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WENO reconstructions of unconditionally optimal high order

Antonio Baeza, Raimund Bürger, Pep Mulet, David Zorío

TL;DR

This work tackles the problem of accuracy loss in high-order WENO reconstructions near critical points of any order. It introduces OWENO, a reconstruction with non-dimensional, scale-independent weights that rely only on local stencil data, supported by theoretical bounds on smoothness indicators and a novel indicator-based construction. The key contributions include a new smoothness indicator $d_2$ and a harmonic-mean-type combination $D_r$ that ensure unconditionally optimal order $2r-1$ for $r\ge3$, plus extensive numerical validation on smooth and discontinuous problems in one and multi-dimensions. The approach yields robust, high-accuracy reconstructions for hyperbolic conservation laws with competitive computational cost, and it opens avenues for extending unconditional optimality to lower-order schemes and further parameter optimization.

Abstract

A modified Weighted Essentially Non-Oscillatory (WENO) reconstruction technique preventing accuracy loss near critical points (regardless of their order) of the underlying data is presented. This approach only uses local data from the reconstruction stencil and does not rely on any sort of scaling parameters. The key novel ingredient is a weight design based on a new smoothness indicator, which defines the first WENO reconstruction procedure that never loses accuracy on smooth data, regardless of the presence of critical points of any order, and is therefore addressed as optimal WENO (OWENO) method. The corresponding weights are non-dimensional and scale-independent. The weight designs are supported by theoretical results concerning the accuracy of the smoothness indicators. The method is validated by numerical tests related to algebraic equations, scalar conservation laws, and systems of conservation laws.

WENO reconstructions of unconditionally optimal high order

TL;DR

This work tackles the problem of accuracy loss in high-order WENO reconstructions near critical points of any order. It introduces OWENO, a reconstruction with non-dimensional, scale-independent weights that rely only on local stencil data, supported by theoretical bounds on smoothness indicators and a novel indicator-based construction. The key contributions include a new smoothness indicator and a harmonic-mean-type combination that ensure unconditionally optimal order for , plus extensive numerical validation on smooth and discontinuous problems in one and multi-dimensions. The approach yields robust, high-accuracy reconstructions for hyperbolic conservation laws with competitive computational cost, and it opens avenues for extending unconditional optimality to lower-order schemes and further parameter optimization.

Abstract

A modified Weighted Essentially Non-Oscillatory (WENO) reconstruction technique preventing accuracy loss near critical points (regardless of their order) of the underlying data is presented. This approach only uses local data from the reconstruction stencil and does not rely on any sort of scaling parameters. The key novel ingredient is a weight design based on a new smoothness indicator, which defines the first WENO reconstruction procedure that never loses accuracy on smooth data, regardless of the presence of critical points of any order, and is therefore addressed as optimal WENO (OWENO) method. The corresponding weights are non-dimensional and scale-independent. The weight designs are supported by theoretical results concerning the accuracy of the smoothness indicators. The method is validated by numerical tests related to algebraic equations, scalar conservation laws, and systems of conservation laws.
Paper Structure (16 sections, 12 theorems, 94 equations, 3 figures, 6 tables)

This paper contains 16 sections, 12 theorems, 94 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Scope
  3. Related work
  4. Outline of this paper
  5. Regularity properties of functions and smoothness indicators
  6. Preliminaries
  7. WENO reconstructions
  8. Design of WENO weights
  9. Motivation
  10. Novel smoothness indicator
  11. Summary of the algorithm
  12. Numerical experiments
  13. Algebraic test cases
  14. Experiments for conservation laws
  15. Conclusions
  16. ...and 1 more sections

Key Result

Theorem 1

\newlabelis0 Let $z,\alpha\in\mathbb{R}$, $h>0$ and $x_i=z+(\alpha+i)h$, $-r+1\leq i\leq r-1$ define a stencil of equally-spaced nodes. If $f$ has a critical point of order $k$ at $z$, then the Jiang-Shu smoothness indicator eq:SI satisfies ${I_{i}=\bar{\mathcal{O}}(h^{2\kappa})}$, where

Figures (3)

  • Figure 1: Example 5 (Burgers equation, discontinuous solution at $T=12$): fifth-order schemes.
  • Figure 2: Example 7 (Euler equations, Shu-Osher problem): numerical solutions at $T=1.8$ by fifth-order schemes: (a) simulated density for spatial discretization $N=200$, (b--d) enlarged views, (e) simulated density for $N=400$, (f) efficiency plot.
  • Figure 3: Example 8 (Double Mach reflection problem, $2560\times640$, 2D Euler equations of gas dynamics): enlarged views of the turbulent zone of the numerical solutions at $T=0.2$ (Schlieren plot).

Theorems & Definitions (25)

  • Theorem 1
  • Proof 1
  • Proposition 1
  • Proof 2
  • Proposition 2
  • Proof 3
  • Theorem 3
  • Proof 4
  • Theorem 4
  • Proof 5
  • ...and 15 more