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首页> 外文期刊>Journal of Computational Physics >The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
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The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems

机译：在局部RBF Hermitian方法中使用PDE中心解决3D对流扩散问题

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摘要

In this work an extension is proposed to the Local Hermitian Interpolation (LHI) method; a meshless numerical method based on interpolation with small and heavily overlapping radial basis function (RBF) systems. This extension to the LHI method uses interpolation functions which themselves satisfy the partial differential equation (PDE) to be solved. In this way, a much improved reconstruction of partial derivatives can be obtained, resulting in significantly improved accuracy in many cases. The implementation algorithm is described, and is validated via three convection-diffusion-reaction problems, for steady and transient situations. A Crank-Nicolson implicit time stepping technique is used for the time-dependent problems. In the proposed approach, a form of 'analytical upwinding' is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function, which includes the desired information about the convective velocity field. The implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one-dimensional travelling front problem at Péclet numbers of 500, 1000, 2000, 5000 and infinity, which corresponds to a shock front in the case of infinity. In addition, the accuracy of the numerical scheme is validated against a one-dimensional steady state solution exhibiting strong boundary layer effects, and also against a steady and a transient three-dimensional convection-diffusion problem on irregular datasets. All the test cases are validated against the corresponding analytical solutions. Finally, the effect of various interpolation stencil configurations is investigated, and some important limitations on local data-centre distribution are identified.
机译：在这项工作中，提出了对局部埃尔米特插值（LHI）方法的扩展。一种基于插值的无网格数值方法，该方法具有较小且高度重叠的径向基函数（RBF）系统。 LHI方法的扩展使用了本身满足待求解偏微分方程（PDE）的内插函数。以此方式，可以获得大大改进的偏导数重构，从而在许多情况下可显着提高精度。描述了实现算法，并通过三个对流扩散反应问题对稳态和瞬态情况进行了验证。 Crank-Nicolson隐式时间步进技术用于与时间有关的问题。在提出的方法中，通过在插值函数中使用控制方程的偏微分算子隐式地实现了“分析迎风”的形式，其中包括有关对流速度场的所需信息。通过解决500、1000、2000、5000和无穷大的佩克莱特数时的一维行进前沿问题，测试了所提出的数值方法固有的隐式迎风方案，该问题对应于无穷大情况下的激波前沿。此外，该数值方案的准确性针对一维表现出强边界层效应的稳态解，也针对非规则数据集的稳态和瞬态三维对流扩散问题进行了验证。所有测试用例均根据相应的分析解决方案进行了验证。最后，研究了各种插值模板配置的影响，并确定了对本地数据中心分布的一些重要限制。

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