High performance computational algorithms for a class of integer and fractional evolutionary models

Evolutionary models that depend on space and time variables occur in many physical processes. A standard approach for such systems is based on a classical diffusion modeling which leads to integer derivatives in the time and spatial variables. However, it has been observed in the literature that in many single- and multi-phase flow cases, especially in complex porous media, it is appropriate to use anomalous sub-diffusion models. Such models can be described by a class of non-local in time fractional derivative partial differential equations (FPDEs). In various applications, such as reservoir management, understanding the long-time behavior and resolving fines structures of processes governed by such models are crucial from early design phase to production phase. Therefore, fine meshes with large degrees of freedom (DoF) are needed in associated computer models to obtain relatively accurate simulated physical processes. Consequently, for long time simulation, implicit time-stepping discretization methods (such as the Crank-Nicolson and implicit Euler) require a computationally prohibitive number of discrete time-steps. Such industrial standard approaches are inherently serial-in-time and require several days of simulation even using efficient parallel-in-space algorithms on high performance computing (HPC) environments. HPC systems provide a large number of processing cores with various limitations, in particular on the amount of memory available per compute node. The memory limitation leads to severe constraints for resolving fine spatial structures that require large DoF. Accordingly, long time simulation cannot be achieved within reasonable simulation time and computational cost. In this work, we avoid the time-stepping computational bottleneck by developing a class of efficient hybrid HPC algorithms that combines parallel in time and space tasks. Our approach facilitates careful balancing between parallel performance and the memory constraint to efficiently simulate evolutionary FPDEs. We demonstrate the parallel HPC performance of the algorithm for several space-time evolutionary models using several millions of spatial DoF. We validate our HPC framework for efficient simulation of a class of fractional-Darcy's law based single-phase flow models, with potential application to develop a new generation of reservoir simulators.