The direct numerical simulation (DNS) offers useful information about the understanding and modeling of turbulent flow. Few direct numerical simulations of wall-bounded compressible turbulent flows have been performed however. The objective of our study is to construct the DNS algorithm which can simulate the compressible turbulent flow between adiabatic and isothermal walls. The flow is selected since it is the simplest turbulent flow with an adiabatic wall, and therefore it is ideal for the modeling of compressible turbulent flow near the adiabatic wall. From the applicability and efficiency, we construct the new DNS algorithm based on the B-spline collocation method in the wall-normal direction. The validity and efficiency of the DNS algorithm are confirmed by comparing our results with those of an existing DNS in the compressible turbulent flow between isothermal walls. Then the DNS of compressible turbulent flow between the adiabatic and isothermal walls exhibits the applicability and usefulness of the algorithm.