High-precision weighted tight nonlinear formats based on Euler’s system of equations

Abstract

The study focuses on enhancing the accuracy of numerical solutions for Euler’s equations, essential for aerospace, engineering, and scientific simulations. To improve the numerical solution accuracy and efficiency of nonlinear problems, this study proposes a high-precision and high stability scheme for the Euler equation system, combining the fifth-order explicit weighted compact nonlinear scheme with the third-order semi-implicit implicit-explicit Runge-Kutta method. The results show that the scheme maintains design order accuracy in both high and low Mach number flows, with a spatial discretization error of 4.95 orders and a time discretization error convergence order of 5.86 orders. The width of the shock transition zone is reduced by 40% compared to traditional weighted compact nonlinear scheme. This demonstrates the accuracy advantage of this format. And this format remains stable even when Courant-Friedrichs-Lewy>1, making it suitable for large-scale industrial simulation, which is beneficial for improving efficiency. In one-dimensional shock tubes, bimodal acoustic pulse collisions, and Lax/Sod problems, the density, velocity, and pressure results of this format are consistent with the reference solution, with no oscillation phenomenon. The rigid processing error of the pressure term is less than 1.134 × 10⁻⁸. It has good stability, which provides a new approach for numerical research to further understand complex flow phenomena.