This paper proposes a flexible low-density parity-check (LDPC) decoder which leverages graphic processor units (GPU) to provide high decoding throughput. LDPC codes are widely adopted by the new emerging standards for wireless communication systems and storage applications due to their near-capacity error correcting performance. To achieve high decoding throughput on GPU, we leverage the parallelism embedded in the check-node computation and variable-node computation and propose a parallel strategy of partitioning the decoding jobs among multi-processors in GPU. In addition, we propose a scalable multi-codeword decoding scheme to fully utilize the computation resources of GPU. Furthermore, we developed a novel adaptive performance-tuning method to make our decoder implementation more flexible and scalable. The experimental results show that our LDPC decoder is scalable and flexible, and the adaptive performance-tuning method can deliver the peak performance based on the GPU architecture.