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Efficient WENO schemes for nonuniform grids

M. C. Martí, P. Mulet, D. F. Yáñez, D. Zorío

TL;DR

This work develops arbitrarily high-order WENO reconstructions on nonuniform grids with simplified, linearly-costly smoothness indicators, enabling efficient, ENO-like behavior in nonuniform settings. The authors prove that the scheme attains maximum accuracy for smooth data and retains essentially non-oscillatory properties near discontinuities, while keeping the cost growth linear in the order and lower than classical Jiang-Shu-based indicators. The approach is validated through mimetic algebraic tests and a suite of one-dimensional conservation-law problems using a finite-volume framework, including linear advection, Burgers, and Shu-Osher-type scenarios; nonuniform grids often yield higher accuracy with far fewer cells than uniform grids. The practical impact lies in providing a robust, efficient reconstruction tool for high-order finite-volume methods on nonuniform meshes, suitable for shock capturing and problems with sharp gradients, with promising directions for unconditional optimality and boundary-condition handling.

Abstract

A set of arbitrarily high-order WENO schemes for reconstructions on nonuniform grids is presented. These non-linear interpolation methods use simple smoothness indicators with a linear cost with respect to the order, making them easy to implement and computationally efficient. The theoretical analysis to verify the accuracy and the essentially non-oscillatory properties are presented together with some numerical experiments involving algebraic problems in order to validate them. Also, these general schemes are applied for the solution of conservation laws and hyperbolic systems in the context of finite volume methods.

Efficient WENO schemes for nonuniform grids

TL;DR

This work develops arbitrarily high-order WENO reconstructions on nonuniform grids with simplified, linearly-costly smoothness indicators, enabling efficient, ENO-like behavior in nonuniform settings. The authors prove that the scheme attains maximum accuracy for smooth data and retains essentially non-oscillatory properties near discontinuities, while keeping the cost growth linear in the order and lower than classical Jiang-Shu-based indicators. The approach is validated through mimetic algebraic tests and a suite of one-dimensional conservation-law problems using a finite-volume framework, including linear advection, Burgers, and Shu-Osher-type scenarios; nonuniform grids often yield higher accuracy with far fewer cells than uniform grids. The practical impact lies in providing a robust, efficient reconstruction tool for high-order finite-volume methods on nonuniform meshes, suitable for shock capturing and problems with sharp gradients, with promising directions for unconditional optimality and boundary-condition handling.

Abstract

A set of arbitrarily high-order WENO schemes for reconstructions on nonuniform grids is presented. These non-linear interpolation methods use simple smoothness indicators with a linear cost with respect to the order, making them easy to implement and computationally efficient. The theoretical analysis to verify the accuracy and the essentially non-oscillatory properties are presented together with some numerical experiments involving algebraic problems in order to validate them. Also, these general schemes are applied for the solution of conservation laws and hyperbolic systems in the context of finite volume methods.
Paper Structure (22 sections, 2 theorems, 80 equations, 5 figures, 6 tables)

This paper contains 22 sections, 2 theorems, 80 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Scope
  3. Outline of this paper
  4. Preliminaries
  5. Reconstructions on nonuniform grids
  6. Generalized WENO schemes
  7. Construction of the method and theoretical results
  8. Remarks regarding the computational cost
  9. Summary of the algorithms
  10. Algorithm for reconstructions from point values
  11. Algorithm for reconstructions from cell average values
  12. Numerical experiments
  13. Algebraic numerical experiments
  14. Test 1: Reconstruction of a smooth function
  15. Test 2: Reconstruction of a discontinuous function
  16. ...and 7 more sections

Key Result

Proposition 3.1

For the standard case, and keeping the same notation, if $f$ has a critical point of order $k$ at $z$, $0\leq k\leq R-2$, then it holds $0\leq\omega_{R,i,h}\leq1$, $0\leq\omega_{R,h}\leq1$ and with $m=\max\limits_{0\leq i<R-1}m_i$, where

Figures (5)

  • Figure 1: Representation of the different nonuniform meshes considered in Tests 1 and 2 for $h=1$, $z=0$ and the values of $c*$ indicated in each case. Note that to simplify the notation we have used $x_i$ to denote $x_{i,h}$. (a) Mesh for Test 1 and smooth initial data, (b) mesh for Test 1 and discontinuous initial data, (c) mesh for Test 2 and smooth initial data and (d) mesh for Test 2 and discontinuous initial data.
  • Figure 2: Test 3: Numerical solution of the linear advection equation with discontinuous initial data for (a) $n=40$, (b) $n=100$
  • Figure 3: Test 4: Numerical solution of the Burgers equation with discontinuous initial data: (a) $n=20$, (b) for $n=100$
  • Figure 4: Test 5: Numerical solution of the scalar equation converging to a Dirac delta function in the interval $[3.138,3.145]$, obtained using: (a) uniform meshes with $n=25599, 51199, 102399$, (b) non-uniform mesh with $n=198$ and $q=1.1$. (c) and (d) are enlarged views of (a) and (b) respectively.
  • Figure 5: Test 6: Numerical solution of the Shu-Osher problem with CWENO-type schemes with uniform (blue) and non-uniform (red) grids with $n=256$ cells.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 4.1