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Center for Nonlinear Studies |
We present novel iterative methods for the time integration of partial differential equations (PDEs) with strong advection. First we consider fully implicit Runge-Kutta (FIRK) methods, and then parallel-in-time methods. As time integration schemes, FIRK methods are desirable from an accuracy standpoint; for example, they have arbitrarily high-order accuracy on differential algebraic equations, while diagonally implicit Runge-Kutta methods may limit to just first or second order for such problems. However, FIRK methods are rarely used in the numerical simulation of PDEs because each time step requires the solution of a large system of block-coupled nonlinear algebraic equations. We introduce an efficient algorithmic framework for solving this system, complete with theoretical robustness guarantees under minimal assumptions on the underlying problem. Numerical examples are shown for both compressible and incompressible Navier-Stokes equations. Traditionally, time-dependent PDEs are simulated sequentially in time via time-stepping; however, they may also be simulated in a parallel-in-time fashion where by the solution is computed at all times simultaneously. To date, many effective parallel-in-time strategies have been developed for diffusion-dominated problems, but there has been markedly less success for hyperbolic problems. We present a framework for the parallel-in-time solution of hyperbolic PDEs based on applying multigrid techniques in the time direction. The key component of our methodology is a carefully designed coarse-grid problem making use of semi-Lagrangian technology. Numerical examples are shown for Burgers equation and the acoustic equations.
Host: Ben Southworth