High order difference schemes using the local anisotropic basis function method
High order difference schemes using the local anisotropic basis function method
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使用局部各向异性基函数方法的高阶差分格式
DOI：
10.1016/j.jcp.2020.109549
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发表时间：
2020
影响因子：
4.1
通讯作者：
King J
中科院分区：
文献类型：
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作者：
King J
无网格方法在复杂几何形状的模拟中具有巨大的潜力，因为它避免了网格生成的耗时过程。光滑粒子流体动力学（SPH）是最广泛使用的无网格方法，但缺乏一致性。高阶，一致的，和本地（使用紧凑的计算支架）无网格方法是特别可取的。在这里，我们提出了一种新的框架，用于生成任意节点分布的局部高阶差分算子，称为局部各向异性基函数方法（LABFM）。权重由各向异性基函数（ABF）的线性和构造，选择这些基函数是为了确保多项式域的精确再现达到给定阶。ABF是基于一个基本的径向基函数（RBF），基本RBF的选择对精度的影响不大，但影响稳定性。LABFM能够生成具有紧凑计算规模的高阶差分算子（二维中具有N <$25个节点的4阶，具有N <$60个节点的8阶）。在域边界（不完全支持）LABFM自动提供与内部方案相同阶数的单侧差异，最高可达4阶。我们使用的方法来解决椭圆，抛物和混合双曲抛物偏微分方程（PDE），显示了8阶收敛。包含高粘度是直接的，并且在求解双曲问题时可以有效地提供稳定性。LABFM是一种新的无网格方法，用于求解复杂几何条件下的偏微分方程。该方法具有高度的可扩展性，对于欧拉格式，在给定精度下，计算效率与RBF-FD具有竞争力。一个特别有吸引力的特点是，在低阶限制，LABFM崩溃光滑粒子流体动力学（SPH），并有可能为任意拉格朗日-欧拉计划与自然适应的分辨率和精度。
Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack of consistency. High order, consistent, and local (using compact computational stencils) mesh-free methods are particularly desirable. Here we present a novel framework for generating local high order difference operators for arbitrary node distributions, referred to as the Local Anisotropic Basis Function Method (LABFM). Weights are constructed from linear sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction of polynomial fields up to a given order. The ABFs are based on a fundamental Radial Basis Function (RBF), and the choice of fundamental RBF has small effect on accuracy, but influences stability. LABFM is able to generate high order difference operators with compact computational stencils (4th order with N≈ 25 nodes, 8th order with N≈ 60 nodes in two dimensions). At domain boundaries (with incomplete support) LABFM automatically provides one-sided differences of the same order as the internal scheme, up to 4th order. We use the method to solve elliptic, parabolic and mixed hyperbolic-parabolic partial differential equations (PDEs), showing up to 8th order convergence. The inclusion of hyperviscosity is straightforward, and can effectively provide stability when solving hyperbolic problems. LABFM is a promising new mesh-free method for the numerical solution of PDEs in complex geometries. The method is highly scalable, and for Eulerian schemes, the computational efficiency is competitive with RBF-FD for a given accuracy. A particularly attractive feature is that in the low order limit, LABFM collapses to Smoothed Particle Hydrodynamics (SPH), and there is potential for Arbitrary Lagrangian-Eulerian schemes with natural adaptivity of resolution and accuracy.
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影响因子：
4.1
作者：
Lind, S. J.;Xu, R.;Rogers, B. D.
通讯作者：
Rogers, B. D.
DOI：
10.1002/fld.1650200824
发表时间：
1995-04-30
影响因子：
1.8
作者：
LIU, WK;JUN, S;ZHANG, YF
通讯作者：
ZHANG, YF
影响因子：
4.1
作者：
Lind, S. J.;Stansby, P. K.
通讯作者：
Stansby, P. K.
Hiroshi Taniguchi：国际工程数值方法杂志。
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