Massive parallelism across space in ODEs

This paper examines a number of previously proposed methods for the parallel integration of differential equations from the perspective of computation graphs. The inherent structure of the computation graph is imposed by the differential equation, but it may not permit adequate parallelism. Many methods can be viewed as modifications of the graph to introduce parallelism at the expense of additional computation, and this viewpoint allows us to consider alternate approaches. The various approaches to parallelism can be classified as method parallelism, parallelism across space, or parallelism across time. Method parallelism is suitable for low-degree parallelism only. This paper, which is based partly on two earlier papers [5,6] concentrates on parallelism across space, both by direct and waveform methods. A companion paper considers parallelism across time.