Parallel Stencil Computation is numerical computation applying local patterns (stencils) to grid points independently, combining neighboring values according to kernel coefficients, commonly used for solving PDEs via finite differences — fundamental to scientific computing with excellent parallelism properties. Stencil algorithms expose massive data parallelism. Stencil Patterns and Kernels define finite difference coefficients combining neighbors—2D 5-point stencil (center, up, down, left, right), 2D 9-point stencil (8 neighbors), 3D 7-point, etc. Coefficients determine output: u_new[i,j] = c0u[i,j] + c1u[i+1,j] + c2*u[i-1,j] + ... Domain Decomposition partitions grid among processors: regular decomposition assigns rectangular regions. Interior points need only local data, boundary points need from neighbors. Halo Exchange before computation: interior processors send boundary rows/columns to neighbors, receive neighbors' boundaries (halos). Halo width depends on stencil radius. All-to-all exchange is bottleneck in weak scaling. Memory Layout Optimization stores data in cache-friendly order: row-major for typical stencils accessing neighbors in same row. Padding to cache line boundaries avoids false sharing. Time Stepping iterates stencil application: time step n uses values from n-1, producing n. Multiple time steps amortize communication overhead. Temporal blocking processes multiple time steps before new halo exchange. GPU Implementation threads process single grid point, block computes larger region with shared memory for halo elements. Block synchronization ensures halo data availability. Overlapping Communication and Computation send interior boundary regions immediately, compute interior while receiving halo, reducing communication latency. Vectorization via SIMD within each grid point dimension—stencil computation on multiple points in parallel (row or column vectors). Stencil Chain Fusion fuses multiple dependent stencil operations to reduce memory traffic. Weak Scaling with fixed problem size per processor maintains constant stencil computation despite increasing processors—limited by halo exchange latency which doesn't decrease. Strong Scaling with fixed total size faces diminishing returns when halo exchange dominates computation. Fundamental limit: stencil communication-to-computation ratio determines scalability—small ratio (many stencil applications) scales well, large ratio (few applications) doesn't requiring algorithmic or hardware improvements.