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 Weak Galerkin Finite Element Method Coupled with Mortar Spectral Element Method for Schrödinger Eigenvalue Problem with an Inverse Square Potential
In this paper, we introduce a weak Galerkin (WG) finite element method coupled with mortar spectral element method (MSEM) to solve the Schrödinger eigenvalue problem with an inverse square potential. For the domain around the inverse square potential, we use the mortar spectral element method to simulate the singularities in eigenfunctions caused by the inverse square potential, while we employ the WG method in the remaining domain. This coupled method can effectively handle the singularity arising from the inverse square potential. Notably, hanging nodes are allowed on the coupled interface. Compared to the conforming finite element method coupled with MSEM, our approach is not constrained by the mesh size of the mortar spectral element. This flexibility permits the use of fine meshes in the WG domain, thereby enhancing accuracy. We provide $hp$ error analysis for both eigenfunctions and eigenvalues. Numerical experiments demonstrate the $hp$ convergence of the theoretical results.
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A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems Based on a Reconstruction Operator
The weak Galerkin (WG) finite element method has shown great potential in solving various type of partial differential equations. In this paper, we propose an arbitrary order locking-free WG method for solving linear elasticity problems, with the help of an appropriate $H(div)$-conforming displacement reconstruction operator. Optimal order locking-free error estimates in both the $H^1$-norm and the $L^2$-norm are proved, i.e., the error is independent of the Lamé constant $\lambda$. Moreover, the term $\lambda \|\nabla \cdot \mathbf{u}\|_k$ does not need to be bounded in order to achieve these estimates. We validate the accuracy and the robustness of the proposed locking-free WG algorithm by numerical experiments.
$H(div)$-conforming displacement reconstruction
An Exponential-Free Runge–Kutta Framework for Developing Third-Order Unconditionally Energy Stable Schemes for the Cahn–Hilliard Equation
In this work, we develop a class of up to third-order energy-stable schemes for the Cahn–Hilliard equation. Building on Lawson’s integrating factor Runge–Kutta method, which is widely used for stiff semilinear equations, we discuss its limitations, such as the inability to preserve the equilibrium state and the oversmoothing of interfacial layers in the solution’s profile because of the exponential damping effects. To overcome this drawback, we approximate the exponential term using a class of sophisticated Taylor polynomials, leading to a novel Runge–Kutta framework called exponential-free Runge–Kutta. By incorporating stabilization techniques, we analyze the energy stability of the proposed schemes and demonstrate that they preserve the original energy dissipation without time-step restrictions. Furthermore, we conduct an analysis of the linear stability and establish an error estimate in the $ℓ^2$ norm. A series of numerical experiments validate the high-order accuracy, mass conservation, and energy dissipation of our schemes.
Network Embedding Using Sparse Approximations of Random Walks
In this paper, we propose an efficient numerical implementation of Network Embedding based on commute times, using sparse approximation of a diffusion process on the network obtained by a modified version of the diffusion wavelet algorithm. The node embeddings are computed by optimizing the cross entropy loss via the stochastic gradient descent (SGD) method with sampling of low-dimensional representations of green functions. We demonstrate the efficacy of this method for data clustering and multi-label classification through several examples, and compare its performance over existing methods in terms of efficiency and accuracy. Theoretical issues justifying the scheme are also discussed.
A Diffusion Model Based Iterative Convolution Thresholding Method for Structural Topological Optimization
In this study, we introduce the Diffusion Model with Iterative Convolution Thresholding Method (DICTM), a novel hybrid approach designed to address the minimum compliance problem in topology optimization. DICTM synergistically combines the robustness of diffusion models with the precision of threshold dynamics to tackle the complexities inherent in linear elasticity problems, while substantially enhancing computational efficiency. Our approach facilitates the generation of initial configurations via the diffusion model, which dramatically improves the efficiency of the subsequent threshold dynamics process, reducing the iteration count to about one-tenth of that required by traditional methods. This significant reduction in computational effort also enables more effective hyperparameter tuning without added cost. The integration of deep-generative models with a rigorous threshold dynamics framework positions DICTM as a powerful tool in topology optimization, producing designs not only with low compliance, but also in a computationally efficient way.
