This paper presents two time-splitting methods for integrating the elastic equations using forward-time schemes. The methods are based on a third-order Runge–Kutta time scheme and the Crowley advection schemes. These schemes are combined with a forward–backward scheme for integrating high-frequency acoustic and gravity modes to create stable split-explicit schemes for integrating the compressible Navier–Stokes equations. The time-split methods allow for both centered and upwind-biased discretizations for the advection terms, larger time steps, and more accurate solutions than existing approaches. The time-split Crowley scheme demonstrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge–Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models. The RK3 advection scheme is stable for Courant numbers less than 1.73, allowing a time step 1.73 times larger than the leapfrog scheme. The RK3 scheme permits the use of even- and odd-ordered spatial discretizations and allows time steps with Courant numbers equal to or greater than one. The RK3 scheme provides a better combination of simplicity, accuracy, and stability compared to the RK2 scheme. The Crowley splitting methodology is also an improvement over the RK2 scheme but is somewhat more expensive. The RK3 scheme allows for higher-order centered and upwind-biased spatial discretizations with reasonably large stable time steps for the advection terms. The RK3 time-splitting method is applied to the evolution of x-momentum and pressure equations. The method combines the forward–backward scheme for the pressure gradient and divergence with the RK3 scheme used on the advection terms. The method is stable and allows for larger time steps. The RK3 scheme is tested using two-dimensional simulations of a cold-bubble downburst problem. The results show that the RK3 scheme is more accurate than the leapfrog scheme and allows for larger time steps. The RK3 scheme is also more efficient than the RK2 scheme. The Crowley time-splitting scheme is another method for integrating the elastic equations. The scheme is applied to the translating downburst problem and shows similar results to the RK3 scheme. The Crowley scheme is more accurate than the RK3 scheme at low resolutions but is more expensive. The RK3 scheme is more efficient and accurate than the Crowley scheme. The paper concludes that the RK3 splitting scheme represents the best combination of accuracy and algorithmic simplicity, allowing for the largest time step of all three schemes. The RK3 scheme is accurate, robust, and permits a time step up to twice as large as that needed to stably integrate leapfrog-based time-split models. The RK3 scheme is also more efficient than the RK2 scheme and is an ideal candidate for numerical weather prediction applications where accuracy, stability, and efficiency are most importantThis paper presents two time-splitting methods for integrating the elastic equations using forward-time schemes. The methods are based on a third-order Runge–Kutta time scheme and the Crowley advection schemes. These schemes are combined with a forward–backward scheme for integrating high-frequency acoustic and gravity modes to create stable split-explicit schemes for integrating the compressible Navier–Stokes equations. The time-split methods allow for both centered and upwind-biased discretizations for the advection terms, larger time steps, and more accurate solutions than existing approaches. The time-split Crowley scheme demonstrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge–Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models. The RK3 advection scheme is stable for Courant numbers less than 1.73, allowing a time step 1.73 times larger than the leapfrog scheme. The RK3 scheme permits the use of even- and odd-ordered spatial discretizations and allows time steps with Courant numbers equal to or greater than one. The RK3 scheme provides a better combination of simplicity, accuracy, and stability compared to the RK2 scheme. The Crowley splitting methodology is also an improvement over the RK2 scheme but is somewhat more expensive. The RK3 scheme allows for higher-order centered and upwind-biased spatial discretizations with reasonably large stable time steps for the advection terms. The RK3 time-splitting method is applied to the evolution of x-momentum and pressure equations. The method combines the forward–backward scheme for the pressure gradient and divergence with the RK3 scheme used on the advection terms. The method is stable and allows for larger time steps. The RK3 scheme is tested using two-dimensional simulations of a cold-bubble downburst problem. The results show that the RK3 scheme is more accurate than the leapfrog scheme and allows for larger time steps. The RK3 scheme is also more efficient than the RK2 scheme. The Crowley time-splitting scheme is another method for integrating the elastic equations. The scheme is applied to the translating downburst problem and shows similar results to the RK3 scheme. The Crowley scheme is more accurate than the RK3 scheme at low resolutions but is more expensive. The RK3 scheme is more efficient and accurate than the Crowley scheme. The paper concludes that the RK3 splitting scheme represents the best combination of accuracy and algorithmic simplicity, allowing for the largest time step of all three schemes. The RK3 scheme is accurate, robust, and permits a time step up to twice as large as that needed to stably integrate leapfrog-based time-split models. The RK3 scheme is also more efficient than the RK2 scheme and is an ideal candidate for numerical weather prediction applications where accuracy, stability, and efficiency are most important