Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

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