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Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions

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Abstract

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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Notes

  1. This is unfortunately generally not true in the nonlinear case, as the energy might increase after one iteration of the explicit Euler method if no dissipation is added to the numerical solution.

  2. A more rigorous study is clearly needed and will be included in future investigations.

  3. The conference presentation [99] by Tolstykh in 2000 seems to be the earliest reference to RBF-FD methods.

  4. For a discussion on the accuracy of infinitely smooth kernels as well as the role of the shape parameter \(\varepsilon \) see for instance [5, 28, 75, 92] and references therein.

  5. In our implementation we are using the MATLAB function integral for their computation so that strictly speaking, none of our integration is exact. This MATLAB function uses global adaptive quadrature and certain (default) error tolerances.

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Acknowledgements

The authors would like to thank Simon-Christian Klein for helpful advice. This work is partially supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) #GL 927/1-1 (Glaubitz), AFOSR #F9550-18-1-0316 (Glaubitz and Gelb), NSF-DMS #1502640, NSF-DMS #1912685, and ONR #N00014-20-1-2595 (Gelb).

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    Jan Glaubitz & Anne Gelb

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Glaubitz, J., Gelb, A. Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions. J Sci Comput 87, 40 (2021). https://doi.org/10.1007/s10915-021-01453-8

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