iterative convolution threshold dynamics
Full-Waveform Inversion with Unbalanced Optimal Transport Metric for Seismic Imaging
Full-waveform inversion (FWI) is an effective method for obtaining high-resolution images of subsurface structures. Conventional full-waveform inversion, which uses the least-square norm $L_2$ to measure the mismatch between observed and synthetic seismograms, frequently suffers from cycle-skipping and local minimum problems. Derived from optimal transport theory, the Wasserstein metric has been proposed to mitigate cycle-skipping issue. However, due to the requirement for mass conservation, the classical quadratic Wasserstein metric is not ideally suited for FWI applications. In this study, we introduce two unbalanced optimal transport (UOT) distances for use in FWI: the regularized UOT and the unbalanced Sinkhorn divergence. An entropy regularization approach and a truncation approximation are employed to guarantee the efficiency of calculating distances and gradients. These unbalanced optimal transport distances preserve the desirable properties of the quadratic Wasserstein metric, particularly its convexity and insensitivity to noise, while overcoming issues related to mass conservation. We compare the unbalanced optimal transport distances with the $L_2$ distance and the classical quadratic Wasserstein metric using the Camembert model and the crustal root model. Our numerical experiments demonstrate the superiority of the unbalanced optimal transport distances over traditional methods.
A Complete Error Analysis of PINNs for Elliptic Equations Using Projected Stochastic Gradient Descent
Physics-informed neural networks (PINNs) have recently gained attention as a powerful and efficient tool for solving partial differential equations (PDEs). Despite their empirical success, the theoretical understanding of PINNs, especially in the context of over-parameterization, remains incomplete. This paper presents a complete error analysis of over-parameterized PINNs for elliptic equations using projected stochastic gradient descent (PSGD) optimization. Our analysis rigorously examines the interplay of approximation error, statistical error, and optimization error, offering a unified framework for understanding the convergence behavior of PINNs. By leveraging the properties of PSGD, we establish convergence rates and derive conditions on neural network architecture, training sample requirements, and optimization parameters to ensure specified accuracy.
Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Lévy Equations
We introduce an innovative approach for solving high-dimensional Fokker-Planck-Lévy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model’s derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations. Code is available at https://github.com/zheyuanhu01/Score-fPINN.
high dimensional partial differential equations
A Two-Stage Fourth-Order Gas-Kinetic CPR Method for Subsonic Flows on Hexahedral Meshes
A compact high-order gas-kinetic scheme (GKS) is developed for three-dimensional subsonic inviscid and viscous flows on hexahedral meshes, which achieves fourth-order accuracy in both space and time. The scheme combines a compact and efficient correction procedure via reconstruction (CPR) framework with a time-evolving gas-kinetic flux, in which the inviscid and viscous fluxes are coupled and computed uniformly. With the CPR framework, the current scheme avoids the difficulty of compact fourth-order reconstruction encountered by the traditional finite volume GKS. Moreover, both the flux and its time-derivative are available in the gas-kinetic flux so that an efficient two-stage temporal discretization can be adopted to achieve fourth-order time accuracy, which is more efficient than the traditional Runge-Kutta CPR method. In addition, with the help of isoparametric transformation, the current scheme can treat curved boundaries with high-order curved meshes. Typical numerical tests demonstrate the good performance of the current scheme.
two-stage fourth-order temporal discretization
Preface to the Special Issue in Memory of Prof. Jiequan Li
On May 14, 2023, the community of computational and applied mathematics lost an exceptional scholar and cherished friend, Professor Jiequan Li. His sudden passing left us with both a profound sense of loss and a deep appreciation of his intellectual legacy. To honor his memory and extraordinary contributions, this special issue of Communications in Computational Physics has been dedicated to him. The collection has been carefully assembled by his friends and collaborators, and it reflects both the breadth of his influence and the vitality of the field he helped shape.
Multiscale Simulation of Rarefied Gas Flows in Simplified Divertor Tokamak Test Facility Particle Exhaust
Simulating gas flow within the divertor, which is a crucial component in nuclear fusion reactors, is essential for assessing and enhancing its design and performance. Traditional methods, such as the direct simulation Monte Carlo and the discrete velocity method, often fall short in efficiency for these simulations. In this study, we utilize the general synthetic iterative scheme to simulate a simplified Tokamak divertor model, demonstrating its fast convergence and asymptotic-preserving properties in complex three-dimensional scenarios. A conservative estimate of speedup by three orders of magnitude is achieved by the general synthetic iterative scheme when compared to the direct simulation Monte Carlo method. We further investigate the relationship between pumping efficiency and factors like temperature, absorptivity, and the Knudsen number, providing valuable insights to guide the design and optimization of divertor structures.
