Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions
arXiv:2102.00967 · doi:10.1007/s10915-021-01453-8
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.
To appear in Journal of Scientific Computing
References in corpus (12)
- Review of Summation-by-parts schemes for initial-boundary-value problems
- Summation-by-parts operators for correction procedure via reconstruction
- Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
- Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
- Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
- Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods
- Application of Modal Filtering to a Spectral Difference Method
- Stable high-order cubature formulas for experimental data
- Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
- Shock Capturing by Bernstein Polynomials for Scalar Conservation Laws
- Stable High Order Quadrature Rules for Scattered Data and General Weight Functions
- High Order Edge Sensors with Regularization for Enhanced Discontinuous Galerkin Methods
Cited by in corpus (7)
- Summation-by-parts operators for general function spaces
- Construction and application of provable positive and exact cubature formulas
- Towards stability of radial basis function based cubature formulas
- Towards stability results for global radial basis function based quadrature formulas
- Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
- Summation-by-parts operators for general function spaces: The second derivative
- Stability estimates for radial basis function methods applied to linear scalar conservation laws