Carbon dioxide pipeline transportation is regarded as the most economical, efficient, and safe way for large-quantity, long-distance carbon dioxide transportation. In the process of planning, design, or operation of CO2 pipelines, it is necessary to simulate the steady-state or unsteady operation considering the hydraulic and thermal process of CO2 pipeline systems. In this paper, a steady state operation simulation method and a transient operation simulation method for non-isothermal carbon dioxide pipeline system are proposed. Firstly, based on the PR (Peng Robinson) equation of state and the thermodynamic differential equations, the equations of the physical parameters of CO2 required for the operation simulation are obtained, then the steady simulation method based on the ordinary differential governing equations and the transient simulation method based on time recursion and implicit central difference are proposed. The mathematical models of steady-state and transient-state operation simulation are established, and the mathematical models are solved based on the Newton-Raphson algorithm. The spatiotemporal evolution law of the operating parameters of the carbon dioxide pipeline can be obtained. Based on the proposed simulation method in this paper and OLGA simulation software, the simulation results of steady-state and transient-state operation of CO2 pipelines transported in gaseous and supercritical states are compared and analyzed. The maximum absolute errors of pressure and temperature along the gaseous and supercritical pipeline systems are 0.03 MPa, 1.04 K, and 0.18MPa, 0.71K, respectively. The maximum relative errors of mass flow rate and pressure at different nodes of the CO2 pipeline system are 2.7% and 2.9%, respectively. The hydraulic and thermal simulation methods of steady-state and transient-state operation of non-isothermal CO2 pipeline systems proposed in this paper can provide a reference for the operation control and flow safety assurance of actual CO2 pipelines.