On multi-core clusters or supercomputers, how to get good performance when running high performance computing (HPC) applications is a main concern. In this report, performance oriented auto-tuning strategies and experimental results are presented for stencil HPC applications on multi-core parallel machines. A typical 2D Jacobi benchmark is chosen as the experimental stencil application. The main tuning strategies include data partitioning within a multi-core node, number of threads within a multi-core node, data partitioning for a number of nodes, number of nodes in a multi-core cluster system. The results of the experiments are based on multi-core parallel machines from PRACE or Grid'5000, such as Curie, and Stremi cluster.