The 1992 paper of Bacry, Mallat and Papanicolaou introduced a wavelet-based space–time adaptive method for evolution PDEs in which spatial resolution is adapted by thresholding wavelet coefficients and the time step is dilated by a fixed factor (2 for advection, 4 for heat) from one resolution level to the next, and it posed the questions that its numerical evidence could not settle. Three decades of later work has settled the boundary-condition and multidimensional-implementation issues and has produced a complete stability and convergence theory for implicit, space–time variational adaptive wavelet methods over general bases; however, no general stability theory exists for the explicit, scale-adaptive (multirate) time stepping that the 1992 scheme and its multiresolution–local-time-stepping successors actually use, the observed wavelet-basis dependence of the stability limit (the 'Daubechies anomaly') remains unexplained, and the choice of the per-scale time-dilation factor for nonlinear equations is still heuristic. The problem is to establish a rigorous stability theory for explicit time-adaptive wavelet/multiresolution schemes valid for general wavelet bases, explain and quantify the basis dependence of the admissible time-step limits, and derive principled rules for adapting the time scaling factor to the nonlinear properties of the equation.

Background

Wavelet bases for PDEs. An orthonormal wavelet basis of L2(R)L^2(\mathbb{R}) is a basis of translates and dilates ψj,k(x)=2j/2ψ(2jx−k)\psi_{j,k}(x)=2^{j/2}\psi(2^j x-k) of a single mother wavelet ψ\psi, organized by scale jj; the nested approximation spaces VjV_j generated by the associated scaling function form a multiresolution analysis, so that a function is represented by coarse-level scaling coefficients plus detail (wavelet) coefficients at each scale. Orthogonality makes the communication between resolution levels automatic, and the fast decay of wavelet coefficients near local irregularities allows spatial adaptivity: one keeps only coefficients above a threshold, so fine-resolution computation is concentrated near singularities or sharp transitions. Classical orthonormal bases include the Shannon wavelet (an ideal bandpass filter, exactly diagonalized by the Fourier transform), the Meyer and Battle–Lemarié wavelets (bandlimited or spline-based, symmetric), and the Daubechies wavelets (compactly supported, orthogonal, generally neither symmetric nor antisymmetric).

Explicit time stepping and the 1992 scheme. For time-dependent PDEs solved by explicit time stepping, stability on a uniform grid is conventionally characterized by a CFL (Courant–Friedrichs–Lewy) bound — the admissible time step must be proportional to the smallest spatial grid spacing — and is analysed by von Neumann analysis, which tracks the amplification of a single Fourier mode over one time step. Bacry, Mallat and Papanicolaou (1992) combined the wavelet multiresolution representation with explicit time stepping: the solution is decomposed into wavelet levels, coefficients below a threshold are discarded (spatial adaptivity), and each resolution level is advanced with its own time step, the step being dilated by a fixed factor (2 for the advection equation, 4 for the heat equation) from one level to the next — an early instance of what is now called local time stepping (LTS), in which different spatial regions are advanced with different time steps. The paper proved stability only for the Shannon wavelet, where every operator involved is diagonal in the Fourier domain, and reported numerical evidence, including a basis-dependent stability anomaly, that it could not explain.

