The authors propose a new waveform relaxation algorithm for general semi-linear reaction-diffusion equations. Compared with the classical waveform relaxation algorithm, the new one has two advantages: the first one is that the system is not decomposed into sub-systems; the second one is represented by the fact that the convergence rate of the new waveform relaxation algorithm does not deteriorate when the spatial grid is refined. An upper bound on the iteration errors, which indicates the superlinear convergence of the algorithm, is given. The corresponding discrete waveform relaxation algorithm for reaction-diffusion equations is presented and its parallelism is analysed. Some numerical experiments are presented in order to verify the theoretical approach.