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Kinetic Representation of the Unified Gas-Kinetic Wave-Particle Method and Beyond
The unified gas-kinetic wave-particle (UGKWP) method is a hybrid method for multiscale flow simulations, in which the contributions to the whole gas evolution from deterministic hydrodynamic wave and stochastic particle transport are combined simultaneously. Originally, the UGKWP method was developed as a direct modeling approach at discrete level. In this work, we revisit the time evolution of each part of the involved simulation particles and wave molecules in UGKWP, and present the corresponding kinetic equations. The resultant kinetic system can be viewed as a collision decomposition of the original kinetic equation, which can serve as a basis for developing other kinetic methods for flows in all flow regimes.
A Lagrangian GRP Algorithm for Axisymmetric Problems of Compressible Fluids
We devise a Lagrangian Generalized Riemann Problem (GRP) algorithm for axisymmetric hydrodynamics, which pays attention to high-resolution boundary treatment at the symmetric axis. The numerical boundary condition here is first formulated the same as the scheme on interior cells. Then we uniformly obtain the requisite interface values in constructing numerical fluxes and sources through the newly-tailored GRP solver and its one-sided variant. There also exist some innovations in the other two critical procedures: the derivation of vertex velocities and the compliance with the geometry conservation law (GCL). Several challenging numerical examples are utilized to demonstrate the performance of our algorithm in resolving discontinuities, maintaining symmetry, alleviating overheating phenomena, and dealing with complex fluids.
high-resolution numerical boundary condition
Role of Fluxes in High-Order Godunov Schemes
This paper focuses on the evolution of some mathematical aspects related to high-resolution approximations to nonlinear hyperbolic balance laws. It addresses the crucial role of numerical fluxes in dealing with the three concepts of consistency, stability and convergence. The classical paper [15] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. The seminal paper of van Leer [30] has inaugurated the period of universal interest in high-resolution extensions of Godunov’s scheme. The fundamental step consists of modifying the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise polynomial (rather than piecewise constant) initial data. The GRP (Generalized Riemann Problem) analysis [1] provided analytical solutions (for piecewise linear data) that could be readily implemented in a high-resolution robust code. The treatment utilizes the framework of “balance laws”, a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula), and the associated fluxes are Lipschitz continuous with respect to the spatial coordinates. Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of “flux consistency” must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [17].
Adaptive Reconstruction Method Using Discontinuity Feedback for High-Order Accuracy Schemes
This paper presents efficient adaptive stencil extension reconstruction methods using a discontinuity feedback factor to address weak robustness and high computational costs in high-order schemes (7th-order and above). The method features two innovations: accuracy order adaptively increases from the lowest level based on local stencil smoothness, unlike WENO and MUSCL limiters that reduce order from highest level; and the Discontinuity Feedback Factor detects sub-cell discontinuity strength while serving as a local smoothness measure. This eliminates expensive smoothness indicators in very high-order schemes and generalizes to arbitrary high orders. Rigorous tests, including a Mach 20000 jet, demonstrate exceptional robustness.
Gas-Kinetic Unified Algorithm for Multi-Component Monatomic Gas Mixture
In recent decades, gas-kinetic unified algorithm (GKUA) based on Boltzmann model equation has been proposed, which is suitable for the whole regimes gas flows. The original GKUA is designed for single-component gas flows. In this work, we extend GKUA to multi-component monatomic gas mixture flows based on the Andries-Aoki-Perthame (AAP) model. The algorithm are validated by the steady normal shock wave structure problem, the flow around two-dimensional circular cylinder and the pressure/temperature gradient driven microchannel flow problems. Simulation results are in good agreement with those obtained by other methods, which verifies the reliability of the proposed algorithm for the simulation of multi-component monatomic gas mixture. Then, taking the 25N attitude control engine two-dimensional profile as the object, the study of two-dimensional profile nozzle internal and external mixed flow problem is carried out, and the influences of the component concentration ratio and mass ratio on the molecular transport in the engine internal and external mixed flow field are discussed.
Convergence Analysis for a Finite Volume Evolution Galerkin Method for Multidimensional Hyperbolic Systems
We study the convergence of a finite volume method based on the method of bicharacteristics for multidimensional hyperbolic conservation laws. In particular, we concentrate on the linear wave equation system and nonlinear Euler equations of gas dynamics. We show the stability and the consistency of the numerical approximations. By means of the generalized Lax equivalence principle we prove the convergence of numerical solutions to the strong solution on the lifespan.
