Robust Discontinuity Indicators for High-Order Reconstruction of Piecewise Smooth Functions

Abstract

In many applications, piecewise continuous functions are commonly interpolated over meshes. However, accurate high-order manipulations of such functions can be challenging due to potential spurious oscillations known as the Gibbs phenomena. To address this challenge, we propose a novel approach, Robust Discontinuity Indicators (RDI), which can efficiently and reliably detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged values. We present a detailed analysis focusing on its derivation and the dual-thresholding strategy. A key advantage of RDI is its ability to handle potential inaccuracies associated with detecting discontinuities on non-uniform meshes, thanks to its innovative discontinuity indicators. We also extend the applicability of RDI to handle general surfaces with boundaries, features, and ridge points, thereby enhancing its versatility and usefulness in various scenarios. To demonstrate the robustness of RDI, we conduct a series of experiments on non-uniform meshes and general surfaces, and compare its performance with some alternative methods. By addressing the challenges posed by the Gibbs phenomena and providing reliable detection of discontinuities, RDI opens up possibilities for improved approximation and analysis of piecewise continuous functions, such as in data remap.