respectively as $\{\Omega_k\}_{k=0}^K$, where the exact solution is smooth on subdomain $\Omega_0$ and low-regular on subdomain $\Omega_k$ $(1\leq k\leq K)$. Secondly, the low-regular solutions on different subdomains $\Omega_k$ $(1\leq k\leq K)$ are approximated by neural networks with different scales, while the smooth solution on subdomain $\Omega_0$ is approximated by the initialized neural network. Thirdly, we determine the undetermined coefficients by solving the linear least squares problems directly or the nonlinear least squares problem via the Gauss-Newton method. The proposed method can be extended to multi-level case naturally. Finally, we use this adaptive method for several peak problems in two/three dimensions to show its high-efficient computational performance.
A Unifying Moving Mesh Method for Curves, Surfaces, and Domains Based on Mesh Equidistribution and Alignment
A unifying moving mesh method is developed for general $m$-dimensional geometric objects in $d$-dimensions ($d \geq 1$ and $1 \leq m \leq d$) including curves, surfaces, and domains. The method is based on mesh equidistribution and alignment and does not require the availability of an analytical parametric representation of the underlying geometric object. Mathematical characterizations of shape and size of $m$-simplexes and properties of corresponding edge matrices and affine mappings are derived. The equidistribution and alignment conditions are presented in a unifying form for $m$-simplicial meshes. The equation for mesh movement is defined based on the moving mesh PDE approach, and suitable projection of the nodal mesh velocities is employed to ensure the mesh points are not moved out of the underlying geometric object. The analytical expression for the mesh velocities is obtained in a compact matrix form. The nonsingularity of moving meshes is proved. Numerical results for curves $(m=1)$ and surfaces $(m=2)$ in two and three dimensions are presented to demonstrate the ability of the developed method to move mesh points without causing singularity and control their concentration.
A Multiscale Algorithm to Drastically Reduce Computational Times when Simulating Liquid/Solid Interaction at Atomic Resolution with Realistic Hydrodynamics Effects
A hybrid multiscale model that incorporates both continuum fluid dynamics effects including thermal fluctuations and molecular dynamics is presented for the challenging 3D liquid-solid interface problem of gap water flows between the moving tip of the Atomic Force Microscope (AFM) cantilever and a material substrate surface. Highlights of the method include all-atom resolution in the vicinity of material surfaces, thereby avoiding the empiricism of interface boundary treatments in continuum mechanics approaches, with considerable computational savings compared with direct single-scale approaches such as Non-Equilibrium Molecular Dynamics (NEMD) methods. All components of the multiscale method are systematically described and verified. To optimize the coupling of the continuum fluid dynamics solution with the molecular dynamics solution, a simple iterative method is used. Predictions of the multiscale method are compared with reference equilibrium molecular dynamics solutions and discussed in the context of available AFM measurements in the literature. Computational performance of the proposed multiscale method is analysed in comparison with that of the NEMD method for a relevant range of AFM problem parameters. In particular, compared with the single-resolution MD method, the implemented multiscale model leads to an eight-order-of-magnitude acceleration of the solution for a slowly moving AFM tip typical of existing experiments. Perspectives on the further development and application of the proposed multiscale method are discussed. The current model is implemented in the open-source GROMACS software, and the corresponding source code is also provided.
A Determinant Aided Fixed Point Method for Nonlinear Coupled Constitutive Relation of Rarefied Nonequilibrium Flows
The Nonlinear Coupled Constitutive Relation (NCCR) model is derived from the generalized hydrodynamic equations of Eu and has the capability to describe some significant characteristics of rarefied flows. However, the NCCR model is a complicated nonlinear system, and previous iterative methods for solving the NCCR equations have been observed to be associated with unphysical solutions and instability in some unfavourable conditions. In this study, a new numerical method for solving NCCR equations is proposed to enhance the reliability of the NCCR model. An objective function for a single variable is employed within a fixed point perspective to determine the solution, and the NCCR equation system is reorganized into a smaller linear matrix system for iterative processes. The determinant of the matrix system is used to search the valid solution region, ensuring the method’s robustness. Three typical flow problems in transition regimes are conducted to validate the numerical performance of the proposed method. Results show that the computational time of the proposed method is only approximately 2 to 6 times that of the NS solution, representing efficiency at the same magnitude order of NS solvers and enabling broader engineering applications.