Genuinely multi-dimensional explicit and implicit generalized Shapiro filters for weather forecasting, computational fluid dynamics and aeroacoustics

This paper addresses the extension of one-dimensional filters in two and three space dimensions. A new multi-dimensional extension is proposed for explicit and implicit generalized Shapiro filters. We introduce a definition of explicit and implicit generalized Shapiro filters that leads to very simple formulas for the analyses in two and three space dimensions. We show that many filters used for weather forecasting, high-order aerodynamic and aeroacoustic computations match the proposed definition. Consequently the new multi-dimensional extension can be easily implemented in existing solvers. The new multi-dimensional extension and the two commonly used methods are compared in terms of compactness, robustness, accuracy and computational cost. Benefits of the genuinely multi-dimensional extension are assessed for various computations using the compressible Euler equations.