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.
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
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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.
A Nonnested Augmented Subspace Method for Kohn-Sham Equation
In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires solving the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.
A Novel High-Order Shock-Capturing Limiter for RKDG Method on Unstructured Meshes via TENO Selection Strategies
There is a rapidly growing need for high-order numerical schemes on unstructured meshes in the numerical simulation of compressible flows. However, numerous barriers exist to achieve the desired high-order properties on unstructured meshes. The traditional finite-difference method is inadequate for complex unstructured meshes, whereas the finite-volume method introduces additional complexities, e.g., too large stencil size with too many neighboring cells for high-order reconstruction. The Runge–Kutta Discontinuous Galerkin (RKDG) method possesses inherent advantages for unstructured meshes, since it can ensure the high-order property regardless of the type of mesh within a compact stencil. On the other hand, the main weakness of the original RKDG method is their inability to capture discontinuities without artificial numerical oscillations. To tackle this issue, the troubled cell indicator and the slope limiter are integrated with the RKDG method to capture physical discontinuities, where the slope limiter modifies the numerical solutions of the troubled cells detected by the troubled cell indicator. In this work, we propose a new troubled cell indicator and a new limiter based on Targeted Essentially Non-Oscillatory (TENO) schemes for unstructured meshes. The new TENO troubled cell indicator can be applied to unstructured meshes with the compact property and ability to detect troubled cells. For each interface of the targeted troubled cells, the new TENO limiter is utilized to select a smooth numerical solution between the troubled cell and its immediate neighboring cell. Then, a novel weighting strategy is proposed to obtain the final reconstructed numerical solution from the candidate numerical solutions at each interface. It is worth noting that the conventional weighting strategy computes the final reconstructed numerical solution based on the area of corresponding neighboring cells, whereas the newly proposed weighting strategy determines the final reconstructed numerical solution regarding the smoothness of each candidate numerical solution from different interfaces rather than their areas. A set of benchmark cases, including strong shockwaves and a broad range of flow length scales, is simulated to illustrate the performance of the newly proposed limiter in comparison to the Weighted Essentially Non-Oscillatory (WENO) limiter on unstructured meshes. The newly proposed scheme significantly enhances the WENO limiter, exhibiting superior robustness and low-dissipation properties.
Runge-Kutta Discontinuous Galerkin method
Integral Regularization PINNs for Evolution Equations
Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations remains a significant challenge. While physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long-time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR-PINNs), a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function. This method divides the entire temporal interval into smaller subintervals and enforces integral constraints either within each subinterval or across intervals extending from the initial moment to the current one, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR-PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors, offering a robust and accurate solution for evolution equations.
Inexact Block LU Preconditioners for Incompressible Fluids with Flow Rate Conditions
When studying the dynamics of incompressible fluids in bounded domains, the only available data often provide average flow rate conditions on portions of the domain's boundary. In engineering applications a common practice to complete these conditions is to prescribe a Dirichlet condition by assuming a priori a spatial profile for the velocity field. However, this strongly influences the accuracy of the numerical solution. A more mathematically sound approach is to prescribe the flow rate conditions using Lagrange multipliers, resulting in an augmented weak formulation of the Navier-Stokes problem. In this paper we start from the SIMPLE preconditioner, introduced so far for the standard Navier-Stokes equations, and we derive two preconditioners for the monolithic solution of the augmented problem. This can be useful in complex applications where splitting the computation of the velocity/pressure and Lagrange multipliers numerical solutions can be very expensive. In particular, we investigate the numerical performance of the preconditioners in both idealized and real-life scenarios. Finally, we also highlight the advantages of treating flow rate conditions with a Lagrange multipliers approach instead of prescribing a Dirichlet condition.
