The progressive-order cell-average WENO algorithm of Amat, Ruiz-Álvarez, Shu, and Yáñez provably attains high order of accuracy near a discontinuity only under an isolated-discontinuity hypothesis: exactly one jump, sitting essentially at a grid interface, with the rest of the large stencil smooth. Three regimes are excluded and remain unresolved: (a) several discontinuities lying close together inside one large stencil, so that no smooth sub-stencil of the required size exists; (b) a discontinuity strictly inside a fine cell (mid-cell jump), where every substencil strategy considered attains only O(1)O(1) accuracy at the cell containing the jump; and (c) oblique or curved edges in two dimensions, where tensor-product multiresolution was used to sidestep the issue. A 2026 follow-up has built a non-separable cell-average method that handles oblique edges numerically but proves no progressive-order theorem near them. The problem is to extend the accuracy theory and the reconstruction methodology of progressive-order cell-average WENO in Harten multiresolution to these excluded regimes: characterize the attainable accuracy and design reconstructions achieving it.

Background

Weighted Essentially Non-Oscillatory (WENO) reconstruction is a standard technique for interpolating or reconstructing piecewise smooth data without spurious oscillations (Gibbs phenomena) near jumps. Given a large stencil of 2r−12r-1 cells, classical WENO-(2r−1)(2r-1) forms a convex combination, with data-dependent nonlinear weights, of interpolating polynomials built on smaller sub-stencils of rr cells, where the weights are driven by smoothness indicators — functionals that are small on sub-stencils crossed by a discontinuity. In the cell-average setting the data are averages of a function over grid cells rather than point values; this is the natural data format for conservative numerical schemes and for Harten's multiresolution (MR), a data-compression framework in which a fine grid is represented by coarse cell averages plus detail coefficients encoding the difference between fine data and a prediction computed by a reconstruction operator. A good prediction operator concentrates large details at cells affected by discontinuities and is high-order accurate elsewhere.

For data with a discontinuity, classical cell-average WENO-(2r−1)(2r-1) guarantees order 2r−12r-1 in smooth regions but degrades to order rr near the jump. Amat, Ruiz-Álvarez, Shu, and Yáñez introduced (first for point values, then for cell averages via the primitive function, i.e., the cumulative integral of the data whose point values encode the cell averages) a progressive-order modification: the linear optimal weights of the WENO combination are replaced by nonlinear weights depending on the discontinuity's position, so that the attained order increases progressively with the distance between the jump and the target cell. Their Corollary 1 (published version, Applied Mathematics and Computation 403, 2021) states that if ff is smooth on the large stencil [xj−rl−1,xj+r−1l−1][x_{j-r}^{l-1}, x_{j+r-1}^{l-1}] minus a single grid interval Ω=[xj+l0−1l−1,xj+l0l−1]\Omega=[x_{j+l_0-1}^{l-1}, x_{j+l_0}^{l-1}] containing the discontinuity, the predicted fine cell average has error O(hlr+l0−1)O(h_l^{r+l_0-1}), recovering O(h2r−1)O(h^{2r-1}) away from the jump — the order progressively improves cell by cell as one moves away from the discontinuity.

This guarantee, however, is derived under three restrictions documented in the source paper itself. First, the analysis (and the weight design) assumes an isolated discontinuity: the construction is stated to work for discontinuities "far enough from each other", and the theorems allow exactly one contaminated interval in the stencil; when several jumps lie inside one large stencil, no smooth sub-stencil of rr cells need exist and the attainable accuracy is unknown. Second, the position of the jump matters: when the discontinuity falls at a cell end, some sub-stencil is fully smooth and order O(hr)O(h^r) or better survives near it, but when the jump lies strictly inside a fine cell (a mid-cell discontinuity), every sub-stencil containing that cell is contaminated and the paper's experiments show that neither the classical nor the new algorithm can do better than O(1)O(1) error at the cell containing the jump; no reconstruction strategy improving this is proposed. (An older adjacent technique, Harten's ENO (Essentially Non-Oscillatory) subcell resolution of 1989, localizes a discontinuity inside a cell and repairs the reconstruction there, but it has never been combined with the progressive WENO framework.) Third, the two-dimensional experiments were performed with a tensor-product extension — applying the one-dimensional algorithm along rows and columns — and deliberately used only vertical edges, because tensor-product MR produces large spurious details at oblique or curved edges that mask the WENO improvements; a genuinely multidimensional (non-tensor-product) reconstruction preserving the progressive-order property near edges of general orientation is not addressed by the tensor-product construction used in the paper.