Problem Statement

For wavelet- and multiresolution-based discretizations of time-dependent partial differential equations in which the spatial grid is adapted by wavelet-coefficient thresholding (keeping only coefficients above a prescribed tolerance, so that fine-resolution computation is concentrated near singularities) and the time step is adapted per resolution level — dilated by a fixed factor from one scale to the next, e.g. a factor 2 per octave for the advection equation and 4 for the heat equation, as in the scheme introduced by Bacry, Mallat and Papanicolaou (1992) — three questions are theoretically unresolved and constitute the open target. (1) Establish a general mathematical stability theory for this explicit time-adaptive scheme, valid for general orthonormal wavelet bases and not only for bases diagonalized by the Fourier transform (such as the Shannon basis), covering at least the heat equation and the linear advection equation and their thresholded, spatially adaptive variants; the accepted notion of stability is the standard one for explicit schemes, namely uniform boundedness of the powers of the one-step amplification operator, i.e. a CFL-type (Courant–Friedrichs–Lewy) bound on the admissible time step; for the thresholded, spatially adaptive variants, whose retained active set changes from step to step, the same notion is understood as uniform boundedness of the solution norm along the active-set evolution, so that bounded powers of the fixed linear amplification operator alone do not discharge this requirement. (2) Explain and quantify the observed wavelet-basis dependence of the admissible time-step limits — in particular why, for the 1992 linear-advection scheme, the maximum stable time step of the time-adaptive scheme coincides with that of the non-adaptive scheme for the bandlimited or symmetric bases (Shannon, Meyer, Battle–Lemarié) but differs for the compactly supported, non-symmetric Daubechies wavelets. (3) Derive a principled, non-heuristic rule for adapting the per-level time-dilation factor to the nonlinear properties of the equation — e.g. the balance between the observed growth of low-frequency errors and the transfer of energy to high-frequency components through nonlinear coupling in the Burgers equation — with a guaranteed stability/accuracy trade-off. Two further issues raised by the same 1992 source — efficient treatment of boundary conditions and efficient implementation in several space dimensions — are recorded for completeness as part of the original compound question, but are not part of the open target because later literature has substantially resolved them. The common scientific objective of (1)–(3) is a theoretically founded space–time adaptive wavelet method for time-dependent PDEs whose stability and accuracy are guaranteed rather than empirically observed. Parts (1)–(3) are each independently required; resolving any one of them does not by itself resolve the task, and the basis-dependence explanation and quantification of part (2) is not merely the advection corollary of the general stability theory sought in part (1).

