Mixed-Precision in High-Order Methods: the Impact of Floating-Point Precision on the ADER-DG Algorithm

We present a mixed-precision implementation of the high-order discontinuous Galerkin method with ADER time stepping (ADER-DG) for solving hyperbolic systems of partial differential equations (PDEs) in the hyperbolic PDE engine ExaHyPE. The implementation provides a simple API extension for specifying the numerical precision for individual kernels, and thus allows for testing the effect of low and mixed precision on the accuracy of the solution. To showcase this, we study the impact of precision on the overall convergence order and actual accuracy of the method as achieved for four common hyperbolic PDE systems and five relevant scenarios that feature an analytic solution. For all scenarios, we also assess how sensitive each kernel of the ADER-DG algorithm is to using double, single or even half precision. This addresses the question where thoughtful adoption of mixed precision can mitigate hurtful effects of low precision on the overall simulation.