This work presents solutions of hyperbolic partial differential equations on periodic domains using a high-order numerical scheme. A Totally Volume-Integrated Discontinuous Galerkin (TVI-DG) method is developed explicitly for spatial discretization of the periodic interval [-π, π], utilizing an orthogonal polynomial basis inspired by Fourier. The integration of semi-discrete systems in time using Second and third-order Strong Stability Preserving (SSP) schemes, also defined on the same periodic interval, which are designed to have better stability properties compared to classical SSP methods. The spatial and temporal discretizations are derived from a unified weighted residual formulation using a standard Galerkin approach. The efficiency of the proposed method is proved through the linear advection equation and the one-dimensional heat equation. The numerical results of the conducted benchmark test cases confirm the accuracy and efficiency of the proposed methods. A comparison between the shape function of this work [-π, π] and those for the standard interval [0, 1] shows that the numerical results are equivalent at a time equal to 2π, highlighting the impact of the computational domain on solution behavior. This work is part of a series extending high-order DG methods to general intervals using generalized polynomial bases................................ Keywords:.............. Polynomial Discretization, Hyperbolic Conservation Laws, TVI-DG Method, SSP Time Schemes, Orthogonal Polynomials, Periodic Intervals.