Article
Mathematics, Applied
Yan-Qun Jiang, Shu-Guang Zhou, Xu Zhang, Ying-Gang Hu
Summary: This paper presents a fifth-order weighted compact nonlinear scheme for solving one- and two-dimensional Hamilton-Jacobi equations. The performance of the method in suppressing numerical oscillations is demonstrated through numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2022)
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
Engineering, Multidisciplinary
Rooholah Abedian
Summary: This paper presents a modified WENO scheme that can adapt between linear upwind and WENO schemes and calculate the numerical flux through reconstruction. By comparing with other schemes and conducting examples, the robustness and efficiency of this scheme are demonstrated.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Wonho Han, Kwangil Kim, Unhyok Hong
Summary: In this study, high-order numerical methods for solving Hamilton-Jacobi equations are investigated. Firstly, new clear concise nonlinear weights are introduced and their convex combination is improved to develop WENO schemes based on Zhu and Qiu (2017). Secondly, an algorithm for constructing a convergent adaptive WENO scheme is presented by applying a simple adaptive step on the proposed WENO scheme, which utilizes a new singularity indicator. Detailed numerical experiments on various problems, including nonconvex ones, are conducted to demonstrate the convergence and effectiveness of the adaptive WENO scheme.
APPLICATIONS OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Chang Ho Kim, Youngsoo Ha, Hyoseon Yang, Jungho Yoon
Summary: In this study, a novel third-order weighted essentially non-oscillatory (WENO) method is proposed to solve Hamilton-Jacobi equations. By incorporating exponential polynomials to construct numerical fluxes and smoothness indicators, the method efficiently distinguishes singular regions from smooth regions and yields improved results around steep gradients. The scheme maintains optimal order of accuracy in smooth areas, even near critical points, as demonstrated with numerical results and comparisons with other WENO schemes.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Ruo Li, Wei Zhong
Summary: In this work, a simple modified mapping technique is proposed to address the issue of spurious oscillations in the component-wise WENO-IM(2, 0.1) scheme, without causing any extra computational cost. The proposed scheme is validated and evaluated through benchmark tests using one-dimensional and two-dimensional Euler equations. The results demonstrate that the new scheme achieves the formal order of accuracy in smooth regions and is as robust as the characteristic-wise WENO-IM(2, 0.1) scheme. Additionally, the new scheme is about 30% to 50% faster in simulating two-dimensional problems compared to the characteristic-wise WENO-IM(2, 0.1) scheme, indicating its excellent efficiency.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Fuxing Hu
Summary: This paper explores a latent advantage of WENO-Z schemes, demonstrating their optimal accuracy in both smooth and non-smooth regions. The fifth-order WENO-Z scheme is shown to be a nonlinear combination of a five-cell stencil and three three-cell stencils, with different dominance in different regions to compress nonphysical oscillations. The study reveals that unlike traditional WENO schemes, WENO-Z schemes do not require linear optimal weights for their performance.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
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
William M. Feldman, Jean-Baptiste Fermanian, Bruno Ziliotto
Summary: The article provides an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with a non-convex Hamiltonian.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
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
Computer Science, Interdisciplinary Applications
Xin Luo, Song-ping Wu
Summary: The study discusses the limited improvement of the WENO-Z+ scheme in solving multiscale problems and proposes the WENO-Z+I scheme, which significantly improves multiscale resolution. The spectral properties of WENO-Z+ type schemes are found to be better than the corresponding linear upwind scheme.
COMPUTERS & FLUIDS
(2021)
Article
Computer Science, Interdisciplinary Applications
Rooholah Abedian
Summary: The authors of this research paper have introduced a new method for solving Hamilton-Jacobi equations, called third-order WENO method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. Numerical results demonstrate the effectiveness of the new method.
INTERNATIONAL JOURNAL OF MODERN PHYSICS C
(2023)
Article
Computer Science, Interdisciplinary Applications
Xin Luo, Song-ping Wu
Summary: Increasing the weight of the less-smooth substencil is an effective way to improve the resolution of the WENO scheme. By assigning weights greater than the corresponding ideal weights to the less-smooth substencils, WENO-Z+ type schemes can achieve spectral properties and accuracy superior to the underlying linear scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics
Kaizhi Wang, Lin Wang, Jun Yan
Summary: The paper provides necessary and sufficient conditions for the existence of viscosity solutions of nonlinear first order PDEs, proving compactness of the set of solutions. Furthermore, it explores the long-term behavior of viscosity solutions for Cauchy problems using weak KAM theory and dynamic methods.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Julian Fischer, Stefan Neukamm
Summary: This study derives optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. The results show error estimates of different orders in different dimensions, indicating the importance of considering dimensionality in homogenization analysis.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