Problem Statement

Extend the progressive-order WENO reconstruction theory and methodology for cell-average data in Harten multiresolution to the regimes excluded by the isolated-discontinuity hypothesis of Amat, Ruiz-Álvarez, Shu, and Yáñez (2021). The common objective is a reconstruction operator for cell-average multiresolution that provably concentrates its error near discontinuities while attaining the best achievable order of accuracy cell by cell. Each of the three excluded situations below is an independently standing sub-result that can be resolved on its own, and together the three complete the same accuracy program:

  1. Close discontinuities in one stencil. Characterize the attainable local order of accuracy when two or more discontinuities lie within a single large stencil of 2r−12r-1 cells, so that the separation ('far enough from each other') hypothesis fails and possibly no smooth sub-stencil of rr cells exists; design nonlinear weights and smoothness indicators that achieve this attainable order.

  2. Mid-cell discontinuities. When a jump lies strictly inside a fine cell, the considered substencil strategies cannot beat O(1)O(1) error at the cell containing the discontinuity. Design a reconstruction strategy that provably improves the accuracy attained at such cells in the cell-average multiresolution prediction, by characterizing the best achievable local order of accuracy at cells near a mid-cell jump and attaining that order.

  3. Genuinely multidimensional edges. For two-dimensional cell-average multiresolution, provide a non-tensor-product reconstruction that handles oblique and curved discontinuity edges and carries a proven progressive-order accuracy guarantee near such edges (extending the existing numerical evidence of Garcés et al. 2026 and the axis-aligned, point-value theory of Mulet et al. 2024), so that errors are concentrated at the edge while the order of accuracy recovers progressively with distance from it.

References

  1. Sergio Amat, Juan Ruiz-Álvarez, Chi-Wang Shu, Dionisio F. Yáñez, 'Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing', Applied Mathematics and Computation 403 (2021) 126131. DOI: 10.1016/j.amc.2021.126131; arXiv:2009.09476, https://arxiv.org/abs/2009.09476 (source paper)
  2. Sergio P. Amat, Juan Ruiz, Chi-Wang Shu, Dionisio F. Yáñez, 'A New WENO-2r Algorithm with Progressive Order of Accuracy Close to Discontinuities', SIAM Journal on Numerical Analysis 58(6) (2020) 3448–3474. DOI: 10.1137/20M1337090, https://doi.org/10.1137/20M1337090 (point-value predecessor)
  3. Pep Mulet, Juan Ruiz-Álvarez, Chi-Wang Shu, Dionisio F. Yáñez, 'A non-separable progressive multivariate WENO-2r point value', Applied Numerical Mathematics 204 (2024) 26–47. DOI: 10.1016/j.apnum.2024.05.025; arXiv:2404.16694, https://arxiv.org/abs/2404.16694
  4. Inmaculada Garcés, Pep Mulet, Juan Ruiz-Álvarez, Chi-Wang Shu, Dionisio F. Yáñez, 'A cell-average non-separable progressive multivariate WENO method for image processing applications', Mathematics and Computers in Simulation 248 (2026) 270–290. DOI: 10.1016/j.matcom.2026.04.023; arXiv:2603.04633, https://arxiv.org/abs/2603.04633
  5. Sergio Amat, Juan Ruiz-Álvarez, Chi-Wang Shu, Dionisio F. Yáñez, 'On the approximation of derivative values using a WENO algorithm with progressive order of accuracy close to discontinuities', Computational and Applied Mathematics 41 (2022), article 297. DOI: 10.1007/s40314-022-02004-z, https://doi.org/10.1007/s40314-022-02004-z
  6. Ami Harten, 'Multiresolution Representation of Data: A General Framework', SIAM Journal on Numerical Analysis 33(3) (1996) 1205–1256. DOI: 10.1137/0733060, https://doi.org/10.1137/0733060
  7. Ami Harten, 'ENO schemes with subcell resolution', Journal of Computational Physics 83(1) (1989) 148–184. DOI: 10.1016/0021-9991(89)90226-X, https://doi.org/10.1016/0021-9991(89)90226-X
  8. M. Carmen Martí, Pep Mulet, Dionisio F. Yáñez, David Zorío, 'Efficient WENO Schemes for Nonuniform Grids', Journal of Scientific Computing 100 (2024), article 6. DOI: 10.1007/s10915-024-02558-6, https://doi.org/10.1007/s10915-024-02558-6
  9. Sergio Amat, Sonia Busquier, Juan Ruiz, 'On the convergence of WENO schemes for scalar hyperbolic conservation laws', The Journal of Analysis 33 (2025) 1937–1956. DOI: 10.1007/s41478-025-00909-6, https://doi.org/10.1007/s41478-025-00909-6
  10. Bojan Crnković, Jerko Škifić, Tina Bosner, 'WENO based adaptive image zooming algorithm', Applied Mathematics and Computation 492 (2025) 129228. DOI: 10.1016/j.amc.2024.129228, https://doi.org/10.1016/j.amc.2024.129228