RBMD: A Molecular Dynamics Package Enabling to Simulate 10 Million All-Atom Particles in a Single Graphics Processing Unit
This paper introduces a random-batch molecular dynamics (RBMD) package for simulations of particle systems at the nano/micro scale. Different from existing packages, the RBMD uses random batch methods for nonbonded interactions of particle systems. The long-range part of Coulomb interactions is calculated in Fourier space by the random batch Ewald algorithm, which achieves linear complexity and superscalability, surpassing classical lattice-based Ewald methods. For the short-range part, the random batch list algorithm is used to construct neighbor lists, significantly reducing computational and memory costs. The RBMD is implemented on GPU-CPU heterogeneous architectures, with classical force fields for all-atom systems. Benchmark systems are used to validate the accuracy and performance of the package. Comparison with the particle-particle particle-mesh and the Verlet list methods in the LAMMPS package is performed on three different NVIDIA GPUs, demonstrating high efficiency of the RBMD on heterogeneous architectures. Our results also show that the RBMD enables simulations on a single GPU with a CPU core up to 10 million particles. Typically, for systems of one million particles, the RBMD allows simulating all-atom systems with a high efficiency of 8.20 ms per step, demonstrating the attractive feature for running large-scale simulations of practical applications on a desktop machine.
A Novel Structure-Preserving Algorithm with Green’s Function Approach for Computing Band Structures of Photonic Quasicrystals
A novel bi-infinite approach to compute the band structures of 2D photonic superlattices with 1D quasicrystal sequences is devised. Leveraging strategically the bi-infinite characteristic, the approach first transforms the infinite-dimensional eigenvalue problem into a finite-dimensional nonlinear eigenvalue problem (NEVP) on a single cell for efficient numerical solution. Challengingly, the NEVP is built upon the solutions to two systems of cyclic nonlinear matrix equations (NMEs) that have to be solved repeatedly during iteratively solving the NEVP. The solutions are efficiently calculated by a newly developed highly efficient coalescing technique followed by a structure-preserving doubling algorithm. It is showed that the cost of coalescing is proportional to the logarithm of $N,$ the length of the truncated quasicrystal sequence, which is significant as the cost of coalescing becomes more noticeable as $N$ gets bigger for highly accurate simulations. Finally, through mathematical analysis, inclusion intervals for eigenvalue of interest are estimated so as to significantly narrow down the scope of search, and that significantly contributes to the overall efficiency of the approach, as the NEVP is nonlinear in nature and has to be solved iteratively.
Total Variation Distance-Enhanced Selective Segmentation for Medical Images
In this paper, we propose an enhanced local intensity clustering energy functional designed for selective segmentation of medical images, particularly those affected by intensity inhomogeneity. The functional includes an area constraint term based on a total variation (TV) distance function derived from the single-scale Retinex output image. This TV distance function measures an unusual distance between points in the image domain and specified marker points, ensuring accurate localization of the selected objects. By combining this with local intensity clustering fitting energy and contour length regularization, the resulting minimization model achieves precisely selective segmentation and tight object wrapping. Moreover, instead of solving the Euler-Lagrange equation or using the level set method, we introduce an efficient iterative convolution-thresholding method to implement the model numerically. This method guarantees energy decay and enables faster convergence to a stable partition. Numerical experiments on some medical images demonstrate the effectiveness and efficiency of our proposed approach for selective image segmentation.
iterative convolution-thresholding method
Computational Method of Grain Boundary Energy Consistent for Different Orientations and Applications to Double Gyroid Beyond Twinning
We develop the method for computing grain boundary (GB) energy consistent for different grain orientations for phase-field type models. The computation is carried out in a parallelepiped domain such that public periodicity is attained along two edges parallel to the dividing plane of two grains. The width of the normal direction is carefully chosen such that the part of each grain in the computational domain can be reassembled into several unit cells. In this way, the bulk energy per volume in the computational domain is identical to the energy density of a unit cell, so that GB energy of different orientations can be obtained in a unified standard. We apply this method to the study of four double-gyroid (DG) GBs with different orientations numerically using the Landau–Brazovskii free energy, including the (422) twin boundary studied recently, a network switching GB, and two tilt GBs. Topological variations and geometric deformations are investigated. It is found that deviations in strut lengths and dihedral angles from the bulk DG substantially exceed changes in strut angles and nodal coplanarity. We also examine the spectra along the contact plane of two grains and utilize them to evaluate the GB widths. Of the four GBs we study, the network switching GB changes to the least extent topologically and geometrically, meanwhile has the lowest energy and the smallest GB width.
