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The present article is concerned with the implementation of the compact finite difference scheme, in the space and the optimal four-stage, order three strong stability-preserving time-stepping Runge-Kutta (SSP-RK43) scheme, in time for computation of one dimensional Kuramoto-Sivashinsky equation (KSE), arises in the study of flame front propagation, phase turbulence in reaction-diffusion system and in many other biological and chemical processes. The efficiency of proposed scheme is confirmed by six test problems with known exact solutions. The numerical results demonstrate the reliability and efficiency of the algorithm developed. Keywords: Kuramoto-Sivashinsky equation, Compact finite difference scheme, SSP-RK43 scheme, Thomas algorithm