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arxiv: 2604.11163 · v1 · submitted 2026-04-13 · 🧮 math.NA · cs.NA· hep-lat· hep-th· physics.comp-ph

From Exact Space-Time Symmetry Conservation to Automatic Mesh Refinement in Discrete Initial Boundary Value Problems

Alexander Rothkopf , W.A. Horowitz , Jan Nordstr\"om This is my paper

Pith reviewed 2026-05-10 15:43 UTC · model grok-4.3

classification 🧮 math.NA cs.NAhep-lathep-thphysics.comp-ph
keywords variational formulationinitial boundary value problemsNoether chargesautomatic mesh refinementsummation-by-parts operatorsspace-time symmetriesdiscrete conservation lawsadaptive discretization
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The pith

Promoting coordinate maps to dynamical degrees of freedom in variational IBVPs preserves exact Noether charge conservation after discretization and induces automatic mesh refinement.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that elevating coordinate maps to dynamical variables alongside physical fields in the variational action for initial boundary value problems protects space-time symmetries even after discretization. This protection yields exact conservation of the associated Noether charges for discrete systems. Because the maps themselves evolve, the space-time grid resolution adjusts automatically in a manner guided by those conservation laws, producing a built-in form of adaptive mesh refinement. The construction relies on summation-by-parts operators and remains valid at high order whether the dynamics are integrated from the action or from the governing equations, as demonstrated for scalar waves in one spatial dimension plus time.

Core claim

By including coordinate maps as dynamical degrees of freedom together with propagating fields in a variational action formulation of IBVPs, space-time symmetries remain protected even after discretization. This leads to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. As long as SBP operators are used for the discretization, the results hold independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order.

What carries the argument

Coordinate maps promoted to dynamical degrees of freedom in the variational action, discretized with summation-by-parts operators that enforce exact Noether charge conservation and thereby drive automatic resolution adjustment.

If this is right

  • Exact Noether charge conservation holds for arbitrary discrete IBVPs when summation-by-parts operators are used.
  • Automatic adaptive mesh refinement emerges directly from the dynamics of the coordinate maps guided by charge conservation.
  • The conservation and refinement properties remain valid at high orders of accuracy.
  • The results apply equally when the problem is solved from the action or from the governing equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational treatment could allow mesh evolution in higher-dimensional or nonlinear systems without manual intervention.
  • Noether charge residuals might serve as built-in monitors of numerical fidelity in general discretizations.
  • The approach could reduce the need for a priori mesh design in problems with evolving features.

Load-bearing premise

That promoting coordinate maps to dynamical degrees of freedom preserves the variational structure without introducing inconsistencies or altering physical content, and that SBP operators suffice to guarantee exact conservation for arbitrary IBVPs beyond the 1+1 scalar-wave case.

What would settle it

A 1+1 scalar-wave simulation using the method in which the discrete Noether charges fail to be conserved to machine precision or the mesh fails to adjust in accordance with the charge dynamics.

Figures

Figures reproduced from arXiv: 2604.11163 by Alexander Rothkopf, Jan Nordstr\"om, W.A. Horowitz.

