Abstract
Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose a high-order ENO method based on radial basis function (RBF) to solve hyperbolic conservation laws on general two-dimensional grids. The radial basis function reconstruction offers a flexible way to deal with ill-conditioned cell constellations. We introduce a smoothness indicator based on RBFs and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting which circumvents the stagnation of the error and keeps the condition number of the reconstruction bounded. We conclude with several challenging numerical examples in two dimensions to show the robustness of the method.
Highlights
Solving systems of hyperbolic conservation laws with high-order methods continues to attract substantial interest
A generalization of the finite volume method is the class of Discontinuous Galerkin (DG) finite element methods [7], for which it is necessary to add limiters to ensure non-oscillatory approximations [22]
There exist several approaches that combine radial basis function (RBF) with finite volume methods, e.g. a highorder weighted ENO (WENO) approach based on polyharmonics [1], a high-order WENO approach based on multiquadratics [6], a high-order RBF based CWENO method [21] and an entropy stable RBF based essentially non-oscillatory (ENO) method [20]
Summary
Solving systems of hyperbolic conservation laws with high-order methods continues to attract substantial interest.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
