A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE Problems

Rana, Md. Masud; Howle, Victoria E.; Long, Katharine; Meek, Ashley; Milestone, W.

doi:10.48550/arxiv.2010.11377

Cited by 3 publications

(9 citation statements)

References 3 publications

(11 reference statements)

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“…It is worth pointing out that while writing this paper, at least three preprints have been posted online studying the use of IRK methods for numerical PDEs. Two papers develop new block preconditioning techniques for parabolic PDEs [24,43], and one focuses on a high-level numerical implementation of IRK methods with the Firedrake package [16].…”

Section: Motivation and Previous Workmentioning

confidence: 99%

Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting

Southworth¹,

Krzysik²,

Pazner³

et al. 2021

Preprint

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Fully implicit Runge-Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but are rarely used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for the fast, parallel solution of the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). This framework also naturally applies to discontinuous Galerkin discretizations in time. The new method can be used with arbitrary existing preconditioners for backward Euler-type time stepping schemes, and is amenable to the use of three-term recursion Krylov methods when the underlying spatial discretization is symmetric. Under quite general assumptions on the spatial discretization that yield stable time integration, the preconditioned operator is proven to have conditioning ∼ O(1), with only weak dependence on number of stages/polynomial order; for example, the preconditioned operator for 10th-order Gauss integration has condition number less than two. The new method is demonstrated to be effective on various high-order finite-difference and finite-element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th order accuracy. In several cases, the new method can achieve 4th-order accuracy using Gauss integration with roughly half the number of preconditioner applications as required using standard SDIRK techniques.

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Section: Motivation and Previous Workmentioning

confidence: 99%

Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting

Southworth¹,

Krzysik²,

Pazner³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It is worth pointing out that while writing this paper, at least three preprints have been posted online studying the use of IRK methods for numerical PDEs. Two papers develop new block preconditioning techniques for parabolic PDEs [22,37] ( [22] also appeals to the Schur decomposition as will be used in this paper), and one focuses on a high-level numerical implementation of IRK methods with the Firedrake package [12].…”

Section: Why Fully Implicit and Previous Workmentioning

confidence: 99%

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

Southworth¹,

Krzysik²,

Pazner³

2021

Preprint

View full text Add to dashboard Cite

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.

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“…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).…”

Section: Constructing Imex Polynomial Integrators Based On Radau Iiamentioning

confidence: 99%

“…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.…”

Section: Linear Stabilitymentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Buvoli¹,

Southworth²

2021

Preprint

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In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with arbitrary order-of-accuracy and varying degree of implicitness. In this first work, we focus on a new class of implicit-explicit polynomial block methods that are based on fully-implicit Runge-Kutta methods with Radau nodes. We show that the new integrators have improved stability compared to existing IMEX Runge-Kutta methods, while also being more computationally efficient due to recent developments in preconditioning techniques for solving the associated systems of nonlinear equations. For PDEs on periodic domains where the implicit component is trivial to invert, we will show how parallelization of the right-hand-side evaluations can be exploited to obtain significant speedup compared to existing serial IMEX Runge-Kutta methods. For parallel (in space) finite-element discretizations, the new methods obtain accuracy several orders of magnitude lower than existing IMEX Runge-Kutta methods, and/or obtain a given accuracy as much as 16 times faster in terms of computational runtime.

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A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE Problems

Cited by 3 publications

References 3 publications

Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting

Fast solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting

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

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

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