GPU-accelerated solution of activated sludge model's system of ODEs with a high degree of stiffness By adopting a parallel Gauss-Jordan algorithm in CUDA to accelerate matrix inversion

Simulation of activated sludge model (ASM) including detailed biokinetic reaction network often requires the solution of a large system of ordinary differential equations (ODEs) at each time frame, which requires long computing times. In this study, an adaptive time step backward differentiation formula (BDF) is proposed to solve the ASM's system of ODEs that mainly contains a high degree of stiffness. A multi-tile CUDA-based parallel GaussJordan (GJ) algorithm for matrix inversion is applied as a part of the adaptive BDF algorithm to accelerate the overall simulation of the ASM. The results indicate that the performance of parallel GJ algorithm on GPU is highly dependent on the size of the matrix and is effective when the size of the matrix is greater than 128x128. Matrix inversion runtime analysis showed that the parallel matrix inversion on GPU could be achieved speedups of up to 15-fold and 430-fold over comparable serial single-CPU implementations for arbitrary matrices of size 256x256 and 2048x2048, respectively. A range of bioreactor configurations in ASM with different number of ODEs are assessed, and the combination of adaptive BDF with GPU-based parallel matrix inversion achieved an overall runtime reduction up to a 50% compared to single CPU implementations.