Title: 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 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 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 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 aremore » 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 fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.« less

Authors:

Southworth, Ben S.  ^[1];

• Search DOE PAGES for author "Southworth, Ben S."
• Search DOE PAGES for ORCID "0000-0002-0283-4928"

Krzysik, Oliver  ^[2];

• Search DOE PAGES for author "Krzysik, Oliver"
• Search DOE PAGES for ORCID "0000-0001-7880-6512"

Pazner, Will  ^[3]

• Search DOE PAGES for author "Pazner, Will"
• Search DOE PAGES for ORCID "0000-0003-4885-2934"

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+ Show Author Affiliations

1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2. Monash Univ., Melbourne, VIC (Australia)
3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing

Publication Date:

Mon Mar 14 00:00:00 EDT 2022

Research Org.:

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:

USDOE

OSTI Identifier:

1873332

Report Number(s):

LA-UR-20-30444
Journal ID: ISSN 1064-8275

Grant/Contract Number:  

89233218CNA000001

Resource Type:

Accepted Manuscript

Journal Name:

SIAM Journal on Scientific Computing

Additional Journal Information:

Journal Volume: 44; Journal Issue: 2; Journal ID: ISSN 1064-8275

Publisher:

Society for Industrial and Applied Mathematics (SIAM)

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Runge-Kutta; numerical PDEs; preconditioning; Navier-Stokes; time integration