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Keywords = unequal-sized stencil

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16 pages, 3006 KB  
Article
A New Type of High-Order Mapped Unequal-Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations
by Zhengwei Hou and Liang Li
Computation 2025, 13(8), 182; https://doi.org/10.3390/computation13080182 - 1 Aug 2025
Viewed by 591
Abstract
In this paper, we propose the MUSWENO scheme, a novel mapped weighted essentially non-oscillatory (WENO) method that employs unequal-sized stencils, for solving nonlinear degenerate parabolic equations. The new mapping function and nonlinear weights are proposed to reduce the difference between the linear weights [...] Read more.
In this paper, we propose the MUSWENO scheme, a novel mapped weighted essentially non-oscillatory (WENO) method that employs unequal-sized stencils, for solving nonlinear degenerate parabolic equations. The new mapping function and nonlinear weights are proposed to reduce the difference between the linear weights and nonlinear weights. Smaller numerical errors and fifth-order accuracy are obtained. Compared with traditional WENO schemes, this new scheme offers the advantage that linear weights can be any positive numbers on the condition that their summation is one, eliminating the need to handle cases with negative linear weights. Another advantage is that we can reconstruct a polynomial over the large stencil, while many classical high-order WENO reconstructions only reconstruct the values at the boundary points or discrete quadrature points. Extensive examples have verified the good representations of this scheme. Full article
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21 pages, 4191 KB  
Article
New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations
by Yan Tan, Hui Lv and Jun Zhu
Mathematics 2023, 11(23), 4842; https://doi.org/10.3390/math11234842 - 1 Dec 2023
Viewed by 1874
Abstract
In this paper, new third-order finite volume unequal-sized weighted essentially non-oscillatory (US-WENO) Lagrangian schemes are designed to solve Euler equations in two and three dimensions. The spatial reconstruction procedures are implemented by using a convex combination of a quadratic polynomial with several linear [...] Read more.
In this paper, new third-order finite volume unequal-sized weighted essentially non-oscillatory (US-WENO) Lagrangian schemes are designed to solve Euler equations in two and three dimensions. The spatial reconstruction procedures are implemented by using a convex combination of a quadratic polynomial with several linear polynomials specified on unequal-sized stencils, so the new US-WENO Lagrangian schemes can achieve the designed third-order accuracy and maintain an essentially non-oscillatory property near strong discontinuities in multi-dimensions. Unlike the traditional WENO reconstruction procedures specified on unstructured meshes, the linear weights of these new two-dimensional and three-dimensional US-WENO spatial reconstructions can be selected as any positive numbers as long as their summation equals one and they are not related to the local mesh topology or the location of quadrature points. Moreover, the linear weights do not have to be recalculated even if the grid moves with the fluid, avoiding the appearance of negative linear weights, thus improving computation efficiency and robustness in multi-dimensional Lagrangian numerical simulations. Finally, extensive benchmark numerical cases are employed to display the excellent capability of the presented US-WENO Lagrangian schemes. Full article
(This article belongs to the Section E: Applied Mathematics)
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12 pages, 907 KB  
Article
A New Fifth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Fractional Differential Equations
by Yan Zhang and Jun Zhu
Fractal Fract. 2022, 6(6), 294; https://doi.org/10.3390/fractalfract6060294 - 27 May 2022
Cited by 2 | Viewed by 2467
Abstract
This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain [...] Read more.
This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order β(0<β<1) is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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