Improved Randomized Neural Network Methods with Boundary Processing for Solving Elliptic Equations
We present two improved randomized neural network methods, namely the RNN-Scaling and RNN-Boundary-Processing (RNN-BP) methods, for solving elliptic equations such as the Poisson equation and the biharmonic equation. The RNN-Scaling method modifies the optimization objective by increasing the weight of boundary equations, resulting in a more accurate approximation. We propose the boundary processing techniques for the rectangular domain that enforce the RNN method to satisfy the non-homogeneous Dirichlet and clamped boundary conditions exactly. We further prove that the RNN-BP method is exact for solutions with specific forms and validate it numerically. Numerical experiments demonstrate that the RNN-BP method is the most accurate among the three methods, with the error reduced by up to 6 orders of magnitude for some tests.
An Efficient Time-Splitting Method to Simulate the Dynamics of Spin-Orbit Coupled Spin-1 Bose-Einstein Condensates
In this paper, an efficient time-splitting Fourier spectral method is proposed to simulate the dynamics of spin-orbit coupled spin-1 Bose-Einstein condensates (SOC spin-1 BECs). We split the Hamiltonian into a linear part, which consists of the Laplace and SOC terms, and a nonlinear part that includes all the remaining terms. The linear subproblem is integrated analytically in phase space by solving an ordinary differential system of constant coefficient matrix. While, for the nonlinear subproblem, it is proved the coefficient matrix is actually time-independent in physical space, therefore, the nonlinear subproblem can be integrated exactly. Based on such two-step splitting, we construct high-order schemes to simulate the dynamics. Our method is spectrally accurate in space and high order in time. It is efficient, explicit, unconditionally stable and simple to implement. In addition, we derive some dynamical properties for SOC spin-1 BECs. Extensive numerical results are presented to confirm the accuracy and efficiency, illustrate the dynamical properties at discrete level, and show interesting physics of SOC spin-1 BECs, such as the SOC effects and different wave patterns.
Vortex Spline Method for Two-Dimensional Rayleigh-Taylor Interfaces Between Inviscid and Perfect Dielectric Fluids with External Electric Fields
Interfacial fluid instabilities are widespread in industrial applications. They often lead to mixing between different fluids and play an important role in industrial processes. How to control such interfacial instabilities between dielectric fluids by external electric fields have been actively explored. To understand the effects of external electric fields on the unstable interface and precisely control it, a numerical method capable of providing accurate results is indispensable and highly desired. In this paper, we present a numerical method for systems containing unstable material interfaces between incompressible, inviscid and perfect dielectric fluids in the presence of gravity and external electric fields in two dimensions. We extend the formulation of vortex sheets in the literature from hydrodynamics to electrohydrodynamics, and derive a numerical method in which the computation of both velocity and electric field is only conducted at the one-dimensional material interface. Non-uniform one-dimensional meshes are used to represent the interface shape, which captures the dominant features with fewer nodes. High-order regularization for the Birkhoff-Rott integral in the literature [1] is implemented to control the numerical instabilities. We implement a dynamic mesh adjustment algorithm to further improve the efficiency and robustness of numerical solutions. Validation studies on the convergence and accuracy are conducted.