Figure 1
Figure 1. Figure 1: Initial Boundary Value Problems: unique predictions for the outcome of exper￾iments from the interplay of dynamics, informed by experimental input in the form of initial and boundary conditions. experimental setup, representing the initial conditions for our prediction. The dynamics of the subsequent evolution in time is encoded in the classical action of the system, often represented in the form of govern… view at source ↗
Figure 2
Figure 2. Figure 2: The variational formulation of classical mechanics. (left) Traditional Hamilton’s principle formulated as boundary value problem. It applies to problems where the final state of the system is known priori. (right) Modern Schwinger-Keldysh-Galley principle formulated as genuine initial value problem. It is based on a doubling of d.o.f. akin to a double shooting method. (figure from [14]) Let us start by con… view at source ↗
Figure 2
Figure 2. Figure 2: fig. 2. By supplying [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (left) Solution for x1 (red open circles) and x2 (blue crosses), obtained from finding the extremum of the naively discretized action eq. (10) shows contamination by the highly oscillatory π-mode. Analytic solution given as gray solid line. (right) Spectrum of the unregularized lowest order SBP operator D [1,2,1] (blue open circles) and the regularized SBP operator (green triangles). Note that the latter d… view at source ↗
Figure 4
Figure 4. Figure 4: (left) Solution for x1 (red open circles) and x2 (blue crosses), obtained from finding the extremum of the discretized action eq. (10) using the regularized SBP operator D¯ [1,2,1]. No more contamination by the π-mode appears in the comparison to the analytic solution (gray solid). (right) Competitive convergence of the solution under grid refinement showing a powerlaw behavior with ∆t2.03 as expected for … view at source ↗
Figure 5
Figure 5. Figure 5: (top row) Conventional formulation of IBVPs using space-time coordinates (t, x) as independent variables. The dynamical degree of freedom ϕ(t, x) evolves as function of space-time coordinates. When naively discretized, the grid structure in space-time coordinates breaks space-time symmetries and continuum Noether charges fail to be conserved. (bottom row) Proposed formulation of IBVPs in the presence of dy… view at source ↗
Figure 6
Figure 6. Figure 6: The setup for our proof-of-principle in 1+1 dimensions. (left) The conventional action approach to scalar wave evolution. The action is expressed in terms of space￾time coordinates and its variation leads to the well-known wave equation. (right) Our new action is expressed in abstract τ, σ variables. The Nambu-Goto term (blue-green shaded) takes on a simple form involving only coordinate map derivatives. T… view at source ↗
Figure 7
Figure 7. Figure 7: (left) Numerical solution of the field evolution as a function of τ and σ, ob￾tained from the optimization of the discretized action eq. (28). Wave propagation is initialized from a bump centered in the spatial domain from which both a left- and right-moving wave-package emanates. The packages approach the Dirichlet boundary, reflect, including a phase flip, and approach the center of the simulation domain… view at source ↗
Figure 8
Figure 8. Figure 8: Discrete Noether charge associated with infinitesimal time translations defined in eq. (26) and evaluated on the solution of the discretized action eq. (28). Note that it is preserved exactly at each τ step. (figure from [13]) The exact conservation of Noether charges actually provides the guiding prin￾ciple for the automatic mesh refinement we observe. This can be understood al￾ready in the continuum form… view at source ↗
read the original abstract

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes treating coordinate maps as dynamical degrees of freedom within a variational action formulation of classical IBVPs. Using SBP discretizations, space-time symmetries are claimed to remain exactly protected at the discrete level, yielding machine-precision conservation of Noether charges independent of whether the system is discretized at the action or equation level. This conservation is asserted to drive automatic adaptive mesh refinement. The approach is presented as a proof-of-principle for the 1+1 scalar wave equation and stated to hold at high order for general discrete IBVPs.

Significance. If the exact discrete Noether conservation generalizes beyond the provided example, the framework would constitute a notable advance in structure-preserving discretizations for hyperbolic systems, offering a symmetry-based route to adaptive refinement without external error indicators. The independence from action-versus-equation formulation and compatibility with high-order SBP operators are potential strengths that could influence numerical methods in relativity, acoustics, and other fields where exact conservation laws are desirable.

major comments (2)
  1. [Proof-of-principle application and abstract] The central claim that the results hold for arbitrary discrete IBVPs (provided only that SBP operators are used) is not supported by a general derivation. The manuscript demonstrates the construction only for the 1+1 scalar wave equation as proof-of-principle; no discrete Noether theorem is supplied that covers higher-dimensional systems, constraints, or non-trivial boundary conditions where the dynamical coordinate maps must simultaneously satisfy the discrete Euler-Lagrange equations and boundary closure without introducing symmetry-violating truncation errors.
  2. [Variational formulation and discretization sections] The assertion that the exact conservation is independent of action-versus-equation level requires explicit verification once the coordinate maps become unknowns. While SBP guarantees spatial telescoping, the time discretization and the coupling of the dynamical coordinates to the fields must still be shown to inherit the continuous symmetry exactly; the 1+1 example does not isolate whether this independence survives for general IBVPs.
minor comments (2)
  1. Clarify the precise definition and evolution equation for the dynamical coordinate maps, including how their degrees of freedom are counted and discretized alongside the physical fields.
  2. Add a brief comparison to existing adaptive-mesh or moving-mesh methods that also exploit variational principles or conservation laws.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the recognition of the potential advance in structure-preserving discretizations and address the major comments point by point below, indicating planned revisions.

read point-by-point responses

  1. Referee: [Proof-of-principle application and abstract] The central claim that the results hold for arbitrary discrete IBVPs (provided only that SBP operators are used) is not supported by a general derivation. The manuscript demonstrates the construction only for the 1+1 scalar wave equation as proof-of-principle; no discrete Noether theorem is supplied that covers higher-dimensional systems, constraints, or non-trivial boundary conditions where the dynamical coordinate maps must simultaneously satisfy the discrete Euler-Lagrange equations and boundary closure without introducing symmetry-violating truncation errors.

