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

Abstract

Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but are rarely 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 the fast, parallel solution of the nonlinear equations that arise from IRK methods applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. This framework also naturally applies to discontinuous Galerkin discretizations in time. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes. 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 2 × 2 systems. Under quite general assumptions on the spatial discretization, it is proven that the preconditioned operator has a condition number of ∼ O(1), with only weak dependence on the number of stages or integration accuracy. 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 integration, while in all cases 4th-order Gauss integration requires roughly half the number of preconditioner applications as required by standard SDIRK methods.

Addititive Polynomial Block Methods, Part I: Framework and Fully-Implicit Methods

“…When q = 2, the IMEX-Radau method is equivalent to the IMEX-Euler method (2.3), while IMEX-Radau* treats the the implicit component with backwards Euler and the explicit component with second-order Adams-Bashforth. For q > 2, the nonlinear implicit equations that arise in both IMEX-Radau methods (5.7) are analogous to those that arise in standard Radau IIA integration, with a modified right-hand side derived from the explicit component (i.e., linear and nonlinear solvers developed for fully implicit RK [18,35,25,30,36,41,40] naturally apply to IMEX-Radau).…”

“…First, we study the performance of the integrators on PDEs with periodic domains where inverting the implicit component is trivial. Then, we compare both method families on a finite-element discretization of a non-periodic problem, where solving the fully implicit system requires special care [25,30,36,41,40]. Lastly, we numerically investigate order reduction on the singularly perturbed Van der Pol equation.…”

“…For the Radau schemes, we must solve an implicit system analogous to that which arises in fully implicit Runge Kutta methods. We wrap the AMG solver already mentioned with the block preconditioning for fully implicit Runge Kutta methods developed in [41,40] to solve the Radau stage equations. We use 4th-order elements in space, with mesh spacing h x ≈ 0.0078 leading to expected spatial accuracy ∼ O(10 −9 ) and solve the implicit equations to 10 −12 relative residual to ensure 2 -error is not affected by an inexact implicit solve.…”

“…This choice may seem peculiar since fully-implicit methods are often considered too slow to be competitive. However, recent developments in block linear and nonlinear solvers make their use in numerical PDEs quite tractable [18,35,25,30,36,41,40], even outperforming diagonally implicit RK methods in many cases [41,40]. Despite these developments, it is not possible to derive IMEX RK methods based on the fully implicit RK methods since the explicit stages would be nonlinearly coupled to the fully implicit stages.…”

Assessment of high-order IMEX methods for incompressible flow

“…The application of implicit Runge-Kutta (IRK) methods to incompressible flows is more complicated due to the quasi-static nature of the continuity equation, but was investigated in [4,24,40]. In most of these studies, the order of convergence reached 3 to 5, although methods up to order 10 were considered in [48]. As an alternative to IRK, variational methods such as Discontinuous Galerkin methods in time and space-time element methods are gaining interest [1,54].…”