In this thesis we develop techniques to efficiently solve numerical Partial Differential Equations (PDEs) using Graphical Processing Units (GPUs). Focus is put on both performance and re--usability of the methods developed, to this end a library, cuSten, for applying finite--difference stencils to numerical grids is presented herein. On top of this various batched tridiagonal and pentadiagonal matrix solvers are discussed. These have been benchmarked against the current state of the art and shown to improve performance in the solution of numerical PDEs. A variety of other benchmarks and use cases for the GPU methodologies are presented using the Cahn--Hilliard equation as a core example, but it is emphasised the methods are completely general. Finally through the application of the GPU methodologies to the Cahn--Hilliard equation new results are presented on the growth rates of the coarsened domains. In particular a statistical model is built up using batches of simulations run on GPUs from which the growth rates are extracted, it is shown that in a finite domain that the traditionally presented results of 1/3 scaling is in fact a distribution around this value. This result is discussed in conjunction with modelling via a stochastic PDE and sheds new light on the behaviour of the Cahn--Hilliard equation in finite domains.

翻译：在此论文中,我们开发了使用图形处理器(GPUs)有效解决数字部分等分法(PDEs)的技术。焦点放在所开发方法的性能和可重新使用性上,为此,这里展示了对数字网格应用有限差异线的库库( custen) 。除了这些分批的三对角和五对角矩阵解答器之外,还讨论了这些技术。这些技术是参照目前工艺状态的基准,并显示可以改进数字PDE的解决方案的性能。使用卡赫- 希利亚方程式作为核心示例,展示了各种其他通用方法的性能和使用案例。最后,通过对卡赫- 希利亚方程式应用 GPU 方法,展示了粗略域的增长率方面的新结果。 特别是利用在GPUS中提取增长率的几批模拟来建立统计模型。 在有限的域中,展示了以卡赫- 赫- 平方程式的正方程式中传统结果,通过1/3 平方程式的平方程式的平面分布在新的平方块中,这是以新的平方形的平方块的平方。