    Authors: We agree that the manuscript provides an explicit construction and verification only for the 1+1 scalar wave equation as a proof-of-principle and does not contain a fully general discrete Noether theorem. The generality claim rests on the observation that SBP operators guarantee exact discrete telescoping (integration by parts) in any spatial dimension together with compatible boundary closures, while the variational treatment of the coordinate maps ensures that the discrete action remains invariant under the relevant space-time symmetries. This structure is intended to carry over to higher-dimensional systems and more complex settings without introducing symmetry-violating truncation errors from the spatial operators. Nevertheless, we acknowledge that explicit verification for constraints and non-trivial boundaries lies beyond the current scope. We will revise the abstract and add a dedicated discussion subsection that (i) clarifies the scope of the proof-of-principle, (ii) outlines how the SBP-plus-variational mechanism extends, and (iii) states the additional analysis required for the cases mentioned by the referee. revision: partial

  2. Referee: [Variational formulation and discretization sections] The assertion that the exact conservation is independent of action-versus-equation level requires explicit verification once the coordinate maps become unknowns. While SBP guarantees spatial telescoping, the time discretization and the coupling of the dynamical coordinates to the fields must still be shown to inherit the continuous symmetry exactly; the 1+1 example does not isolate whether this independence survives for general IBVPs.

    Authors: In the 1+1 scalar-wave example we explicitly compare the two routes: direct discretization of the action (with dynamical coordinates included variationally) versus discretization of the resulting Euler-Lagrange equations. In both cases the Noether charges are conserved to machine precision, because the discrete action is constructed to be invariant under the discrete symmetry transformations and the SBP property supplies exact spatial telescoping. The time discretization is a variational integrator that inherits the same invariance. The independence therefore follows from the fact that the Noether identity is a direct consequence of the variational principle plus the SBP telescoping property, independent of whether one solves the discrete action or the discrete equations. We will add a short clarifying paragraph in the discretization section that isolates this argument, references the 1+1 verification, and indicates why the same reasoning applies to general SBP-based IBVPs. revision: yes

standing simulated objections not resolved
  • A complete general discrete Noether theorem covering arbitrary dimensions, constraints, and non-trivial boundary conditions is not supplied in the manuscript and would require substantial additional work beyond the present proof-of-principle.

Circularity Check

0 steps flagged

No circularity: derivation follows from variational structure plus external SBP properties

full rationale

The paper starts from a standard variational action principle for classical dynamics, promotes coordinate maps to dynamical degrees of freedom, and discretizes using summation-by-parts (SBP) operators. Exact Noether conservation then follows from the telescoping property of SBP, which is an established discrete analogue of integration by parts and is independent of the present work. The 1+1 scalar-wave example serves as explicit verification rather than a tautological fit; no parameter is tuned to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the claimed independence from action-versus-equation level is justified directly by the SBP construction. The automatic mesh-refinement consequence is a derived outcome of the dynamical coordinates, not a redefinition of the input symmetry. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on the standard variational action principle and the use of summation-by-parts operators; the key addition is the dynamical treatment of coordinates, which is introduced without independent empirical support beyond the 1+1 wave example.

axioms (2)
  • domain assumption Modern variational action formulation of classical dynamics
    The paper builds upon this formulation to treat IBVPs directly on the action level.
  • domain assumption Summation-by-parts (SBP) operators preserve the necessary discrete properties
    The exact conservation and independence from action vs. equation level are stated to hold whenever SBP operators are used.
invented entities (1)
  • Dynamical coordinate maps no independent evidence
    purpose: Serve as additional degrees of freedom that protect space-time symmetries and drive automatic mesh refinement
    Coordinate maps are promoted to dynamical variables inside the variational formulation.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Journal of Mathematical Physics61(11) (2020)

    Anerot, B., Cresson, J., Hariz Belgacem, K., Pierret, F.: Noether’s-type theorems on time scales. Journal of Mathematical Physics61(11) (2020)

    work page 2020

  2. [2]

    Cambridge University Press (2019)

    Carroll, S.: Spacetime and Geometry. Cambridge University Press (2019)

    work page 2019

  3. [3]

    Communications on pure and applied math- ematics5(3), 243–255 (1952)

    Courant, R., Isaacson, E., Rees, M.: On the solution of nonlinear hyperbolic differ- ential equations by finite differences. Communications on pure and applied math- ematics5(3), 243–255 (1952)

    work page 1952

  4. [4]

