This paper studies numerical scheme and suppression method of wall heating error for elastic-plastic flow with cell-centered Lagrange Godunov method. Provide the viscosity correction equation of Godunov scheme, describe the procedure of a viscous shock formation and propagation with a jump type initial data, and analyze the relationship between the viscosity behavior of the correction equation and wall heating error. On this basis, a new HLLC-type approximate Riemann solver is proposed. In this solver, an adaptive heat conduction viscosity is introduced to suppress wall heating error of internal energy and density at the interface; What's more, an additional contact velocity is proposed to suppress the over-heating phenomenon of deviatoric stress.