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Parallel-in-time solution of scalar nonlinear conservation laws

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

Abstract

We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton's method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration, however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley--Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.

Parallel-in-time solution of scalar nonlinear conservation laws

Abstract

We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton's method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration, however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley--Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.
Paper Structure (40 sections, 9 theorems, 127 equations, 15 figures, 1 algorithm)

This paper contains 40 sections, 9 theorems, 127 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. PDE discretization
  3. Numerical flux
  4. WENO reconstructions
  5. Nonlinear iteration scheme
  6. Linearization
  7. Linearization I: Time integration
  8. Linearization II: Numerical flux
  9. Linearization III: Reconstructions
  10. Linearization IV: Summary
  11. Approximate MGRIT solution of linearized problems
  12. MGRIT overview
  13. Coarse-grid operator
  14. Semi-Lagrangian discretization and its truncation error
  15. Numerical results
  16. ...and 25 more sections

Key Result

Lemma SM1.1

\newlabelSMlem:standard-poly-interp-estSM1 Let $Q_k(x)$ be the unique polynomial of degree at most $k$ that interpolates the function $U(x)$ at the $k+1$ nodes $\{ x_{j+1/2} \}_{j = -\ell-1}^{r}$, $r = k - \ell -1$. Suppose that $U$ is at least $k+1$ times differentiable over the interpolation int with $\varphi(x) \in (x_{-\ell - 1/2}, x_{r+1/2})$ some (unknown) point in the interior of the stenc

Figures (15)

  • Figure 1: Schematic illustration of the conservation property \ref{['eq:cons-lin-SL-cons-local']} that underlies the FV semi-Lagrangian scheme \ref{['eq:SL-FV-scheme']}: The integrals of the solution of \ref{['eq:cons-lin']} over the bold magenta lines are equal. The east-bounding characteristic curve satisfies the final-value problem \ref{['eq:depart-SL-FV']}, and the west-bounding characteristic curve satisfies the same problem with the arrival point shifted from $x_{i+1/2}$ to $x_{i-1/2}$. At the departure time $t_n$, the departure points $\widetilde{\xi}_{i\pm 1/2} := \xi_{i \pm 1/2}(t_n)$ are decomposed into the sum of their east neighbouring cell interfaces $\widetilde{x}_{i \pm 1/2}$ and their mesh-normalized distances $\varepsilon_{i \pm 1/2}$ from these interfaces, $\widetilde{\xi}_{i\pm 1/2} = \widetilde{x}_{i \pm 1/2} - h \varepsilon_{i \pm 1/2}$. \newlabelfig:SL-FV-conservation0
  • Figure 1: Solution plots for numerical test problems: Burgers equation \ref{['eq:burgers']} (left column) and Buckley--Leverett equation \ref{['eq:buck']} (right column). Plots in the top row are space-time contours, with the black lines being 20 contour levels evenly spaced between 0 and 1. Plots in the bottom row are solution cross-sections at the times indicated in the legends. For the Burgers equation, the initial discontinuity at $x = -0.5$ propagates as a rarefaction wave with speed $1$ while the one at $x = 0$ propagates as a shock wave with speed $1/2$. At $t = 1$ these two waves merge and propagate at a reduced speed thereafter (see Supplementary Material Section \ref{['SMsec:over-solving']} for further details). For the Buckley--Leverett equation, both discontinuities propagate as compound waves: They contain both a shock and an attached rarefaction. \newlabelfig:test-prob0
  • Figure 2: Residual convergence histories for the nonlinear solver \ref{['alg:richardson']} applied to 1st-order accurate discretizations. Solid lines correspond to linear systems being approximately solved by one MGRIT iteration, and broken lines correspond to linear systems being solved exactly. Left column: Burgers \ref{['eq:burgers']}. Right column: Buckley--Leverett \ref{['eq:buck']}. The GLF numerical flux is used in the top row, and the LLF numerical flux in the bottom row. All solves use nonlinear F-relaxation with $m = 8$ (Line \ref{['ln:nonlin-relax']} in \ref{['alg:richardson']}). Dotted lines in the bottom left plot show residual histories for $n_x = 64, 128, 256, 1024$ corresponding to using the MGRIT coarse-grid operator \ref{['eq:Psi']} without a truncation error correction, that is, ${\@fontswitch{}{\mathcal{}} T}_{\rm ideal}^{n} = {\@fontswitch{}{\mathcal{}} T}_{\rm direct}^{n} = 0$; note that the $k=2$ data point for $n_x = 1024$ is not visible since $\Vert \bm{r}_2 \Vert / \Vert \bm{r}_0 \Vert \approx 10^{76}$. \newlabelfig:num-res-inexact-1st0
  • Figure 3: Residual convergence histories for the nonlinear solver \ref{['alg:richardson']} applied to 3rd-order accurate discretizations using LLF numerical fluxes. Solid lines correspond to linear systems being approximately solved by one MGRIT iteration, and broken lines correspond to linear systems being solved exactly. Left column: Burgers \ref{['eq:burgers']}. Right column: Buckley--Leverett \ref{['eq:buck']}. All solves use nonlinear F-relaxation with $m = 8$ (Line \ref{['ln:nonlin-relax']} in \ref{['alg:richardson']}). Tow row: Reconstructions are linearized with \ref{['eq:weighted-reconstruct-grad-zero']}, corresponding to Picard linearization of the WENO weights. Bottom row: Reconstructions are linearized with \ref{['eq:weighted-reconstruct-grad-FD']}, corresponding to (approximate) Newton linearization of the WENO weights. \newlabelfig:num-res-inexact-3rd-LLF0
  • Figure SM1: Residual convergence histories for the nonlinear solver Algorithm \ref{['alg:richardson']} when the linearized problems are solved directly. Left column: Burgers \ref{['eq:burgers']}. Right column: Buckley--Leverett \ref{['eq:buck']}. The GLF numerical flux is used in the top row, and the LLF numerical flux in the bottom row. Solid lines correspond to no nonlinear relaxation, and broken lines to nonlinear F-relaxation using a CF-splitting factor $m = 8$. All discretizations are 1st-order accurate. \newlabelSMfig:num-res-exact-1stSM1
  • ...and 10 more figures

Theorems & Definitions (20)

  • Remark 4.1: Newton vs. Picard iteration
  • Remark 6.1: Over-solving
  • Lemma SM1.1: Error estimate for polynomial interpolation
  • Remark SM2.1: Limiting for Buckley--Leverett
  • Lemma SM5.1: Error estimate for semi-Lagrangian method
  • Proof 1
  • Lemma SM5.2: Error estimate for semi-Lagrangian numerical flux
  • Proof 2
  • Lemma SM6.2: Exact solution of integral form of conservation law
  • Proof 3
  • ...and 10 more