Incorporating neural networks for the inverse problem of solution of Ordinary Differential Equations (ODEs) represents a pivotal research direction within computational mathematics. Within neural network architectures, the integration of the intrinsic structure of ODEs offers advantages such as enhanced prediction accuracy and reduced data utilization. Among these structural ODE forms, the Lagrangian representation stands out due to its significant physical underpinnings. Building upon this framework, Bhattoo introduced the concept of Lagrangian Neural Networks (LNNs). Then in this article, we introduce an extension (Generalized Lagrangian Neural Networks) to Lagrangian Neural Networks (LNNs) mainly based on mathematics and physics, innovatively tailoring them for non-conservative systems. By leveraging the foundational importance of the Lagrangian within Lagrange’s equations, we formulate the model based on the generalized Lagrange’s equation. This modification not only enhances prediction accuracy but also guarantees Lagrangian representation in nonconservative systems. Furthermore, we perform various experiments, encompassing 1-dimensional and 2-dimensional examples, along with an examination of the impact of network parameters, which proved the superiority of Generalized Lagrangian Neural Networks (GLNNs).
Bound-Preserving Point-Average-Moment PolynomiAl-Interpreted (PAMPA) Scheme: One-Dimensional Case
We propose a bound-preserving (BP) Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme by blending third-order and first-order constructions. The originality of the present construction is that it does not need any explicit reconstruction within each element, and therefore the construction is very flexible. The scheme employs a classical blending approach between a first-order BP scheme and a high-order scheme that does not inherently preserve bounds. The proposed BP PAMPA scheme demonstrates effectiveness across a range of problems, from scalar cases to systems such as the Euler equations of gas dynamics. We derive optimal blending parameters for both scalar and system cases, with the latter based on the recent geometric quasi-linearization (GQL) framework of [Wu & Shu, SIAM Review, 65 (2023), pp. 1031–1073]. This yields explicit, optimal blending coefficients that ensure positivity and control spurious oscillations in both point values and cell averages. This framework incorporates a convex blending of fluxes and residuals from both high-order and first-order updates, facilitating a rigorous BP property analysis. Sufficient conditions for the BP property are established, ensuring robustness while preserving high-order accuracy. Numerical tests confirm the effectiveness of the BP PAMPA scheme on several challenging problems.
Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme
First-Principle-Like Reinforcement Learning of Nonlinear Numerical Schemes for Conservation Laws
In this study we present a universal nonlinear numerical scheme design method for nonlinear conservation laws, enabled by multi-agent reinforcement learning (MARL). Unlike contemporary approaches based on supervised learning or reinforcement learning, our method does not rely on reference data or empirical design. Instead, a first-principle-like approach using fundamental computational fluid dynamics (CFD) principles, including total variation diminishing (TVD) and $k$-exact reconstruction, is employed to design nonlinear numerical schemes. The third-order finite volume scheme is employed as the workhorse to test the performance of the MARL-based nonlinear numerical scheme design method. Numerical results demonstrate that the new MARL-based method can strike a balance between accuracy and numerical dissipation in nonlinear numerical scheme design, and outperforms the third-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) with the van Albada limiter for shock capturing. Furthermore, we demonstrate for the first time that a numerical scheme trained from one-dimensional (1D) Burgers’ equation simulations can be directly used for numerical simulations of both 1D and 2D (two-dimensional constructions using the tensor product operation) Euler equations. The framework of the MARL-based numerical scheme design concepts can incorporate, in general, all types of numerical schemes as simulation machines.
Numerical Modeling of Flocking Dynamics with Topological Interactions
In this paper, we propose a numerical investigation of topological interactions in flocking dynamics. Starting from a microscopic description of the phenomena, mesoscopic and macroscopic models have been previously derived under specific assumptions. We explore the role of topological interactions by describing the convergence speed to consensus in both microscopic and macroscopic dynamics, considering different forms of topological interactions. Additionally, we compare mesoscopic and macroscopic dynamics for monokinetic and non-monokinetic initial data. Finally, we illustrate with some simulations in one- and two-dimensional domains the sensitive dependence of solutions on initial conditions, including the case where the system exhibits two solutions starting with the same initial data.
Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem
Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.
