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

The accurate reconstruction of piecewise continuous functions on meshes is challenging due to potential spurious oscillations—namely the Gibbs phenomenon—especially for high-order methods. This paper introduces the Robust Discontinuity Indicators (RDI) method, a novel technique for constructing discontinuity indicators. These indicators can effectively identify both (Formula presented.) and (Formula presented.) discontinuities in a single pass using a new comprehensive theoretical analysis combined with cell-based overshoot–undershoot indicators and node-based oscillation indicators. In addition to detecting discontinuities, these indicator values can also facilitate the construction of adaptive weighting schemes to mitigate the Gibbs phenomenon. Due to its flexibility, RDI can accommodate complex geometries and applies to nonuniform unstructured meshes and general surfaces, broadening its utility. Through experiments, we show that RDI can accurately capture discontinuities while producing fewer false positives than two-pass methods. By providing a more rigorous method for discontinuity detection, RDI has the potential to offer significant improvements in computational simulations and data remapping.