Modified Parareal Algorithm for Solving Time-Dependent Differential Equations

Parallel algorithms are implemented to compute the solutions of partial differential equations and ordinary differential equations of complex dynamical systems to achieve near real-time solutions. One of the parallel algorithms widely implemented is the Parareal algorithm to solve time-dependent differential equations for various scientific applications. Parareal algorithm has shown promising speedups in achieving near real-time solutions using accelerators. However, it has been observed that the sequential predictorcorrector step of the Parareal algorithm impacts the computational performance. This paper analyses the Parareal algorithm and proposes modification to the predictor-corrector step of the Parareal algorithm to exploit data parallelism more and reduce the computation time. The modified algorithm is implemented to solve two systems of interdependent ODEs. The numerical accuracy and performance analysis of the modified algorithm is shown to be same as the original Parareal. The performance analysis of the modified algorithm on two accelerator computing architectures: Intel Xeon Phi CPU and Graphical processing units with OpenMP, OpenACC, and CUDA programming models are presented. The modified algorithm demonstrates performance improvement ranging from 1.2x-2x with respect to the original Parareal algorithm.