A GPU-efficient local ADER finite-difference method for compressible-flow DNS

Abstract

Algorithm design tailored to graphics processing units (GPUs) can substantially improve the efficiency of computational fluid dynamics (CFD) simulations. Efforts to adapt CFD solvers to GPUs often focus on spatial discretization, while time advancement commonly relies on multistage Runge–Kutta (RK) methods. However, these RK methods require repeated residual evaluations and full-domain operations, increasing the cost of compressible-flow direct numerical simulation (DNS). To reduce this overhead, we develop a local arbitrary high-order derivatives (ADER) finite-difference method that couples a compact ten-point quadratic predictor to a conservative central corrector through operator matching. With two quadrature nodes, one Picard correction, and selective dissipation, smooth convective tests recover third-order convergence under fixed-CFL refinement and near-seventh-order spatial behavior when background spatial error dominates. Temporal states remain kernel-local, and halo communication overlaps interior computation. Tests of smooth advection, periodic turbulence, and wall-bounded flows assess accuracy and flow statistics. On a $302^3$-node viscous Taylor–Green benchmark without heat conduction, throughput reaches 230.87 million node updates per second on one V100 GPU in double precision, 1.944 times that of spatially matched three-stage Runge–Kutta integration. Weak-scaling efficiency reaches 99.44% from 4 to 32 V100 GPUs at 1.812 billion nodes.