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

01/05/2021
by   Ben S. Southworth, et al.
0

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 2x2 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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

11/02/2020

A scalable exponential-DG approach for nonlinear conservation laws: with application to Burger and Euler equations

We propose an Exponential DG approach for numerically solving partial di...
02/19/2020

Structure-Preserving and Efficient Numerical Methods for Ion Transport

Ion transport, often described by the Poisson–Nernst–Planck (PNP) equati...
02/06/2020

Enhancement of shock-capturing methods via machine learning

In recent years, machine learning has been used to create data-driven so...
06/16/2021

Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

We propose fully explicit projective integration and telescopic projecti...
09/07/2021

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

In this paper we generalize the polynomial time integration framework to...
12/09/2021

Preconditioning Richards Equations: spectral analysis and parallel solution at very large scale

We consider here a cell-centered finite difference approximation of the ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.