Full Resolution Full Field Simulation Using Localized Computations

摘要

Locality is inherent to all transient flow and transport phenomena. Superposition of the two disparate spotiotemporal scales that underlie flow and transport leads to the problem of dimensionality. While the spatial locality in the temporal evolution of state variables is well-studied, little is known about the locality that is present over the course of individual Newton iterations that arise in the solution of an implicit timestep. This work derives a priori sharp and conservative estimates for the Newton update before its corresponding linear system has been solved. The current focus is the sequential implicit procedure for general nonlinear and heterogeneous two-phase flow in multiple dimensions. Due to the analytical nature of the work, the estimates are independent of the underlying spatial discretization. There are numerous applications of these estimates, including their use towards characterizing the convergence rate of Newton’s method as a function of timestep size, or towards the development of scalable linear preconditioning strategies. This work focuses on applying the estimates to reduce the size of the linear system that is to be solved at each Newton iteration. The key to the derivation of these estimates is in forming and solving the infinite-dimensional Newton iteration for the semidiscrete residual equations. In an implicit simulator, the Newton updates are accurate approximations to the continuous in space infinitedimensional updates. While the infinite dimensional updates are obtained by solving linear PDEs analytically, the discrete approximations are obtained by inverting a linear system. For flow, the analytical estimate for the Newton update is the solution of a linear constant coefficient screened Poisson equation. For transport, the estimate is the solution of a first-order linear PDE. The analytical solutions for these estimates are derived, and they can be applied computationally to any discrete problem. In a simulation, at each nonlinear iteration, the estimates are evaluated over parts of the domain and they are subsequently used to identify the unknowns and cells that will undergo a Newton update that is larger than any prescribed tolerance. The reduced linear system is then formed for these unknowns only, and is solved to obtain the Newton update. Since the estimates are conservative, there is no degradation of the nonlinear convergence rate.