用弱强迫边界条件稳定化守恒律方程的径向基函数方法
Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions
摘要：
众所周知，边界条件（BC）可能会导致全局径向基函数（RBF）方法变得不稳定的双曲守恒律（CL）。在这里，我们调查这一现象，并确定强有力的执行BC的机制，触发这种稳定性问题。基于这一观察，我们提出了一种技术，在RBF方法中弱执行BC。在双曲CL的情况下，这是通过从CL的弱形式而不是通常强制的强形式仔细构建RBF方法来实现的。此外，我们证明了全局RBF方法可能违反守恒，当近似不考虑这些因素时，会产生物理上不合理的解决方案。数值实验验证了我们的理论结果。
英文摘要：
It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.
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