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A Third Order Bound-Preserving Nodal Discontinuous Galerkin Method for Miscible Displacements in Porous Media
In this paper, we develop a provable third order bound-preserving (BP) nodal discontinuous Galerkin (DG) method for compressible miscible displacements. We consider the problem with a multi-component fluid mixture and physically the volumetric concentration of each component, $c_j(j=1,···,N)$, is between 0 and 1. The main idea is to apply a positivity-preserving (PP) method to all $c′_js,$ while enforce $∑_jc_j=1$ simultaneously. First, we treat the time derivative of the pressure as a source and choose suitable “consistent” numerical fluxes in the pressure and concentration equations to construct a nodal interior penalty DG (IPDG) method to enforce $∑_jc_j =1.$ For PP, we represent the cell average of $c_j$ as a weighted summation of Gaussian quadrature point values, and transform which to some other specially chosen point values. We prove that by taking appropriate parameters in the nodal IPDG method and a suitable time stability condition, the cell average can be kept positive, which further implies that the cell averages of all components are between 0 and 1. Finally, we apply a polynomial scaling limiter to obtain physically relevant numerical approximations without sacrificing accuracy. Numerical experiments are given to demonstrate desired accuracy, BP and good performances of our proposed approach.
An Efficient Numerical Algorithm for Solving Coupled Time-Dependent Ginzburg-Landau Equation for Superconductivity and Elasticity
A decoupled finite element algorithm is developed for simulating the vortex dynamics on an elastic superconductor which couples the time-dependent Ginzburg-Landau equation with the complex-valued superconducting order parameter and the vector-valued magnetic potential, and the elasticity equation. We present an iterative algorithm for the decoupled system arising from the time and spatial discretization using a combination of preconditioner, algebraic multigrid method (AMG) and preconditioned conjugate gradient method (PCG). The iterative algorithm allows us to perform large-scale three-dimensional simulations of mesoscale pattern formation during superconducting phase transitions with arbitrary elastic boundary conditions. The performance and efficiency of the algorithm are numerically verified by several benchmark problems, exhibiting up to two orders of magnitude improvement depending on the scale of discrete system compared to the exact solver.
Adaptive Basis-Inspired Deep Neural Network for Solving Partial Differential Equations with Localized Features
This paper proposes an Adaptive Basis-inspired Deep Neural Network (ABI-DNN) for solving partial differential equations with localized phenomena such as sharp gradients and singularities. Like the adaptive finite element method, ABI-DNN incorporates an iteration of "solve, estimate, mark, enhancement", which automatically identifies challenging regions and adds new neurons to enhance its capability. A key challenge is to force new neurons to focus on identified regions with limited understanding of their roles in approximation. To address this, we draw inspiration from the finite element basis function and construct the novel Basis-inspired Block (BI-block), to help understand the contribution of each block. With the help of the BI-block and the famous Kolmogorov Superposition Theorem, we first develop a novel fixed network architecture named the Basis-inspired Deep Neural Network (BI-DNN), and then integrate it into the aforementioned adaptive framework to propose the ABI-DNN. Extensive numerical experiments demonstrate that both BI-DNN and ABI-DNN can effectively capture the challenging singularities in target functions. Compared to PINN, BI-DNN attains significantly lower relative errors with a similar number of trainable parameters. When a specified tolerance is set, ABI-DNN can adaptively learn an appropriate architecture that achieves an error comparable to that of BI-DNN with the same structure.
Adaptive basis-inspired deep neural network
The Multidomain Hybrid Discontinuous Galerkin Method and Finite Difference Method for Solving Compressible Navier-Stokes Equations on Hybrid Meshes
In this work, we develop a multidomain hybrid discontinuous Galerkin (DG) method and finite difference(FD) method for solving two-dimensional compressible Navier-Stokes equations on the hybrid meshes. The direct discontinuous Galerkin (DDG) method and central difference(CD) scheme are utilized to discretize the viscous fluxes respectively. This approach combines the flexibility for the complex geometries of the DG method on the unstructured meshes, and the computational efficiency of the FD method on Cartesian grids. At the artificial interfaces between the DG subdomain and FD subdomain, the square ghost cells are generated and the weighted essentially non-oscillatory (WENO) interpolation is employed to reconstruct the degrees of freedom of these ghost cells. To ensure the accuracy in smooth regions and the correct position of the shock wave, the troubled cell indicator is adopted to determine the nonconservative or conservative coupling modes. The construction process of the numerical fluxes at the artificial interfaces is described specifically and the corresponding WENO interpolation coefficients are given in detail. Numerous numerical results demonstrate that the multidomain hybrid method achieves high-order accuracy in smooth regions, robustness in shock simulations, flexibility in handling complex geometries, and significant computational cost savings compared to the traditional DG method on the hybrid meshes.
Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares
The principal goal of 1-bit compressive sampling is to decode $n$-dimensional signals with a sparsity level of $s$ from $m$ binary measurements. This task presents significant challenges due to nonlinearity, noise, and sign flips. In this paper, we propose the use of the cardinality-constrained least squares decoder as an optimal solution. We establish that, with high probability, the proposed decoder achieves a minimax estimation error, up to a constant $c$, as long as $m ≥ \mathcal{O}(s{\rm log} \ n)$. In terms of computational efficiency, we employ a generalized Newton algorithm (GNA) to solve the cardinality-constrained minimization problem. At each iteration, this approach incurs the cost of solving a least squares problem with a small size. Through rigorous analysis, we demonstrate that, with high probability, the $ℓ_∞$ norm of the estimation error between the output of GNA and the underlying target diminishes to $\mathcal{O}( \sqrt{\frac{{\rm log} \ n}{m}})$ after at most $\mathcal{O}({\rm log} \ s)$ iterations. Furthermore, provided that the target signal is detectable, we can recover the underlying support with high probability within $\mathcal{O}({\rm log} \ s)$ steps. To showcase the robustness of our proposed decoder and the efficiency of the GNA algorithm, we present extensive numerical simulations and comparisons with state-of-the-art methods.
least square with cardinality constraint
Dimension-Free Ergodicity of Path Integral Molecular Dynamics
The quantum thermal average plays a central role in describing the thermodynamic properties of a quantum system. Path integral molecular dynamics (PIMD) is a prevailing approach for computing quantum thermal averages by approximating the quantum partition function as a classical isomorphism on an augmented space, enabling efficient classical sampling, but the theoretical knowledge of the ergodicity of the sampling is lacking. Parallel to the standard PIMD with $N$ ring polymer beads, we also study the Matsubara mode PIMD, where the ring polymer is replaced by a continuous loop composed of $N$ Matsubara modes. Utilizing the generalized $\Gamma$ calculus, we prove that both the Matsubara mode PIMD and the standard PIMD have uniform-in-$N$ ergodicity, i.e., the convergence rate towards the invariant distribution does not depend on the number of modes or beads $N$.
A $p$-Adaptive Treecode Solution of the Poisson Equation in the General Domain
Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive method is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics.
The Effective Use of BLAS Interface for Implementation of Finite-Element ADER-DG and Finite-Volume ADER-WENO Methods
Numerical methods of the ADER family, in particular finite-element ADER-DG and finite-volume ADER-WENO methods, are among the most accurate numerical methods for solving quasilinear hyperbolic PDE systems. The internal structure of ADER-DG and ADER-WENO numerical methods contains a large number of basic linear algebra operations related to matrix multiplications. The main interface of software libraries for matrix multiplications for high-performance computing is BLAS. An effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods is presented. The calculated matrices are small matrices; and this allows to use effectively JIT technologies. The proposed approach immediately operates on AoS, which allows to efficiently calculate flux, source and non-conservative terms without transposition. The obtained computational costs demonstrated that the effective implementation, based on the use of the JIT functions of the BLAS, outperformed both the implementation based on the general BLAS functions and the vanilla implementations by several orders of magnitude. The complexity of developing an implementation based on the proposed approach does not exceed the complexity of developing a vanilla implementation. Performance analysis using roofline partly explains the observed features of the decreasing of computational costs.