RBF WENO Reconstructions with Adaptive Order and Applications to Conservation Laws
RBF WENO Reconstructions with Adaptive Order and Applications to Conservation Laws
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DOI：
10.1007/s10915-022-01827-6
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发表时间：
2022-04
影响因子：
2.5
通讯作者：
T. Arbogast;Chieh-Sen Huang;Ming-Hsien Kuo
中科院分区：
文献类型：
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作者：
T. Arbogast;Chieh-Sen Huang;Ming-Hsien Kuo
本文提出了一种有限体积径向基函数（RBF）逼近多维网格单元模板上的函数的框架。给出了逼近的存在性定理。在一维情况下，当胞元直径趋于零时，数值结果表明当径向基函数无穷可微时，径向基函数逼近收敛到与多项式逼近相同的阶数.具体的多二次径向基函数上的2和3网格单元的支架被证明具有这种收敛性质。提出了一种基于两层径向基函数的自适应阶数加权基本无振荡（韦诺）重构算法（RBF-WENO-AO）。WENO-AO重建使用任意线性权重，因此可以很容易地开发RBF近似，即使在多维非均匀网格上。在经典的基于多项式的韦诺之后，为重建定义了平滑度指标。在一维情形下，给出了光滑和间断两种情形的收敛定理。这些重建适用于发展有限体积格式的双曲型守恒律的非均匀网格在多维空间。重点是基于多二次径向基函数的重建，当解是光滑的时，多二次径向基函数是三阶的，否则是二阶的，即，RBF-WENO-AO（3，2）.数值算例表明，该格式在求解双曲型守恒律方程时保持了适当的精度，并具有基本无振荡性。
This paper develops a framework for finite volume radial basis function (RBF) approximation of a functionuon a stencil of mesh cells in multiple dimensions. The theory of existence of the approximation is given. In one dimension, as the cell diameters tend to zero, numerical evidence is given to show that the RBF approximation converges touto the same order as a polynomial approximation when the RBF is infinitely differentiable. Specific multiquadric RBFs on stencils of 2 and 3 mesh cells are proven to have this convergence property. A two-level RBF based weighted essentially non-oscillatory (WENO) reconstruction with adaptive order (RBF-WENO-AO) is developed. WENO-AO reconstructions use arbitrary linear weights, and so they can be developed easily for RBF approximations, even on nonuniform meshes in multiple dimensions. Following the classical polynomial based WENO, a smoothness indicator is defined for the reconstruction. For one dimension, the convergence theory is given regarding the cases whenuis smooth and whenuhas a discontinuity. These reconstructions are applied to develop finite volume schemes for hyperbolic conservation laws on nonuniform meshes over multiple space dimensions. The focus is on reconstructions based on multiquadric RBFs that are third order when the solution is smooth and second order otherwise, i.e., RBF-WENO-AO(3,2). Numerical examples show that the scheme maintains proper accuracy and achieves the essentially non-oscillatory property when solving hyperbolic conservation laws.