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An efficient third-order WENO scheme with unconditionally optimal accuracy

Antonio Baeza, Raimund Bürger, Pep Mulet, David Zorío

TL;DR

This work identifies a fundamental limitation of traditional 3-point third-order WENO schemes: accuracy can degrade near smooth extrema at critical points. It introduces an unconditionally optimal third-order WENO method (OWENO3) that uses an extended four-point data stencil, where the extra node is employed solely in smoothness-weight calculations, not in the flux evaluation. The authors prove the scheme achieves optimal third-order accuracy on smooth data and degrades gracefully to second-order near discontinuities, and extend the approach to systems of hyperbolic conservation laws with consistent flux reconstruction. Numerical experiments across algebraic tests and conservation-law problems demonstrate superior efficiency and accuracy of OWENO3 relative to standard 3rd-order schemes and competitive performance against fifth-order WENO schemes, highlighting practical benefits for weak solutions and complex flow problems.

Abstract

A novel scheme, based on third-order Weighted Essentially Non-Oscillatory (WENO) reconstructions, is presented. It attains unconditionally optimal accuracy when the data is smooth enough, even in presence of critical points, and second-order accuracy if a discontinuity crosses the data. The key to attribute these properties to this scheme is the inclusion of an additional node in the data stencil, which is only used in the computation of the weights measuring the smoothness. The accuracy properties of this scheme are proven in detail and several numerical experiments are presented, which show that this scheme is more efficient in terms of the error reduction versus CPU time than its traditional third-order counterparts as well as several higher-order WENO schemes that are found in the literature.

An efficient third-order WENO scheme with unconditionally optimal accuracy

TL;DR

This work identifies a fundamental limitation of traditional 3-point third-order WENO schemes: accuracy can degrade near smooth extrema at critical points. It introduces an unconditionally optimal third-order WENO method (OWENO3) that uses an extended four-point data stencil, where the extra node is employed solely in smoothness-weight calculations, not in the flux evaluation. The authors prove the scheme achieves optimal third-order accuracy on smooth data and degrades gracefully to second-order near discontinuities, and extend the approach to systems of hyperbolic conservation laws with consistent flux reconstruction. Numerical experiments across algebraic tests and conservation-law problems demonstrate superior efficiency and accuracy of OWENO3 relative to standard 3rd-order schemes and competitive performance against fifth-order WENO schemes, highlighting practical benefits for weak solutions and complex flow problems.

Abstract

A novel scheme, based on third-order Weighted Essentially Non-Oscillatory (WENO) reconstructions, is presented. It attains unconditionally optimal accuracy when the data is smooth enough, even in presence of critical points, and second-order accuracy if a discontinuity crosses the data. The key to attribute these properties to this scheme is the inclusion of an additional node in the data stencil, which is only used in the computation of the weights measuring the smoothness. The accuracy properties of this scheme are proven in detail and several numerical experiments are presented, which show that this scheme is more efficient in terms of the error reduction versus CPU time than its traditional third-order counterparts as well as several higher-order WENO schemes that are found in the literature.
Paper Structure (16 sections, 3 theorems, 52 equations, 11 figures, 5 tables)

This paper contains 16 sections, 3 theorems, 52 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Scope
  3. Related work
  4. Outline of the paper
  5. Optimal third-order scheme
  6. Preliminaries
  7. Third-order WENO reconstructions
  8. On the accuracy loss of third-order WENO schemes
  9. Unconditionally optimal third-order scheme with an additional node
  10. WENO schemes for systems of conservation laws
  11. Hyperbolic systems of conservation laws
  12. Third-order WENO scheme
  13. Numerical experiments
  14. Accuracy tests with algebraic problems
  15. Conservation law experiments
  16. ...and 1 more sections

Key Result

Proposition 2

\newlabelw_accuracy0 If $f$ has a critical point at $z$ of order $k$, $k\in\{0,1\}$, there holds

Figures (11)

  • Figure 1: Example 2b (Burgers equation, discontinuous solution at $T=12$): third-order schemes.
  • Figure 2: Example 3a: Shu-Osher problem. $T=1.8$. $n=200$.
  • Figure 3: Example 3a: Shu-Osher problem. $T=1.8$. $n=400$.
  • Figure 4: Example 3b: Ratio Error/CPU comparison for Shu-Osher problem.
  • Figure 5: Example 4a: Blast wave problem. $T=0.038$.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Remark 2.1
  • Definition 1
  • Proposition 2
  • Proof 1
  • Proposition 3
  • Proof 2
  • Theorem 4
  • Proof 3
  • Remark 2.2