This paper presents two time-splitting methods for integrating the elastic equations, based on a third-order Runge–Kutta (RK3) time scheme and Crowley advection schemes. These methods combine a forward–backward scheme for high-frequency acoustic and gravity modes with the RK3 or Crowley schemes to create stable split-explicit schemes for the compressible Navier–Stokes equations. The RK3 scheme allows for both centered and upwind-biased spatial discretizations, enabling larger time steps and more accurate solutions compared to existing methods. The Crowley scheme, while slightly less accurate, is more computationally efficient and can handle high-order spatial discretizations. Both schemes are shown to be stable and accurate through linear and nonlinear tests, with the RK3 scheme offering the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models. The paper also discusses the application of these schemes to two-dimensional simulations, demonstrating their effectiveness in capturing complex flow features.This paper presents two time-splitting methods for integrating the elastic equations, based on a third-order Runge–Kutta (RK3) time scheme and Crowley advection schemes. These methods combine a forward–backward scheme for high-frequency acoustic and gravity modes with the RK3 or Crowley schemes to create stable split-explicit schemes for the compressible Navier–Stokes equations. The RK3 scheme allows for both centered and upwind-biased spatial discretizations, enabling larger time steps and more accurate solutions compared to existing methods. The Crowley scheme, while slightly less accurate, is more computationally efficient and can handle high-order spatial discretizations. Both schemes are shown to be stable and accurate through linear and nonlinear tests, with the RK3 scheme offering the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models. The paper also discusses the application of these schemes to two-dimensional simulations, demonstrating their effectiveness in capturing complex flow features.