Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Abstract

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley-Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.

A New Class of High-Order Methods for Fluid Dynamics Simulations Using Gaussian Process Modeling: One-Dimensional Case

“…On the contrary, RBF models in the context of interpolations and reconstructions follow the parametric modeling paradigm, requiring to solve a linear system Aλ = F (see e.g., [4,31]) whose size is determined and fixed by the number of desired N interpolation points f = [f (x 1 ), . .…”

“…These studies focused on designing their RBF methods with the use of adaptive shape parameters to control local errors. Also in [4], two types of multiquadrics and polyharmonic spline RBFs were used to model the Euler equations with a strategy of selecting optimal shape parameters for different RBF orders. Stability analysis on the fully discretized hyperbolic PDEs in both space and time using the multiquadrics RBF is reported in [9].…”

Entropy stable essentially nonoscillatory methods based on RBF reconstruction

“…The fourth order entropy conservative flux with coefficients α 2 p 4 3 , ¡ 1 6 q and the sixth order scheme with α 3 p 3 2 , ¡ 3 10 , 1 30 q present two explicit examples.…”

“…The combination of the RBF interpolation with finite volume methods works analogeously to the RBF-TeCNOp Method. Aboiyar et al [1] combine in their work a high-order WENO approach with a polyharmonic spline reconstruction and Bigoni et al [3] apply a high-order WENO approach to multiquadratics.…”

Two-Dimensional RBF-ENO Method on Unstructured Grids

“…There exist several approaches that combine RBFs with finite volume methods, e.g. a highorder 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 ENO method [20]. However, most of these are suitable only for one-dimensional grids or are at most second order accurate.…”