We study in this paper the convergence behavior of the Schwarz waveform relaxation (SWR) algorithm for a class of non-dissipative evolutional equations.Both the Dirichlet and Robin transmission conditions (TCs) are investigated.For the Dirichlet TC,we analyze the algorithm in the case of many subdomains.For the nonlinear initial-boundary value problem (a)tu = μ(a)xxu + f(u),the allowed upper bound of f(u) which guarantees convergence of the algorithm in the case of N(≥ 3) subdomains is obtained and it is proved that it is smaller than the one with N= 2.For the Robin TC,we consider the initial value problem (a)tu = μ(a)xxu+au with a > 0,and focus on deriving the optimized parameter involved in the Robin TC and analyzing the asymptotical properties of the algorithm with respect to T,the length of the time interval,and the meshes △t and △x.The optimization procedure towards determining the optimal choice of the parameter is different from the case a < 0,which has been deeply studied in the literatures.In particular,it is shown that for overlap size l=O(△x) small and △t = O(△xr) with r < 4/3,the equioscillation principle which works for a < 0 does not hold when a > 0.We generalize the optimization procedure to other non-dissipative problems,such as advection reaction diffusion problems and reaction diffusion equations with time delay.Finally,numerical results are presented to support our theoretical conclusions.