References

  1. E. Bacry, S. Mallat, G. Papanicolaou, 'A wavelet based space-time adaptive numerical method for partial differential equations', M2AN — Mathematical Modelling and Numerical Analysis (ESAIM: Modélisation mathématique et analyse numérique) 26(7):793–834, 1992. DOI: 10.1051/m2an/1992260707931, https://doi.org/10.1051/m2an/1992260707931 (open full text: https://numdam.org/item/M2AN_1992__26_7_793_0/ and PDF https://numdam.org/item/M2AN_1992__26_7_793_0.pdf)
  2. A. Cohen, I. Daubechies, P. Vial, 'Wavelets on the Interval and Fast Wavelet Transforms', Applied and Computational Harmonic Analysis 1(1):54–81, 1993. DOI: 10.1006/acha.1993.1005, https://doi.org/10.1006/acha.1993.1005
  3. A. Cohen, W. Dahmen, R. DeVore, 'Adaptive wavelet methods for elliptic operator equations — convergence rates', Mathematics of Computation 70(233):27–75, 2001 (published online 2000). DOI: 10.1090/S0025-5718-00-01252-7, https://doi.org/10.1090/S0025-5718-00-01252-7
  4. A. Cohen, W. Dahmen, R. DeVore, 'Adaptive Wavelet Methods II — Beyond the Elliptic Case', Foundations of Computational Mathematics 2(3):203–245, 2002. DOI: 10.1007/s102080010027, https://doi.org/10.1007/s102080010027
  5. W. Dahmen, A. Kunoth, K. Urban, 'Biorthogonal Spline Wavelets on the Interval — Stability and Moment Conditions', Applied and Computational Harmonic Analysis 6(2):132–196, 1999. DOI: 10.1006/acha.1998.0247, https://doi.org/10.1006/acha.1998.0247
  6. C. Schwab, R. Stevenson, 'Space-time adaptive wavelet methods for parabolic evolution problems', Mathematics of Computation 78(267):1293–1318, 2009. DOI: 10.1090/S0025-5718-08-02205-9, https://doi.org/10.1090/S0025-5718-08-02205-9
  7. S. Kestler, K. Steih, K. Urban, 'An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations', Mathematics of Computation 85(299):1309–1333, 2016. DOI: 10.1090/mcom/3009, https://doi.org/10.1090/mcom/3009
  8. M. D. Gunzburger, A. Kunoth, 'Space-Time Adaptive Wavelet Methods for Optimal Control Problems Constrained by Parabolic Evolution Equations', SIAM Journal on Control and Optimization 49(3):1150–1170, 2011. DOI: 10.1137/100806382, https://doi.org/10.1137/100806382
  9. M. O. Domingues, S. M. Gomes, O. Roussel, K. Schneider, 'An adaptive multiresolution scheme with local time stepping for evolutionary PDEs', Journal of Computational Physics 227(8):3758–3780, 2008. DOI: 10.1016/j.jcp.2007.11.046, https://doi.org/10.1016/j.jcp.2007.11.046
  10. Müller Moreira Lopes, M. O. Domingues, K. Schneider, O. Mendes, 'Local time-stepping for adaptive multiresolution using natural extension of Runge–Kutta methods', Journal of Computational Physics 382:291–318, 2019. DOI: 10.1016/j.jcp.2018.10.052, https://doi.org/10.1016/j.jcp.2018.10.052
  11. T.-T.-P. Hoang, L. Ju, W. Leng, Z. Wang, 'High order explicit local time stepping methods for hyperbolic conservation laws', Mathematics of Computation 89(324):1807–1842, 2020. DOI: 10.1090/mcom/3507, https://doi.org/10.1090/mcom/3507
  12. M. J. Grote, S. Michel, S. A. Sauter, 'Stabilized leapfrog based local time-stepping method for the wave equation', Mathematics of Computation 90(332):2603–2643, 2021. DOI: 10.1090/mcom/3650, https://doi.org/10.1090/mcom/3650
  13. E. Montseny, S. Pernet, X. Ferriéres, G. Cohen, 'Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell's equations', Journal of Computational Physics 227(14):6795–6820, 2008. DOI: 10.1016/j.jcp.2008.03.032, https://doi.org/10.1016/j.jcp.2008.03.032
  14. M. Krumpholz, L. P. B. Katehi, 'MRTD: new time-domain schemes based on multiresolution analysis', IEEE Transactions on Microwave Theory and Techniques 44(4):555–571, 1996. DOI: 10.1109/22.491023, https://doi.org/10.1109/22.491023
  15. O. V. Vasilyev, S. Paolucci, 'A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs', Journal of Computational Physics 138(1):16–56, 1997. DOI: 10.1006/jcph.1997.5814, https://doi.org/10.1006/jcph.1997.5814
  16. O. V. Vasilyev, C. Bowman, 'Second-Generation Wavelet Collocation Method for the Solution of Partial Differential Equations', Journal of Computational Physics 165(2):660–693, 2000. DOI: 10.1006/jcph.2000.6638, https://doi.org/10.1006/jcph.2000.6638
  17. J. M. Alam, N. K.-R. Kevlahan, O. V. Vasilyev, 'Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations', Journal of Computational Physics 214(2):829–857, 2006. DOI: 10.1016/j.jcp.2005.10.009, https://doi.org/10.1016/j.jcp.2005.10.009
  18. M. Heindl, A. Kunoth, 'An adaptive wavelet viscosity method for systems of hyperbolic conservation laws', Journal of Computational and Applied Mathematics 240:215–224, 2013. DOI: 10.1016/j.cam.2012.07.030, https://doi.org/10.1016/j.cam.2012.07.030
  19. C. Harnish, L. Dalessandro, K. Matouš, D. Livescu, 'An adaptive wavelet method for nonlinear partial differential equations with applications to dynamic damage modeling', Journal of Computational Physics 479:112002, 2023. DOI: 10.1016/j.jcp.2023.112002, https://doi.org/10.1016/j.jcp.2023.112002
  20. C. D. Cochran, K. Matouš, 'Spacetime Wavelet Method for the Solution of Nonlinear Partial Differential Equations', International Journal for Numerical Methods in Engineering 126(13):e70076, 2025. DOI: 10.1002/nme.70076, https://doi.org/10.1002/nme.70076