Mixed-precision parallel linear solver for high-order compact finite difference schemes

Compact finite difference methods are widely used in simulations of fluid mechanics problems for their high spectral resolution. For large-scale simulations on modern high-performance computing systems, it is extremely challenging to efficiently solve linear systems arising from compact numerical schemes with hierarchical parallelism on distributed and shared memory. This work further optimizes the previous scalable direct parallel algorithm published by the authors by introducing conditional mixed-precision operations for data involved in the communication between distributed memory partitions. The feasibility of conducting the mixed-precision operations is based on the mathematical structure of the linear system during the solution process, and the numerical error behavior is characterized. Theoretical analysis and numerical experiments have demonstrated that the solution can still achieve double precision accuracy with mixed-precision operations compared to the direct linear solver. This optimization is particularly beneficial for multi-dimensional computations using higher-order compact schemes where cyclic penta-diagonal systems are solved and performance is limited by the communication bandwidth.