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On the Stability Barrier of Hermite Type Discretizations of Advection Equations

Xianyi Zeng

Abstract

In this paper we establish a stability barrier of a class of high-order Hermite-type discretization of 1D advection equations underlying the hybrid-variable (HV) and active flux (AF) methods. These methods seek numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously. It was shown in earlier work that the HV methods are supraconvergent, providing that the discretization uses more unknowns in the upwind direction than the downwind one, similar to the "upwind condition" of classical finite-difference schemes. Although it is well known that the stencil of finite-difference methods could not be too biased towards the upwind direction for stability consideration, known as "stability barrier", such a barrier has not been established for Hermite-type methods. In this work, we first show by numerical evidence that a similar barrier exists for HV methods and make a conjecture on the sharp bound on the stencil. Next, we prove the existence of stability barrier by showing that the semi-discretized HV methods are unstable given a stencil sufficiently biased towards the upwind direction. Tighter barriers are then proved using combinatorical tools, and finally we extend the analysis to studying other Hermite-type methods built on approximating nodal solutions and derivatives, such as those widely used in Hermite WENO methods.

On the Stability Barrier of Hermite Type Discretizations of Advection Equations

Abstract

In this paper we establish a stability barrier of a class of high-order Hermite-type discretization of 1D advection equations underlying the hybrid-variable (HV) and active flux (AF) methods. These methods seek numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously. It was shown in earlier work that the HV methods are supraconvergent, providing that the discretization uses more unknowns in the upwind direction than the downwind one, similar to the "upwind condition" of classical finite-difference schemes. Although it is well known that the stencil of finite-difference methods could not be too biased towards the upwind direction for stability consideration, known as "stability barrier", such a barrier has not been established for Hermite-type methods. In this work, we first show by numerical evidence that a similar barrier exists for HV methods and make a conjecture on the sharp bound on the stencil. Next, we prove the existence of stability barrier by showing that the semi-discretized HV methods are unstable given a stencil sufficiently biased towards the upwind direction. Tighter barriers are then proved using combinatorical tools, and finally we extend the analysis to studying other Hermite-type methods built on approximating nodal solutions and derivatives, such as those widely used in Hermite WENO methods.
Paper Structure (15 sections, 9 theorems, 154 equations, 3 figures, 2 tables)

This paper contains 15 sections, 9 theorems, 154 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Barriers of stable HV methods
  4. Proof of \ref{['thm:bar_sqrt']}
  5. Combinatorics and ReH
  6. Proof of \ref{['thm:bar_pi']}
  7. Proof of \ref{['thm:bar_asymp']}
  8. Relation to other Hermite type methods
  9. Conclusions
  10. Existing strategies for studying stability barrier of FDMs
  11. Order stars
  12. Integral form of the error function
  13. The proof of remaining three cases in \ref{['thm:bar_sqrt']}
  14. Proofs of combinatoric results
  15. Proof of \ref{['eq:hermite_h_identity']} using Sigma

Key Result

Lemma 2.1

The set $\mathcal{S}'$ is contained in $\mathbb{C}^+$ if and only if for all $0<\theta<2\pi$, the following two conditions are satisfied: Here $H=H(\theta)$ and $F=F(\theta)$.

Figures (3)

  • Figure A.1: Order stars (shaded region) of fifth-order FDMs: the left one is stable with $l=3, r=2$, and the right one is unstable with $l=4, r=1$.
  • Figure A.2: Order stars of two HV methods: (upper) a stable one with stencil $(4,3)$ and (lower) an unstable one with stencil $(5,2)$. The principal sheets are in the left panels and the second sheets in the right ones.
  • Figure D.1: Results generated by Sigma.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Conjecture 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • ...and 5 more