Article
Computer Science, Interdisciplinary Applications
Chuan Fan, Xiangxiong Zhang, Jianxian Qiu
Summary: In this paper, a high order weighted essentially non-oscillatory (WENO) finite difference discretization method is constructed for solving the compressible Navier-Stokes (NS) equations. The method achieves positivity preservation of density and internal energy through a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The core advantages of the proposed method are robustness and efficiency, making it particularly suitable for solving challenging problems involving low density and low pressure flow regime.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Zhuang Zhao, Min Zhang
Summary: In this paper, a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme is proposed for the shallow water equations with non-flat bottom topography in pre-balanced form. The scheme achieves well-balanced property by balancing the flux gradients and source terms using the idea of WENO-XS scheme. The HWENO scheme reconstructs the fluxes in the original equations using nonlinear HWENO reconstructions and approximates other fluxes in the derivative equations using high-degree polynomials directly. An HWENO limiter is applied to control spurious oscillations and maintain well-balanced property.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Yifei Wan, Yinhua Xia
Summary: “The numerical simulation of strong shock waves in steady-state Euler equation solutions is a challenging problem. In this paper, a new hybrid WENO scheme is proposed, which can better simulate steady-state solutions with higher convergence and less error, while maintaining oscillation-free characteristics.”
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Chuan Fan, Zhuang Zhao, Tao Xiong, Jianxian Qiu
Summary: In this paper, a robust fifth order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations is proposed. The proposed scheme performs flux reconstructions in the finite difference framework without using the derivative value of a target cell, resulting in a simpler and more robust scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Xiaokai Huo, Hailiang Liu
Summary: This study introduces a new fully-discretized finite difference scheme for solving quantum diffusion equations, which preserves positivity and energy stability. The challenge of proving positivity preservation for fourth order partial differential equations is addressed by reformulating the scheme as an optimization problem based on a variational structure and utilizing the singular nature of the energy functional near boundary values. The scheme is also shown to be mass conservative and consistent.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Yaguang Gu, Zhen Gao, Guanghui Hu, Peng Li, Qingcheng Fu
Summary: In this paper, a fifth order well-balanced positivity-preserving finite difference scale-invariant AWENO scheme is proposed for the compressible Euler equations. The scheme is designed to ensure well-balancedness and positivity of the density and pressure throughout the computation. The use of the Si-WENO operator and interpolation-based and flux-based positivity-preserving limiters improve computational efficiency.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Deniz A. Bezgin, Steffen J. Schmidt, Nikolaus A. Adams
Summary: Neural networks have been integrated into the WENO scheme to address challenges in achieving maximum-order convergence and ENO property, demonstrating good generalizability and performance in various test cases. The WENO3-NN scheme learns a non-trivial dispersion-dissipation relation and may introduce vanishing dissipation near the cutoff wavenumber, which is counterintuitive to classical discretization-design principles.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Yan Zhang, Weihua Deng, Jun Zhu
Summary: In this paper, a new sixth-order WENO scheme is proposed for solving fractional differential equations, which has lower computational cost and simple engineering applications.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Lavanya V. Salian, Rathan Samala
Summary: In this work, a fifth-order weighted essentially non-oscillatory (WENO) scheme with exponential approximation space is proposed for solving dispersive equations. The scheme uses an exponential approximation space with a tension parameter optimized for the characteristic data, avoiding spurious oscillations. The effectiveness of the scheme is demonstrated through detailed formulation and numerical examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Chen Li, Dong Sun, Qilong Guo, Pengxin Liu, Hanxin Zhang
Summary: A new hybrid central-type scheme is proposed in this paper, which includes WENO4-LC for shock capturing and a low dissipative third-order grid-centered upwind scheme, with the extension of a discontinuity indicator to the four-point stencil version. Numerical tests show that the new indicator is good at detecting discontinuities, and the performance of the hybrid scheme is significantly improved compared to classical WENO schemes, even slightly better in resolving fluid structures.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics
Omer Musa, Guoping Huang, Mingsheng Wang
Summary: The paper proposes a new simple smoothness indicator for fifth-order linear reconstruction, reducing the complexity of the WENO-AO(5,3) scheme. It modifies the WENO-AO(5,3) scheme to the WENO-O scheme with a new and simple formulation, showing through numerical experiments its accuracy and efficacy compared to original schemes. The results indicate that the proposed WENO-O scheme is not only comparable in accuracy and efficacy to the original scheme but also decreases computational cost and complexity.
