Improving locality of explicit one-step methods on GPUs by tiling across stages and time steps

Abstract

The performance of explicit parallel methods solving large systems of ordinary differential equations (ODEs) on GPUs is often memory bound. Therefore, locality optimizations, such as kernel fusion, are desirable. This paper exploits a special property of a large class of right-hand-side (RHS) functions to enable the fusion of computations of blocks of components of dependent stages of the method. This allows the derivation of tilings of the stages not only within one time step, but also spanning several successive time steps. Our approach is based on a representation of the ODE method by a data flow graph and allows efficient GPU code with fused kernels to be generated automatically for user-defined tilings. In particular, we investigate two generalized tiling strategies, trapezoidal and hexagonal tiling, and two different partitionings, which are evaluated experimentally for several different high- and low-order Runge–Kutta (RK) methods.