A kernel-based discretisation method for first order partial differential equations
A kernel-based discretisation method for first order partial differential equations
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DOI：
10.1090/mcom/3265
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发表时间：
2016-01
期刊：
影响因子：
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通讯作者：
Tobias Ramming;H. Wendland
中科院分区：
文献类型：
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作者：
Tobias Ramming;H. Wendland
我们推导出一种新的离散方法的任意空间维度的一阶偏微分方程，这是基于无网格的空间近似。这种空间近似类似于SPH（光滑粒子流体动力学）技术，是一种典型的基于核的方法。然而，它与SPH方法有很大的不同，因为它采用欧拉方法而不是拉格朗日方法。我们证明了所得到的半离散计划的稳定性和收敛性在一定的光滑性假设下的PDE的定义函数。近似阶取决于底层核和解的光滑性。因此，我们还回顾了一个简单的方法来构建光滑的内核产生任意收敛阶。最后，我们给出了一个数值例子，通过测试我们的方法在一维Burgers方程的情况下。
We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique and is a typical kernel-based method. It differs, however, significantly from the SPH method since it employs an Eulerian and not a Lagrangian approach. We prove stability and convergence for the resulting semi-discrete scheme under certain smoothness assumptions on the defining function of the PDE. The approximation order depends on the underlying kernel and the smoothness of the solution. Hence, we also review an easy way of constructing smooth kernels yielding arbitrary convergence orders. Finally, we give a numerical example by testing our method in the case of a one-dimensional Burgers equation.