Article
Computer Science, Interdisciplinary Applications
Nuo Lei, Juan Cheng, Chi -Wang Shu
Summary: In this paper, a high-order positivity-preserving polynomial projection remapping method is developed for the discontinuous Galerkin scheme based on the L2 projection. An indirect arbitrary Lagrangian-Eulerian discontinuous Galerkin method is presented by combining the Lagrangian type DG scheme and the rezoning strategies. The remapping method, which accurately clips the intersections between the old distorted mesh and the new rezoned mesh, is highly accurate and suitable for large deformable problems. A positivity-preserving limiter is added for the physical variables in computational fluid dynamics without compromising the original high-order accuracy and conservation. A multi-resolution weighted essentially non-oscillatory limiter is used to overcome numerical oscillations and maintain high-order accuracy in smooth regions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Xiaofeng Cai, Sebastiano Boscarino, Jing-Mei Qiu
Summary: The paper introduces a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for solving nonlinear Vlasov dynamics, achieving high spatial and temporal accuracy. Inherit advantages from the SLDG method, the proposed method performs well in resolving complex solution structures, conserves mass and positivity, and can evolve with adaptive time stepping.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Xiaofeng Cai, Jing-Mei Qiu, Yang Yang
Summary: The paper introduces a new method called ELDG, which incorporates a modified adjoint problem and integration of PDE over a space-time region partitioned by time-dependent linear functions. By introducing a new flux term to account for errors in characteristics approximation, the ELDG method combines the advantages of SL DG and classical Eulerian RK DG methods. The use of linear functions for characteristics approximation in the EL DG framework simplifies shapes of upstream cells and reduces time step constraints.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Pei Fu, Yinhua Xia
Summary: This paper presents an almost arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve the compressible Euler equations in one and two space dimensions, and considers their positivity preserving property of states density and pressure. The proposed ALE-DG method, coupled with a modified strong stability-preserving Runge-Kutta method and the positivity preserving limiter, ensures the geometric conservation law and the positivity property of the scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ziyao Xu, Chi -Wang Shu
Summary: This study is a follow-up work that improves and extends the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations. Previous methods were only applicable to equations with constant coefficients and achieved second order accuracy. In this study, we propose high order positivity-preserving DG methods for one-dimensional variable coefficient and nonlinear equations, as well as two and three-dimensional stationary hyperbolic equations with constant coefficients. The algorithms are validated through extensive numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Changxin Qiu, Qingyuan Liu, Jue Yan
Summary: In this article, the direct discontinuous Galerkin method with interface correction (DDGIC) was used to solve the chemotaxis Keller-Segel equation, demonstrating a third order accuracy and positivity-preserving property. By appropriately choosing the numerical flux coefficients, the cell density approximation can be kept non-negative at all times, and maintaining uniform third order accuracy with the positivity-preserving limiter.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Hailiang Liu, Zhongming Wang, Peimeng Yin, Hui Yu
Summary: In this paper, we propose and analyze third order positivity-preserving discontinuous Galerkin schemes for solving the time-dependent system of Poisson-Nernst-Planck equations. Our method ensures the positivity of numerical solutions and restores it through a scaling limiter.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Chang Yang, Michel Mehrenberger
Summary: This paper introduces a highly accurate monotonicity-preserving Semi-Lagrangian scheme for Vlasov-Poisson simulations, which uses a limiter to avoid accuracy loss and clipping near extrema while maintaining monotonicity. The scheme preserves the monotonicity of the solution for locally monotonic data, while maintaining good properties and high accuracy similar to the unlimited scheme. Numerical tests show that the limited scheme is more diffusive compared to cubic splines, but has better L-1 conservation, making it advantageous for problems with sharp gradients.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
T. Dzanic, F. D. Witherden
Summary: This work presents a positivity-preserving entropy-based adaptive filtering method for shock capturing in discontinuous spectral element methods. The method adapts the filter strength to enforce positivity and a local discrete minimum entropy principle, allowing it to robustly handle strong discontinuities with sub-element resolution. It does not require problem-dependent parameter tuning and can be easily implemented on general unstructured meshes with relatively low computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Nuo Lei, Juan Cheng, Chi-Wang Shu
