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Stencil computation is computationally intensive and required by many applications. This paper proposes an approach to exploit run-time reconfigurability of field-programmable accelerators for stencil computation. System throughput is optimized by partitioning, analysing and scheduling tasks in applications to remove idle functions. To evaluate the proposed approach, Reverse Time Migration (RTM), a high performance application, is developed. Our optimized runtime reconfigurable solution, which targets a Virtex-6 FPGA in a Maxeler MAX3424A system, can achieves an improved throughput of 102.8 GFlop/s, up to two orders of magnitude faster than the CPU reference designs, 1.59 times faster than the best published GPU and FPGA results, and 1.45 times faster than an optimized static implementation.