The Piecewise-Parabolic Method (PPM) is a higher-order extension of Godunov's method for solving gas dynamics problems. It introduces a more accurate representation of smooth spatial gradients and steeper representation of discontinuities, particularly contact discontinuities. The method uses a higher-order spatial interpolation and a simpler, more robust algorithm for calculating nonlinear wave interactions. Additional dissipation is introduced to ensure accurate results without degrading the solution quality. The PPM scheme is described for scalar advection equations and extended to gas dynamics in both Lagrangian and Eulerian coordinates. In Lagrangian coordinates, the method involves interpolating distributions of dependent variables, solving Riemann problems to calculate time-averaged pressures and velocities, and using these to compute effective fluxes. In Eulerian coordinates, the method uses a single-step formulation, with interpolation functions and Riemann solvers to calculate fluxes. The method is designed to preserve third-order accuracy in space and time, though it sacrifices some time accuracy for computational efficiency. The PPM method includes a discontinuity detection algorithm to identify and handle discontinuities, ensuring that the solution remains sharp and accurate. This algorithm is applied to density interpolation and contact discontinuities. The method is also extended to handle one-dimensional Eulerian hydrodynamics, with modifications to ensure consistency between the Lagrangian step and the remap. The method is tested on various scenarios, including strong shocks, and is shown to produce accurate results with minimal oscillations. The PPM method is characterized by its ability to handle complex fluid dynamics problems with high accuracy and stability. It is particularly effective in capturing sharp discontinuities and maintaining solution quality even in the presence of strong shocks. The method is widely used in computational fluid dynamics for its robustness and accuracy in simulating gas dynamics.The Piecewise-Parabolic Method (PPM) is a higher-order extension of Godunov's method for solving gas dynamics problems. It introduces a more accurate representation of smooth spatial gradients and steeper representation of discontinuities, particularly contact discontinuities. The method uses a higher-order spatial interpolation and a simpler, more robust algorithm for calculating nonlinear wave interactions. Additional dissipation is introduced to ensure accurate results without degrading the solution quality. The PPM scheme is described for scalar advection equations and extended to gas dynamics in both Lagrangian and Eulerian coordinates. In Lagrangian coordinates, the method involves interpolating distributions of dependent variables, solving Riemann problems to calculate time-averaged pressures and velocities, and using these to compute effective fluxes. In Eulerian coordinates, the method uses a single-step formulation, with interpolation functions and Riemann solvers to calculate fluxes. The method is designed to preserve third-order accuracy in space and time, though it sacrifices some time accuracy for computational efficiency. The PPM method includes a discontinuity detection algorithm to identify and handle discontinuities, ensuring that the solution remains sharp and accurate. This algorithm is applied to density interpolation and contact discontinuities. The method is also extended to handle one-dimensional Eulerian hydrodynamics, with modifications to ensure consistency between the Lagrangian step and the remap. The method is tested on various scenarios, including strong shocks, and is shown to produce accurate results with minimal oscillations. The PPM method is characterized by its ability to handle complex fluid dynamics problems with high accuracy and stability. It is particularly effective in capturing sharp discontinuities and maintaining solution quality even in the presence of strong shocks. The method is widely used in computational fluid dynamics for its robustness and accuracy in simulating gas dynamics.