Development of a Mixed-Precision Discontinuous Galerkin Framework for Compressible Flow Calculations

Discontinuous Galerkin (DG) methods combine features of finite element (FE) and finite volume (FV) solvers to solve partial differential equations in an element-local framework. This offers inherent advantages for massively-parallel computing architectures due to the high arithmetic intensity and lower memory requirements of element-local schemes. This paper presents a DG framework built using JAX, which targets several hardware platforms using the XLA compiler. The framework achieves an arbitrarily high order of accuracy and correctly simulates canonical fluid dynamics test cases using single-precision, but the associated convergence rate degrades very quickly using single-precision. Concurrently, single- and double-precision compute time is compared across a variety of hardware architecture including graphics processing units (GPUs), tensor processing units (TPUs), and both x86_64 and ARM computers. It is shown that single-precision produces as much as 2x speedup even on traditional architectures. This motivates the development of a mixed-precision framework that combines the speedup from using single-precision with the improved accuracy from double-precision representations. To this end, this paper demonstrates that selectively increasing the precision of certain operations in a DG solver can lower the machine-precision error floor. Some observations on the contributions of the precision of different operations in a DG solver to its overall accuracy and convergence rate are also presented.