Summary: In this paper, a high-order discontinuous Galerkin method is developed for the numerical solution of the nonlocal electron heat transport model. The method is shown to have high accuracy and positivity-preserving properties through numerical examples. The comparison between the local and nonlocal models reveals additional physical phenomena such as flux reduction and preheat effect.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jian Cheng, Fan Zhang
Summary: We have developed a high-order path-conservative discontinuous Galerkin method for simulating compressible two-medium flows. The method satisfies the equilibrium condition and improves robustness for complex flows with large density and pressure ratios.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Weijie Zhang, Yinhua Xia, Yan Xu
Summary: This paper develops well-balanced arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the shallow water equations, which preserve both static and moving water equilibrium, while maintaining high order accuracy and positivity preservation.Numerical experiments demonstrate the well-balanced property, positivity preservation and high order accuracy of these schemes in different circumstances.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Erica R. Johnson, James A. Rossmanith, Christine Vaughan
Summary: The HyQMOM variant of QMOM is proven to have moment-invertibility over a convex region in solution space. A high-order discontinuous Galerkin (DG) scheme is developed to solve the resulting fluid system, with novel limiters introduced to guarantee the system's hyperbolicity. The scheme is also extended to include a BGK collision operator, which is shown to be asymptotic-preserving in the high-collision limit.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics
Jianqiang Sun, Jiameng Kong, Lijuan Zhang, Jingxian Zhang
Summary: This paper investigates the symplectic structure of fractional coupled nonlinear Schrodinger equations and proposes a new format. Numerical experiments show that the new scheme outperforms the classical symplectic scheme in terms of energy conservation property.
JOURNAL OF MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Evan J. Lieberman, Xiaodong Liu, Nathaniel R. Morgan, Darby J. Luscher
Summary: A new Lagrangian modal discontinuous Galerkin (DG) hydrodynamic method is presented to support a dynamic dislocation based crystal plasticity model for simulating the mechanical behavior of crystallographic materials under dynamic conditions. The method uses varying the position and material properties of material points within an element to represent the heterogeneous behavior of polycrystalline microstructures.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Sergio Amat, Antonio Baeza, Juan Ruiz, Chi-Wang Shu
Summary: This paper aims to redesign an algorithm previously analyzed by Amat et al. to achieve a centered strategy for approximating the solutions of linear and nonlinear systems of conservation laws. The analysis of the new algorithm explains its effectiveness near shocks in conservation law solutions and emphasizes the goal of using the most centered stencil in high gradient conditions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Multidisciplinary
Nuo Lei, Juan Cheng, Chi-Wang Shu
Summary: This paper presents a high-order accurate positivity-preserving conservative remapping algorithm based on WENO reconstruction, which has been demonstrated through a series of numerical experiments to be conservative, positivity-preserving, and efficient.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Computer Science, Interdisciplinary Applications
Yun-Long Liu, Chi-Wang Shu, A-Man Zhang
Summary: A new interface treatment method is proposed for simulating compressible two-medium problems using the RKDG method. The method ensures a smooth transition of the interface while minimizing overshoots or undershoots with the adoption of entropy-fix technique. It demonstrates high accuracy and compactness in handling interfaces with large entropy ratios.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Qi Tao, Yan Xu, Chi-Wang Shu
Summary: This paper presents an ultra-weak local discontinuous Galerkin (UWLDG) method for a class of nonlinear fourth-order wave equations, designed and analyzed. The method demonstrates energy conserving properties and optimal error estimates, which are confirmed through numerical experiments. Compatible high order energy conserving time integrators are also proposed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Tingting Li, Jianfang Lu, Chi-Wang Shu
Summary: This paper investigates the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The study utilizes the simplified inverse Lax-Wendroff procedure and third order TVD Runge-Kutta method, along with two analysis techniques to check algorithm stability, yielding consistent results in both semi-discrete and fully-discrete cases. Numerical experimental results are presented to validate the theoretical findings.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Yong Liu, Jianfang Lu, Chi-Wang Shu
Summary: In this paper, an essentially oscillation-free discontinuous Galerkin method is developed for systems of hyperbolic conservation laws. The method introduces numerical damping terms to control spurious oscillations. Both classical Runge-Kutta method and modified exponential Runge-Kutta method are used in time discretization. Extensive numerical experiments demonstrate the robustness and effectiveness of the algorithm.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Jichun Li, Chi-Wang Shu, Wei Yang
