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Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioning

O. A. Krzysik, H. De Sterck, R. D. Falgout, J. B. Schroder

TL;DR

This work considers the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear, and approximately solves linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used.

Abstract

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.

Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioning

TL;DR

This work considers the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear, and approximately solves linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used.

Abstract

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an -dimensional system of PDEs, applying the preconditioner consists of solving a sequence of scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.
Paper Structure (21 sections, 2 theorems, 71 equations, 11 figures, 2 algorithms)

This paper contains 21 sections, 2 theorems, 71 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Acoustics: Introduction and background
  3. Characteristic variables
  4. Godunov discretization
  5. Acoustics: Parallel-in-time solution
  6. The outer iteration with characteristic-variable block preconditioning
  7. Godunov discretization in characteristic variables
  8. Block preconditioners
  9. Numerical results
  10. Conservative nonlinear systems: Introduction and background
  11. Finite-volume discretization
  12. Conservative nonlinear systems: Parallel-in-time solution
  13. The linearized time-stepping operator
  14. The inner linearized iteration
  15. Constructing $\widehat{{\@fontswitch{}{\mathcal{}} P}}$ and $\widetilde{{\@fontswitch{}{\mathcal{}} P}}$
  16. ...and 6 more sections

Key Result

Lemma 3.2

\newlabellem:god-char-stencils0 Let $\Phi \in \mathbb{R}^{2 n_x \times 2 n_x}$ be the time-stepping operator for the Godunov discretization eq:acoustics-god of eq:acoustic, and $\widehat{\Phi} := {\@fontswitch{}{\mathcal{}} R}_0^{-1} \Phi {\@fontswitch{}{\mathcal{}} R}_0 \in \mathbb{R}^{2 n_x \tim

Figures (11)

  • Figure 1: Left: Material properties \ref{['eq:mat-param-2']}, \ref{['eq:mat-param-3']}, \ref{['eq:mat-param-4']}, \ref{['eq:mat-param-5']} ordered from top to bottom. Middle: Residual history using preconditioners $\widehat{{\@fontswitch{}{\mathcal{}} P}}_L$ (solid lines) and $\widehat{{\@fontswitch{}{\mathcal{}} P}}_D$ (dashed lines) in \ref{['eq:prec-def']}. Right: Residual history using preconditioners $\widetilde{{\@fontswitch{}{\mathcal{}} P}}_L$ (solid lines) and $\widetilde{{\@fontswitch{}{\mathcal{}} P}}_D$ (dashed lines) in \ref{['eq:prec-approx-def']} which are based on approximate diagonal blocks. All preconditioners are applied exactly via sequential time-stepping (SinT). Legend entries correspond to space-time mesh resolutions of $n_x \times n_t$.
  • Figure 1: Small-amplitude \ref{['eq:SWE-idp']} problem for \ref{['eq:SWE']}, with $\varepsilon = 0.1$. Bottom left: Nonlinear residual history with dotted lines corresponding to exact solves of the linearized problems ${\@fontswitch{}{\mathcal{}} A}_k \bm{e}_k^{\rm lin} = \bm{r}_k$, and solid lines to approximate solves with a single iteration of \ref{['alg:char-prec']} using $\widehat{{\@fontswitch{}{\mathcal{}} P}}$. Bottom right: Nonlinear residual history where linearized problems ${\@fontswitch{}{\mathcal{}} A}_k \bm{e}_k^{\rm lin} = \bm{r}_k$ are approximately solved with a single iteration of \ref{['alg:char-prec']} using $\widetilde{{\@fontswitch{}{\mathcal{}} P}}$ with diagonal blocks $\widetilde{{\@fontswitch{}{\mathcal{}} A}}_{ii}$ either inverted exactly (solid lines) or approximately with one MGRIT V-cycle (dashed lines). Legends in the bottom row correspond to space-time mesh resolutions of $n_x \times n_t$. \newlabelfig:SWE-IDP-weakly0
  • Figure 2: Test problems for the acoustics equations \ref{['eq:acoustic']}. Each of the five rows in the figure corresponds to each of the five material parameters in \ref{['eq:acoustics-mat-params-examples']}. Left: Material parameters \ref{['eq:acoustics-mat-params-examples']}. Middle: Space-time contours of the resulting pressure component of the solution vector; note these contours correspond to solves using $n_x = 2048$. Right: Residual histories when using preconditioners $\widetilde{{\@fontswitch{}{\mathcal{}} P}}$ in \ref{['eq:prec-approx-def']} with diagonal blocks approximately inverted with a single MGRIT iteration (PinT). Dashed lines correspond to $\widetilde{{\@fontswitch{}{\mathcal{}} P}}_D$ and solid lines correspond to $\widetilde{{\@fontswitch{}{\mathcal{}} P}}_L$. Legend entries correspond to space-time mesh resolutions of $n_x \times n_t$.
  • Figure 2: Larger-amplitude \ref{['eq:SWE-idp']} problem for \ref{['eq:SWE']}, with $\varepsilon = 0.6$. \newlabelfig:SWE-IDP-fully0
  • Figure 3: Small-amplitude \ref{['eq:euler-idp']} problem for \ref{['eq:euler']} with $\varepsilon = 0.2$. \newlabelfig:euler-IDPP-weakly0
  • ...and 6 more figures

Theorems & Definitions (7)

  • Remark 3.1
  • Lemma 3.2: Godunov discretization in characteristic variables
  • Proof 1
  • Corollary 3.3: Impedance-linearized Godunov discretization in characteristic variables
  • Proof 2
  • Remark 3.4: Schur-complement preconditioning
  • Proof 3