Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs

TL;DR

This paper develops a theoretical and algorithmic framework for efficiently solving nonlinear equations from high-order implicit Runge-Kutta methods applied to complex PDEs, including those with algebraic constraints, demonstrating improved convergence and applicability to fluid dynamics.

Contribution

It introduces new linearizations and preconditioners for IRK methods, enabling faster, more robust solutions for nonlinear PDEs and DAEs, with proven condition number bounds independent of discretization details.

Findings

Achieves up to 10th-order accuracy with Gauss IRK.

Requires roughly half the preconditioner applications compared to SDIRK.

Demonstrates effectiveness on fluid flow problems like Euler and Navier-Stokes.

Abstract

Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2x2 systems. Under quite general assumptions, it is proven that the preconditioned 2x2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases 4th-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard SDIRK methods.