This paper presents a workflow for synthesizing near-optimal FPGA implementations for structured-mesh based stencil applications for explicit solvers. It leverages key characteristics of the application class, its computation-communication pattern, and the architectural capabilities of the FPGA to accelerate solvers from the high-performance computing domain. Key new features of the workflow are (1) the unification of standard state-of-the-art techniques with a number of high-gain optimizations such as batching and spatial blocking/tiling, motivated by increasing throughput for real-world work loads and (2) the development and use of a predictive analytic model for exploring the design space, resource estimates and performance. Three representative applications are implemented using the design workflow on a Xilinx Alveo U280 FPGA, demonstrating near-optimal performance and over 85% predictive model accuracy. These are compared with equivalent highly-optimized implementations of the same applications on modern HPC-grade GPUs (Nvidia V100) analyzing time to solution, bandwidth and energy consumption. Performance results indicate equivalent runtime performance of the FPGA implementations to the V100 GPU, with over 2x energy savings, for the largest non-trivial application synthesized on the FPGA compared to the best performing GPU-based solution. Our investigation shows the considerable challenges in gaining high performance on current generation FPGAs compared to traditional architectures. We discuss determinants for a given stencil code to be amenable to FPGA implementation, providing insights into the feasibility and profitability of a design and its resulting performance.