Article
Computer Science, Interdisciplinary Applications
Meiqi Tan, Juan Cheng, Chi-Wang Shu
Summary: Time discretization is crucial for time-dependent partial differential equations (PDEs). This paper discusses different time-marching methods and their limitations. The EIN method, which involves adding and subtracting a large linear highest derivative term, is proposed as a solution for equations with nonlinear high derivative terms.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Haijin Wang, Anping Xu, Qi Tao
Summary: This paper presents the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations, and analyzes the stability and error estimates of the corresponding fully discrete schemes by coupling with a specific Runge-Kutta type implicit-explicit time discretization. Numerical experiments are conducted to verify the theoretical results.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Haijin Wang, Xiaobin Shi, Qiang Zhang
Summary: In this paper, stability analysis and optimal error estimates are presented for fully discrete schemes solving the one-dimensional linear convection-diffusion equation with periodic boundary conditions. The schemes use local discontinuous Galerkin (LDG) spatial discretization methods coupled with implicit-explicit (IMEX) temporal discretization methods based on backward difference formulas (BDF). A general framework of stability analysis is established for the corresponding fully discrete LDG-IMEX-BDF schemes up to fifth order in time by combining improved multiplier technique and LDG methods. The schemes are proven to be unconditionally stable and achieve optimal orders of accuracy in both space and time by energy analysis. Numerical tests are conducted to validate the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Haijin Wang, Fengyan Li, Chi-wang Shu, Qiang Zhang
Summary: In this paper, the stability of fully discrete methods for the linear convection-diffusion equation in one dimension with periodic boundary conditions is analyzed. Two general frameworks of energy-method based stability analysis are established using forward and backward temporal differences. The fully discrete schemes are shown to have monotonicity stability under certain time step conditions, and numerical experiments are conducted to provide stricter time step conditions for practical use.
MATHEMATICS OF COMPUTATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Chen Liu, Xiangxiong Zhang
Summary: In this paper, a scheme for solving compressible Navier-Stokes equations with desired properties is constructed. The scheme achieves high order spatial accuracy, conservation, and positivity-preserving of density and internal energy. Numerical tests show that higher order polynomial basis produces better numerical solutions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Shashank Jaiswal
Summary: Adaptivity is crucial for addressing practical challenges, especially in computational fluid dynamics workflow. The mixed non-conforming discontinuous Galerkin discretization method is introduced for the full Boltzmann equation, providing optimal convergence for non-linear kinetic systems on non-orthogonal grids. The method allows for analysis of complex problems on massively parallel scales and is applicable to a wide range of rarefied flows. The computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible compared to conforming domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
A. C. W. Creech, A. Jackson
Summary: This paper introduces a hybrid approach for explicitly-filtered Large Eddy Simulation using a Discontinous Galerkin discretisation for velocity, which incorporates information from a Continuous Galerkin version of the velocity field to improve computational performance while maintaining stability and accuracy.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Sohail Reddy, Maciej Waruszewski, Felipe A. V. de Braganca Alves, Francis X. Giraldo
Summary: This work presents IMplicit-EXplicit (IMEX) formulations for discontinuous Galerkin (DG) discretizations of the compressible Euler equations governing non-hydrostatic atmospheric flows. Two different IMEX formulations are proposed to address the stiffness problem caused by the governing dynamics and the domain discretization. Efficient Schur complements are derived for both equation sets, and their performance is studied on 2D and 3D test problems, showing their convergence rates and efficiency in mesoscale and global applications.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Qin Zhou, Binjie Li
Summary: The study focuses on optimal Dirichlet boundary control for a fractional/normal evolution equation with a final observation. It derives the unique existence of the solution and the first-order optimality condition of the optimal control problem. The convergence of a temporally semidiscrete approximation is rigorously established to verify the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Wenjing Feng, Hui Guo, Yue Kang, Yang Yang
Summary: In this paper, we introduce a novel SIPEC time marching method for the coupled system of two-component compressible miscible displacements. By incorporating a correction stage in each time step, we achieve second-order accuracy while maintaining bound preservation for the concentration equation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Hamdullah Yucel
Summary: This study focuses on goal-oriented a posteriori error estimates for the numerical approximation of Dirichlet boundary control problem on a two dimensional convex polygonal domain, using the local discontinuous Galerkin method for discretization. Primal-dual weighted error estimates are derived for the objective functional, with numerical examples presented to demonstrate the performance of the estimator.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaosheng Li, Wei Wang
Summary: In this work, an efficient high-order discontinuous Galerkin (DG) method is developed for solving the Electrical Impedance Tomography (EIT) problem. A new optimization problem is proposed to recover the conductivity from the Dirichlet-to-Neumann map, aiming to minimize the mismatch between the predicted and measured currents on the boundary. The numerical results demonstrate the high accuracy and efficiency of the proposed high-order DG method for various two-dimensional problems, including single and multiple inclusions. Analysis and computation for discontinuous conductivities are also investigated in this work.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Divay Garg, Kamana Porwal
Summary: In this article, the Dirichlet boundary control problem governed by the Poisson equation is studied, and symmetric discontinuous Galerkin finite element methods are designed and analyzed for its numerical approximation. The discrete optimality system is obtained by exploiting the symmetric property of the bilinear forms. By utilizing various intermediate problems, the optimal order convergence rates are obtained for the control in the energy and L-2 norms. Additionally, an a posteriori error estimator is derived using an auxiliary system of equations, which is proven to be reliable and efficient. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
C. Pagliantini, G. Manzini, O. Koshkarov, G. L. Delzanno, V. Roytershteyn
Summary: This study investigates the conservation properties of the Hermite-DG approximation of the Vlasov-Maxwell equations. The total mass is preserved independently for each plasma species in this semi-discrete formulation. The presence of central numerical fluxes in the DG approximation of Maxwell's equations leads to the existence of an energy invariant, while upwind fluxes introduce a dissipative term. We analyze the capability of explicit and implicit Runge-Kutta temporal integrators in preserving these invariants and propose modified explicit RK methods known as relaxation Runge-Kutta methods to ensure energy preservation.
COMPUTER PHYSICS COMMUNICATIONS
(2023)
Article
Mathematics
Martin Dindos, Jungang Li, Jill Pipher
Summary: The notion of p-ellipticity has been extended to second order elliptic systems, with the definition of three new notions of p-ellipticity established and their important roles in solving boundary value problems demonstrated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(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
Sergio Amat, Juan Ruiz-Alvarez, Chi-Wang Shu, Dionisio F. Yanez
Summary: This paper introduces a new algorithm to improve the results of the WENO-(2r - 1) algorithm near singularities, as well as its application in signal processing, with theoretical results confirmed through numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Jun Zhu, Chi-Wang Shu, Jianxian Qiu
Summary: The study applies high-order multi-resolution WENO techniques as limiters to solve steady-state problems, with a new troubled cell indicator designed to detect cells needing limiting procedures. The methods gradually degrade accuracy near strong discontinuities to eliminate oscillations and reduce numerical residuals to machine zero.
APPLIED NUMERICAL 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
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)