Flux reconstruction approach for long-time calculations of the linearized Euler equation

The Linearized Euler Equations (LEE) are one of the fundamental governing equations in acoustics research. Due to the relatively small magnitude of acoustic disturbances compared to the physical quantities of the background flow, ensuring highly precise solutions without dissipation and dispersion over time is essential. The Flux Reconstruction (FR) method, recognized as a high-order numerical method with flexible construction capabilities, has been extensively utilized in recent years for solving partial differential equations numerically. This paper introduces an offsetting numerical flux for one-dimensional LEE. Subsequently, it develops a flux reconstruction scheme with energy-conserving characteristics, referred to as the ECFR scheme, aimed at achieving high-precision numerical solutions for prolonged computations. The energy-conserving feature of the scheme is substantiated through theoretical derivation and further corroborated by numerical validation. Moreover, the impact of Mach numbers and the parameter of the offsetting flux on the scheme's performance is discussed. The stability of the scheme is also analyzed, with the limit of the CFL number being provided. The novel ECFR scheme facilitates obtaining high-precision acoustic numerical solutions, thereby providing strong support for the study of complex engineering problems such as aircraft, submarine, and automobile noise.