To address the shortcomings of the MQC approaches above, other non-adiabatic dynamics methods have been developed in the diabatic representation, several of which are based on mapping formalism. For example, methods such as the partially linearized density matrix (PLDM) method , symmetrical quasiclassical (SQC) method , and the quantum-classical Liouville equation (QCLE) method are developed based on the Meyer-Miller-Stock-Thoss (MMST) mapping formalism . Further, methods such as the spin-mapped linearized semi-classical (spin-LSC) and the spin-mapped partially linearized density matrix (spin-PLDM) method are derived from the recently developed generalized spin-mapping formalism, which uses the generalized spin-mapping relations to describe the electronic DOF 67 while using a linearized approximation for the nuclear DOF . These methods have shown significant improvements in numerical results over existing MQC approaches, such as in characterizing the population dynamics of spin-boson models 66 exciton dynamics in light-harvesting complexes, ab initio on-the-fly simulations, as well as exciton-polariton quantum dynamics and vibrational polariton quantum dynamics. Based on these previous results, these nonadiabatic dynamical methods based on mapping formalism should also outperform MQC methods in simulating the polariton relaxation dynamics, although there have been limited investigations into the efficacy of these methods for polaritonic systems. In this paper, we use the Ehrenfest method, global flux surface-hopping (GFSH) method , SQC method with corrected zero-point energy (γ-SQC) and spin-LSC method, to simulate the population dynamics of a system that is initially excited into the upper polariton state. These methods are benchmarked with the numerically exact hierarchical equations of motion (HEOM) approach . Numerical results are presented for HTC models with different physical parameters, such as the number of molecules that are coupled to the cavity mode, the single-molecule coupling strength, the light-matter detunings, and various parameters for the phonon bath. Our results provide valuable information on the accuracy of the commonly used MQC methods and the recently developed mapping approaches for simulating polariton relaxation dynamics in the HTC model.