The large time step scheme of Euler equations with source terms for nozzle flows

Abstract

Hyperbolic conservation laws with source terms govern critical flow phenomena, including turbulence transport and hypersonic chemically reacting flows. Despite their engineering significance, traditional numerical schemes for these systems face strict CFL (≤1.0) constraints, limiting computational efficiency. This study addresses the computational challenges posed by geometric source terms in Euler equations governing compressible nozzle flows. The proposed approach decomposes the governing equations into convective and source components: while convection is resolved via an LTS Godunov scheme, geometric source terms—arising from nozzle area variations (quasi-1D) or cylindrical coordinate transformations (2D axisymmetric)—are discretized using a novel combination of explicit and exact schemes. Numerical validations include: (1) quasi-1D cases under supersonic start/non-start conditions (internal shocks near outlets) at CFL ≤8.0, demonstrating shock-capturing accuracy with 55 % RMS error reduction at CFL = 8.0; (2) 2D hypersonic nozzle flows (Mach 4.0 design), achieving stable simulations at CFL = 5.0 with <1 % exit Mach deviation and 66 % runtime reduction. Results confirm that the proposed LTS framework overcomes CFL ≤1.0 limitations while enhancing efficiency—computational costs and errors decrease monotonically with increasing CFL numbers. This method establishes a generalizable paradigm for stiff, multidimensional Euler systems, balancing robustness and accuracy without sacrificing explicit computation advantages.