We present the first FPGA implementation of the full simulation pipeline of a shallow water code based on the discontinuous Galerkin method. Using OpenCL and following an algorithm-hardware codesign approach, the software reference is transformed into a dataflow architecture that can process a full mesh element per clock cycle. The novel projection approach on the algorithmic level complements the pipeline and memory optimizations in the hardware design. With this, the FPGA kernels for different polynomial orders outperform the CPU reference by 43x -- 144x in a strong scaling benchmark scenario. A performance model can explain the measured FPGA performance of up to 717 GFLOPs accurately.