We have developed efficient splitting algorithms for high-order compact finite-difference methods to approximate second-order space derivatives. In general, the methods’ high-order compact finite-difference schemes require the inversion of a multidiagonal matrix that is commonly less efficient. To solve this problem, we used ideas from splitting algorithms in one-way wave-equation migration that work by decomposing the multidiagonal matrix into a series of tridiagonal matrices and then subsequently solving the tridiagonal matrices. This approach results in more efficient algorithms with little loss of accuracy. The splitting algorithms can be implemented in three different ways. Our computational complexity analysis verifies that our methods can reduce the calculation burden from exponential to linear growth. Numerical experiments demonstrate the correctness and effectiveness of our algorithms.