Two-Dimensional RBF-ENO Method on Unstructured Grids

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. Keywords Finite volume methods • Euler equations • High-order methods • Unstructured grids • Radial basis functions Mathematics Subject Classification 35L65 • 65M08 • 65D05 • 65M12 This work has been supported by SNSF (200021_165519).

Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

“…The effectiveness of the residual viscosity method when an oversampled RBF-FD method is used, is displayed in Figure 10, where we show the spatial distribution of the residual (24) and the coefficient ε given in (21). We observe that the residual is giving the information about the position of the discontinuity (large oscillations) present in the solution.…”

“…Weighted ENO (WENO) methods introduced in [22] use a convex combination of the candidate interpolation stencils to reconstruct a non-oscillatory function. There exist many subsequent papers on (W)ENO methods, see e.g., [23,24,25], where the authors develop (W)ENO methods using the RBF interpolants and by that allow approximations on unstructured grids.…”

“…where the residuals R * are defined for each equation in (18), analogously to the formulation for the scalar conservation law in (24). We have:…”

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

“…RBFs have become powerful tools in multivariate interpolation and approximation theory, since they are easy to implement, allow arbitrary scattered data, and can be spectrally accurate. They are also often used to solve numerical partial differential equations (PDEs) [67,22,63,57,66,72,83,54,55]. In this regard, although RBFs are considered to be a viable alternative to traditional methods such as finite difference (FD), finite element (FE) and spectral methods, investigations into their stability are still underdeveloped and/or unsatisfactory.…”