The combined approach of linearization and splitting up is used for devising new algorithms to solve a one-dimensional Burgers' equation. Two schemes are discussed and the computed solutions are compared with the exact solution. For this problem it is found that the schemes proposed yield excellent numerical results for Reynolds number R ranges from 50 up to 1500. The schemes were also tested for another problem whose R = 10000. In this case a filtering technique is used to overcome the nonlinear instability.