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Locally conservative and flux consistent iterative methods

Viktor Linders, Philipp Birken

TL;DR

The paper formalizes locally conservative and flux consistent iterations for implicit discretizations of conservation laws, extending the Lax-Wendroff framework to temporally retarded systems via a common flux-modification factor $c$. It shows pseudo-time iterations with explicit RK methods are locally conservative but generally not flux consistent, and provides a flux-consistency correction that restores correct convergence. It then establishes that Newton and Krylov subspace methods are locally conservative (and Newton is flux consistent under bivariate flux assumptions), with the Newton-Krylov combination inheriting local conservation. Numerical experiments on the 2D compressible Euler equations confirm the theory and illustrate that enforcing flux consistency can improve efficiency in some scenarios, while its impact wanes with more iterations and varies with the CFL number.

Abstract

Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow.

Locally conservative and flux consistent iterative methods

TL;DR

The paper formalizes locally conservative and flux consistent iterations for implicit discretizations of conservation laws, extending the Lax-Wendroff framework to temporally retarded systems via a common flux-modification factor . It shows pseudo-time iterations with explicit RK methods are locally conservative but generally not flux consistent, and provides a flux-consistency correction that restores correct convergence. It then establishes that Newton and Krylov subspace methods are locally conservative (and Newton is flux consistent under bivariate flux assumptions), with the Newton-Krylov combination inheriting local conservation. Numerical experiments on the 2D compressible Euler equations confirm the theory and illustrate that enforcing flux consistency can improve efficiency in some scenarios, while its impact wanes with more iterations and varies with the CFL number.

Abstract

Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow.
Paper Structure (15 sections, 10 theorems, 70 equations, 3 figures)

This paper contains 15 sections, 10 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Conservation laws and the Lax-Wendroff theorem
  5. Conservation and consistency of pseudo-time iterations
  6. Systems of conservation laws
  7. Higher order implicit Runge-Kutta methods
  8. Newton's method
  9. Scalar conservation laws
  10. Systems of conservation laws
  11. Krylov subspace methods
  12. Numerical experiments
  13. Convergence tests
  14. Acceleration experiments
  15. Conclusions

Key Result

Theorem 1

Consider a sequence of grids $(\Delta x_\ell, \Delta t_\ell)$ such that $\Delta x_\ell, \Delta t_\ell \rightarrow 0$ as $\ell \rightarrow \infty$. Suppose that the numerical flux $\underline{\mathbf{\boldsymbol{f}}}_{i \pm \frac{1}{2}}$ in eq:FV is consistent with $\underline{\mathbf{\boldsymbol{f}}

Figures (3)

  • Figure 1: Density error upon grid refinement for the compressible Euler equations. Convergence is seen towards the modified conservation laws (Modified), not the original ones (Original), unless flux consistency is enforced (Consistent). (a) Pseudo-time iterations using SSPRK3. (b) Newton's method with SSPRK3 as subsolver for the linear systems.
  • Figure 2: Density error upon grid refinement for the compressible Euler equations. (a) The number of iterations is fixed: One Newton and one GMRES iteration per time step (N1K1); One Newton and two GMRES iterations (N1K2); one Newton with a nearly exact linear solver (N1); nearly exact Newton-GMRES (Exact). (b) Tolerance governed Newton-GMRES with the Eisenstat-Walker procedure.
  • Figure 3: Efficiency study for Newton-GMRES with standard (solid lines) and flux consistent (dotted lines) initial guesses. (a) Function evaluations needed to reach a given residual. (b) Function evaluations per Newton iteration.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Theorem 2
  • Remark 1
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 12 more