Article
Mathematics, Applied
Bao-Shan Wang, Peng Li, Zhen Gao
Summary: In this study, a high-order, well-balanced, and positivity-preserving finite-difference scheme is designed for solving the shallow water equations with or without dry areas. The hydrostatic reconstruction method is used to achieve balance and preserve the positivity of the water height.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mechanics
Peixun Yu, Xiao Han, Heye Xiao, Jiakuan Xu, Junqiang Bai
Summary: The study focuses on the development and application of differential schemes for the numerical simulation of aeroacoustics over complex geometries. A hybrid interpolation method and conservative metric method are used to discretize the acoustic and geometric variables, resulting in higher resolution and lower dispersion. The developed mapping functions effectively suppress spurious oscillations and improve the resolution of discontinuous regions. Numerical results demonstrate that the new scheme has better discontinuity capture capability and is suitable for computational aeroacoustics of complex geometries.
Article
Mathematics, Applied
Liang Li, Jun Zhu
Summary: This paper introduces a new fifth-order finite difference MUS-WENO scheme, which can achieve smaller truncation errors and optimal accuracy in smooth regions when solving multi-dimensional Hamilton-Jacobi equations. The scheme has good convergence, robustness, efficiency, and easy extension to multiple dimensions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Sangkwon Kim, Chaeyoung Lee, Hyun Geun Lee, Hyundong Kim, Soobin Kwak, Youngjin Hwang, Seungyoon Kang, Seokjun Ham, Junseok Kim
Summary: This study introduces an unconditionally stable positivity-preserving numerical method for the Fisher-KPP equation in one-dimensional space, demonstrating its stability, boundedness, and positivity-preserving properties through computational experiments.
DISCRETE DYNAMICS IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Rakesh Kumar, S. Baskar
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2016)
Article
Mathematics, Applied
Rakesh Kumar, Ashok Choudhary, S. Baskar
APPLICABLE ANALYSIS
(2020)
Article
Computer Science, Interdisciplinary Applications
Rakesh Kumar, Praveen Chandrashekar
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Computer Science, Interdisciplinary Applications
Rakesh Kumar, Praveen Chandrashekar
COMPUTERS & FLUIDS
(2019)
Article
Mathematics, Applied
Samala Rathan, Rakesh Kumar, Ameya D. Jagtap
APPLIED MATHEMATICS AND COMPUTATION
(2020)
Article
Mathematics, Applied
Praveen Chandrashekar, Rakesh Kumar
JOURNAL OF SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Dinshaw S. Balsara, Rakesh Kumar, Praveen Chandrashekar
COMMUNICATIONS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE
(2021)
Article
Computer Science, Interdisciplinary Applications
Asha K. Dond, Rakesh Kumar
Summary: The author developed a Modified WENO (MWENO) scheme to address the limitations of the original WENO reconstruction in capturing composite structure for non-convex flux while ensuring entropy convergence. By utilizing a new troubled-cell indicator based on the smoothness indicator of the WENO reconstruction, the MWENO reconstruction procedure effectively handles troubled and non-troubled cells separately, as demonstrated in numerical experiments for 1D and 2D test cases, confirming the effectiveness of the proposed scheme.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Acoustics
Anurag Kumar, Bhavneet Kaur, Rakesh Kumar
Summary: In this work, a new and improved fifth-order finite difference WENO scheme is developed for approximating solutions to hyperbolic conservation laws and associated problems. The scheme incorporates higher-order information and a new global smoothness indicator, resulting in high resolution, fifth-order accuracy, and robustness compared to other modern WENO schemes.
Article
Computer Science, Interdisciplinary Applications
Rakesh Kumar, Praveen Chandrashekar
Summary: The article proposes two new algorithms for multi-level schemes in order to improve computational efficiency while maintaining accuracy. The algorithms utilize the troubled-cell indicator developed from the extra smoothness indicator of multi-level schemes and employ different reconstruction strategies based on the characteristics of the cells. Numerical experiments demonstrate that the proposed algorithms require 15%-75% less computational time compared to multi-level schemes, while preserving the advantageous features of multi-level schemes.
COMPUTERS & FLUIDS
(2022)
Article
Acoustics
Ameya D. Jagtap, Rakesh Kumar
Article
Computer Science, Software Engineering
Rakesh Kumar
BIT NUMERICAL MATHEMATICS
(2018)