Stability Analysis and Structure Preserving Schemes for the Reactive Euler Equations with a New Equation of State
This paper is concerned with the multi-dimensional reactive Euler equations under the new equation of state (EoS) recently proposed in [45]. We show that under this EoS the classical thermodynamic entropy is a strictly convex entropy function in general dimension. Based on this we further prove that the reactive Euler equations satisfy the stability conditions for hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. The eigen-decompositions of the Jacobian matrices in two and three dimensions are also provided. Moreover, we develop a positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme by adapting that in [46] for the EoS of ideal gas to the newly proposed one. A key step in doing so is to prove that the HLL (Harten-Lax-van Leer) flux is entropy stable, which is established by tactfully using a natural assumption on a function in the EoS. The high convergence orders stable entropy, no oscillation and positivity of the scheme are demonstrated with numerical examples.
The Implicit Unified Gas Kinetic Scheme for Neutron Transport with Arbitrary Anisotropic Scattering Model
In previous work, we developed the unified gas dynamics scheme (UGKS) and its corresponding time-implicit variant (IUGKS) for neutron transport. The multi-scale characteristics of these schemes are particularly well-suited for transport equations, which show good performances in large-scale conditions. However, only the isotropic scattering model equation was considered in these studies, which does not meet the application requirements in complex engineering cases. In this paper, we extend our studies of isotropic neutron IUGKS to encompass the transport equation with arbitrary anisotropic scattering. We first clarify the construction principle of IUGKS-PN scheme according to the general form of anisotropic scattering models. However, the construction of the scheme is complex and the versatility is not strong. To address these challenges, a more concise scheme IUGKS-AM is proposed based on the isotropic IUGKS framework, in which the higher-order moments of the neutron distribution function are estimated. We theoretically analyze the asymptotic property of IUGKS-AM, and verify the numerical performances in anisotropic model by numerical experiments. Finally, several complex anisotropic scattering models are computed by using the numerical scheme, demonstrating its general applicability. This study significantly enhances the practical utility of multi-scale neutron transport UGKS in engineering applications.
Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank
Training deep neural networks for scientific computing remains computationally expensive due to the slow formation of diverse feature representations in early training stages. Recent studies [37] identify a staircase phenomenon in training dynamics, where loss decreases are closely correlated with increases in $\varepsilon$-rank, reflecting the effective number of linearly independent neuron functions. Motivated by this observation, this work proposes a structured first-layer initialization (SFLI) pre-training technique to enhance the diversity of neural features at initialization by constructing $\varepsilon$-linearly independent neurons in the input layer. We present systematic initialization schemes compatible with various activation functions and integrate the strategy into multiple neural architectures, including modified multi-layer perceptrons and physics-informed residual adaptive networks. Only needing to add one line of code to conventional stochastic gradient descent algorithms, extensive numerical experiments on function approximation and PDE benchmarks, demonstrate that SFLI significantly improves the initial $\varepsilon$-rank, accelerates convergence, mitigates spectral bias, and enhances prediction accuracy.
A Novel High-Resolution TENO Scheme with Adaptive Artificial Anti-Diffusion
Low-dissipation higher-order approaches are crucial for flows characterized by shocks and turbulence. The broad applicability of conventional WENO/TENO systems for accurately capturing shocks makes them a preferred option for simulating complex flow phenomena. Nonetheless, the inherent numerical diffusion in these techniques poses a significant challenge in simulating multiscale turbulent flow. This study mitigates the inherent numerical damping by using locally adaptive higher-order artificial anti-diffusion. A switching function quantifies the extent of anti-dissipation in the smooth zone and assesses the local variation of flow field variables. To assure stability, the anti-diffusion coefficient is meticulously assessed using two fixed free parameters. Next strategy utilizes a fourth-order central difference scheme in the highly smooth region, incorporating an additional coefficient modified in a binary fashion. The adaptive regulating parameter $C_T$ in the fifth-order TENO scheme is integrated with the anti-diffusive function. It is recognized that the new scheme exhibits less numerical diffusion than the original TENO scheme. This is also reflected in the numerical experiments, encompassing a series of benchmark test scenarios featuring shock and fine-scale structures. It is shown that the proposed scheme could resolve the turbulent scales with much higher resolution and capture the shock sharply. It is worth noting that these strategies can easily be implemented in the other variations of the TENO scheme.
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.
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.