SSE Vectorized and GPU Implementations of Arakawa's Formula for Numerical Integration of Equations of Fluid Motion

The numerical method presented by Arakawa in 1966[3] implements a finite difference scheme of the Jacobian for the solution of the equation of motion for two-dimensional incompressible flows, which diminishes nonlinear computational instability and permits long-term numerical integrations. This paper presents an efficient implementation of Arakawa's formula using vectorized Streaming SIMD Extension (SSE) and Advanced Vector Extension (AVX) instructions. Additionally, we have improved the performance of memory access in the code. Performance measurements show that the vectorized implementation is close to two times more efficient compared to an implementation without SSE. The AVX version will in the near future further improve the vectorized performance with an estimated factor of up to 1.8. Finally we compare our results to an implementation on a general purpose graphics processor (GPGPU) and to auto-vectorization by two compilers.