This paper is concerned with the development and testing of advanced time-stepping methods suited for the integration of time-accurate, real-world applications of computational fluid dynamics (CFD). The performance of several time discretization methods is studied numerically with regards to computational efficiency, order of accuracy, and stability, as well as the ability to treat effectively stiff problems. We consider matrix-free implementations, a popular approach for time-stepping methods applied to large CFD applications due to its adherence to scalable matrix-vector operations and a small memory footprint. We compare explicit methods with matrix-free implementations of implicit, linearly-implicit, as well as Rosenbrock-Krylov methods. We show that RosenbrockKrylov methods are competitive with existing techniques excelling for a number of problem types and settings.

High-level parallel programming approaches have recently become popular in complex fluid dynamics research since they are cross-platform and easy to implement. The parallelism paradigm currently gaining popularity is to shift the careful optimization from programmers to intelligent compilers that ideally could implement an efficient usage of hardware resources. OpenACC is a high-level directive-based parallel library for programming on graphics processing units (GPU). The programming development procedure on CPU and GPU platforms can be effectively unified using the capability of portable programming with OpenACC. The directive-based model stands in contrast with the detailed implementation model of CUDA and OpenCL that suffers from poor portability and maintainability with changes in the accelerator hardware. In this work, we put effort toward drawing some outlines on efficiently annotating the base serial CPU version of the CFD code to achieve acceptable multi-threaded computational performance on the GPU. An Artificial Compressibility Method (ACM) is used for studying steady-state incompressible 2D and 3D heat and fluid flows. We study loop scheduling with careful attention given to the limited on-chip memory available. The possibility of asynchronous calculations in the CFD algorithm is investigated to increase the computing performance. The PGI Fortran 15.7 compiler is used for the following study. Memory management and loop scheduling considerations result in approximately 20% speedup in the GPU computing performance from the base OpenACC implementation. A performance analysis of the thermal flow solver shows that it works approximately 8 times faster on a single NVIDIA Tesla C2075 GPU card compared to a single Xeon processor core and demonstrates comparable speed to 8 cores on a Xeon CPU server using an MPI multi-CPU implementation of the code. A multi-GPU performance analysis is performed using NVIDIA Tesla M2050 GPUs on the Virginia Tech HokieSpeed supercomputer. A weak scalability analysis demonstrates up to 90% efficiency with 32 GPUs. With 16 GPUs the thermal flow solver works up to 190 times faster than a single Xeon processor CPU, 10 times faster than a single GPU, and 12 times faster than 16 Xeon processor CPUs distributed on separate sockets on the HokieSpeed supercomputer using an MPI multi-CPU implementation of the code.

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