Summary: This paper focuses on the time-domain carpet cloak model and proposes two new finite element schemes to address the numerical stability issue of previous schemes. The unconditional stability of the Crank-Nicolson scheme and the conditional stability of the leap-frog scheme are proved, both inheriting the exact form of the continuous stability.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Nuo Lei, Juan Cheng, Chi-Wang Shu
Summary: We propose a high-order positivity-preserving conservative remapping method on 3D tetrahedral meshes based on the weighted essentially non-oscillatory reconstruction method. By accurately calculating the overlaps between meshes, our method allows for wider range of mesh movements and simplifies the remapping process. Utilizing the third order multi-resolution WENO reconstruction procedure, we distribute nonlinear weights based on the smoothness of polynomials to achieve optimal accuracy and avoid numerical oscillations. Our method also incorporates efficient local limiting to preserve positivity without compromising high-order accuracy and conservation. Numerical tests confirm the properties of our remapping algorithm, such as high-order accuracy, conservation, non-oscillatory performance, positivity-preserving, and efficiency.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Dan Ling, Chi-Wang Shu, Wenjing Yan
Summary: The paper focuses on the design of numerical methods for the diffusive-viscous wave equations with variable coefficients and develops a local discontinuous Galerkin (LDG) method. Numerical experiments are provided to demonstrate the optimal convergence rate and effectiveness of the proposed LDG method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jie Du, Chi-Wang Shu, Xinghui Zhong
Summary: In this paper, we propose an improved simple WENO limiter for the Runge-Kutta discontinuous Galerkin method in solving two-dimensional hyperbolic systems on unstructured meshes. The major improvement is reducing the number of polynomials transformed to the characteristic fields for each direction, resulting in reduced computational cost and improved efficiency. The improved limiter provides a simpler and more practical way for the characteristic-wise limiting procedure, while maintaining uniform high-order accuracy in smooth regions and controlling nonphysical oscillations near discontinuities. Numerical results show that the improved limiter outperforms the original one in terms of accuracy and resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Sergio Amat, Juan Ruiz-Alvarez, Chi-Wang Shu, Dionisio F. Yanez
Summary: This article introduces a new WENO algorithm for approximating derivative values of a function on a non-regular grid. The algorithm adapts ideas from a previous study to design nonlinear weights that maximize accuracy near discontinuities. The article provides proofs, discusses stencil selection, and presents explicit formulas for the weights and smoothness indicators. Numerical experiments are also conducted to validate the theoretical results.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Siting Liu, Stanley Osher, Wuchen Li, Chi -Wang Shu
Summary: In this work, a novel framework for numerically solving time-dependent conservation laws with implicit schemes is proposed. The approach involves casting the initial value problem as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. The flexibility in the choice of time and spatial discretization schemes, as well as the large regions of stability gained from the implicit structure, make this approach advantageous. It is highly parallelizable and easy to implement, while avoiding non-linear inversions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Transportation
Liangze Yang, Chi-Wang Shu, S. C. Wong, Mengping Zhang, Jie Du
Summary: This study investigates the existence and uniqueness of solutions to the Hoogendoorn-Bovy (HB) pedestrian flow model. The results show that the HB model can be transformed into a forward conservation law equation and a backward Hamilton-Jacobi equation, ensuring the existence and uniqueness of solutions for both equations when suitable parameters are chosen.
TRANSPORTMETRICA B-TRANSPORT DYNAMICS
(2023)
Article
Mathematics, Applied
Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang
Summary: Fixed-point fast sweeping WENO methods are efficient high-order numerical methods used to solve steady-state solutions of hyperbolic PDEs. They have high-order accuracy and good performance.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Physics, Mathematical
Jiayin Li, Chi-Wang Shu, Jianxian Qiu
Summary: In this paper, a high-order moment-based multi-resolution HWENO scheme is proposed for hyperbolic conservation laws. The scheme reconstructs the function values at Gauss-Lobatto points using the information of the zeroth and first order moments, leading to improved stability and resolution. Compared to general HWENO and WENO schemes, this moment-based scheme has a more compact stencil size and a higher CFL number restriction.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