    Computers & Fluids95, 171–196 (2014)

    Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Computers & Fluids95, 171–196 (2014)

    work page 2014

  5. [5]

    Shackleton, A

    Galley, C.R.: Classical Mechanics of Nonconservative Systems. Physical Review Letters110(17), 174301 (Apr 2013).https://doi.org/10.1103/PhysRevLett. 110.174301,https://link.aps.org/doi/10.1103/PhysRevLett.110.174301, publisher: American Physical Society

    work page doi:10.1103/physrevlett 2013

  6. [6]

    Pearson Education India (2011)

    Goldstein, H.: Classical mechanics. Pearson Education India (2011)

    work page 2011

  7. [7]

    Horowitz, W.A., Rothkopf, A.: Hamilton Revised: The Action Principle for Initial Value Problems (3 2026)

    work page 2026

  8. [8]

    Journal of Computational Physics270, 86–104 (2014)

    Lundquist, T., Nordström, J.: The SBP-SAT technique for initial value problems. Journal of Computational Physics270, 86–104 (2014)

    work page 2014

  9. [9]

    SIAM Journal on Scientific Computing38(3), A1561–A1586 (2016)

    Nordström, J., Lundquist, T.: Summation-by-parts in time: the second derivative. SIAM Journal on Scientific Computing38(3), A1561–A1586 (2016)

    work page 2016

  10. [10]

    Applied Mathematics and Computa- tion272, 403–419 (2016)

    Pinto, M.C., Mounier, M., Sonnendrücker, E.: Handling the divergence constraints in maxwell and vlasov–maxwell simulations. Applied Mathematics and Computa- tion272, 403–419 (2016)

    work page 2016

  11. [11]

    Rothkopf, A., Horowitz, W.A.: Variational approach to nonholonomic and inequality-constrained mechanics. Phys. Rev. E113(2), 024126 (2026).https: //doi.org/10.1103/rch9-wsbw

    work page doi:10.1103/rch9-wsbw 2026

  12. [12]

    Fxhenn: Fpga-based acceleration framework for homomorphic encrypted cnn inference,

    Rothkopf, A.: Exact symmetry preservation and automatic mesh refinement for 1+1d scalar wave propagation (Apr 2024).https://doi.org/10.5281/zenodo. 11082746,https://doi.org/10.5281/zenodo.11082746

    work page doi:10.5281/zenodo 2024

  13. [13]

    Rothkopf, A., Horowitz, W.A., Nordström, J.: Exact symmetry conservation and automatic mesh refinement in discrete initial boundary value problems. J. Comput. Phys.524, 113686 (2025).https://doi.org/10.1016/j.jcp.2024.113686 18 A. Rothkopf et al

    work page doi:10.1016/j.jcp.2024.113686 2025

  14. [14]

    Rothkopf, A., Nordström, J.: A new variational discretization technique for ini- tial value problems bypassing governing equations. J. Comput. Phys.477, 111942 (2023).https://doi.org/10.1016/j.jcp.2023.111942

    work page doi:10.1016/j.jcp.2023.111942 2023

  15. [15]

    Rothkopf, A., Nordström, J.: A symmetry and Noether charge preserving dis- cretization of initial value problems. J. Comput. Phys.498, 112652 (2024).https: //doi.org/10.1016/j.jcp.2023.112652

    work page doi:10.1016/j.jcp.2023.112652 2024

  16. [16]

    Rothkopf, A., Nordström, J.: The crucial role of Lagrange multipliers in a space- time symmetry preserving discretization scheme for IVPs. J. Comput. Phys.511, 113138 (2024).https://doi.org/10.1016/j.jcp.2024.113138

    work page doi:10.1016/j.jcp.2024.113138 2024

  17. [17]

    Journal of Computational Physics268, 17–38 (2014)

    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial– boundary-value problems. Journal of Computational Physics268, 17–38 (2014)

    work page 2014

  18. [18]

    Journal of Computational Physics397, 108819 (2019)

    Svärd, M., Nordström, J.: On the convergence rates of energy-stable finite- difference schemes. Journal of Computational Physics397, 108819 (2019)

    work page 2019

  19. [19]

    Wilson, K.G.: Confinement of Quarks. Phys. Rev. D10, 2445–2459 (1974).https: //doi.org/10.1103/PhysRevD.10.2445

    work page doi:10.1103/physrevd.10.2445 1974

  20. [20]

    Cambridge university press (2004)

    Zwiebach, B.: A first course in string theory. Cambridge university press (2004)

    work page 2004