An Explicit Mapped Tent Pitching Scheme for Maxwell Equations

Christoph Wintersteiger; Jay Gopalakrishnan; Joachim Sch\"oberl; Matthias Hochsteger

arxiv: 1906.11029 · v1 · pith:QUYXORDFnew · submitted 2019-06-26 · 🧮 math.NA · cs.NA

An Explicit Mapped Tent Pitching Scheme for Maxwell Equations

Jay Gopalakrishnan , Matthias Hochsteger , Joachim Sch\"oberl , Christoph Wintersteiger This is my paper

Pith reviewed 2026-05-25 15:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Maxwell equationstent pitchingexplicit schemeshyperbolic equationsfinite elementsparallel methods

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The pith

Mapping causal tents to tensor products yields explicit high-order schemes for Maxwell equations

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

This paper develops Mapped Tent Pitching schemes for solving the time-dependent Maxwell equations. It partitions spacetime into unstructured causal tents and maps them to tensor-product domains to enable local explicit evolution. The method uses a structure-aware Taylor time-stepping technique to achieve high-order accuracy in space and time while supporting variable time steps and local refinements. This results in highly parallel algorithms that work well on modern computer architectures and extends to general linear hyperbolic equations.

Core claim

Provided an approximate solution at the tent bottom, the equations can be evolved to the tent top by mapping each tent to a tensor-product domain, allowing construction of fully explicit schemes that advance the solution through unstructured meshes without compromising high-order accuracy.

What carries the argument

The mapping of each causal tent to a tensor-product domain of space and time, which enables fully explicit time advancement while respecting causality.

If this is right

Variable time steps and local refinements are possible while keeping high order accuracy.
The schemes are suitable for general linear hyperbolic equations.
Highly parallel algorithms result that utilize modern computer architectures well.
Standard explicit Runge-Kutta schemes face difficulties in this context, addressed by Taylor time-stepping.

Where Pith is reading between the lines

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

The approach may allow for better load balancing in parallel computations due to local time steps.
It could be extended to other wave propagation problems beyond Maxwell equations.
Adaptive refinement strategies in spacetime might integrate naturally with this method.

Load-bearing premise

Mapping each causal tent to a tensor-product domain permits construction of a fully explicit scheme while preserving accuracy and stability.

What would settle it

Numerical experiments demonstrating that the mapped scheme loses stability or order of convergence on meshes with significant variation in tent heights.

Figures

Figures reproduced from arXiv: 1906.11029 by Christoph Wintersteiger, Jay Gopalakrishnan, Joachim Sch\"oberl, Matthias Hochsteger.

read the original abstract

We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations. It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent. By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes. This work highlights a difficulty that arises when standard explicit Runge Kutta schemes are used in this context and proposes an alternative structure-aware Taylor time-stepping technique. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. These Mapped Tent Pitching (MTP) schemes lead to highly parallel algorithms, which utilize modern computer architectures extremely well.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete way to make explicit high-order time stepping work with causal spacetime tents on unstructured meshes by mapping to tensor-product domains and using Taylor stepping instead of Runge-Kutta.

read the letter

The main contribution is the mapped tent pitching construction that turns unstructured causal tents into tensor-product domains so a structure-aware Taylor integrator can advance the solution explicitly. This targets the known difficulty that standard Runge-Kutta schemes encounter when tent heights vary. The approach is presented as new in its specific combination of mapping and time stepper, and it aims to keep high order while allowing local time-step variation and parallel execution. That part of the argument is direct and follows from the algorithmic choices described. The paper also notes that the same framework applies to general linear hyperbolic systems, which broadens its scope beyond Maxwell equations. The construction itself shows no internal inconsistency or hidden fitting; the steps are laid out as a straightforward development from existing tent partitioning ideas. A soft spot is that the mapping step itself could introduce overhead or subtle stability effects that are asserted to be harmless but would need explicit checks in the analysis or tests. The claim of no order reduction with variable steps is plausible from the description but rests on the mapping preserving the underlying spatial properties, which is the weakest link until the full error estimates are examined. This work is for people already working on high-order discontinuous Galerkin or finite-element methods for wave problems who need explicit time marching on complex meshes. A reader focused on parallel solvers or spacetime discretizations would get the most out of the algorithmic details. It deserves peer review because the core idea is specific enough and addresses a practical bottleneck in the field.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces Mapped Tent Pitching (MTP) schemes for the time-dependent Maxwell equations and general linear hyperbolic PDEs. Spacetime is partitioned into causal tents respecting causality. Each tent is mapped to a tensor-product domain (spatial domain times time interval), enabling a fully explicit scheme via structure-aware Taylor time-stepping rather than standard Runge-Kutta methods. The resulting methods support variable time steps and local refinements without compromising high-order accuracy in space and time, yielding highly parallel algorithms suited to modern architectures.

Significance. If the central construction holds, the work provides a concrete algorithmic route to explicit, high-order, locally adaptive discretizations for hyperbolic systems that remain stable and accurate under variable steps. The structure-aware Taylor integrator is a targeted fix for the documented difficulty with standard explicit RK schemes inside the tent-pitching framework. The emphasis on parallelism and unstructured-mesh compatibility addresses a practical need in computational electromagnetics. The stress-test assumption that the tensor-product mapping preserves accuracy and stability does not appear to introduce hidden inconsistencies; the paper presents the mapping and integrator choice as directly enabling the claimed properties.

minor comments (2)

The description of how the Taylor time-stepping coefficients are computed after the mapping (likely in the section detailing the explicit scheme) would benefit from an explicit statement of the resulting CFL condition, if any, to confirm it matches the underlying spatial discretization.
A short remark on the precise polynomial degree or order of the Taylor expansion used in the numerical examples would help readers verify the claimed order preservation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the significance of the Mapped Tent Pitching approach for explicit high-order discretizations of hyperbolic systems, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes a direct algorithmic construction: partitioning spacetime into causal tents, mapping each to a tensor-product domain, and applying structure-aware Taylor time-stepping to obtain an explicit scheme. No predictions, fitted parameters, or results are shown to reduce to inputs by construction. No self-citations or uniqueness theorems are invoked in the provided text to justify core steps. The method is presented as following from the mapping and integrator choice without self-referential reduction. This is a standard case of an independent algorithmic development.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented physical entities, or ad-hoc axioms are mentioned; the work rests on standard numerical-analysis assumptions for hyperbolic PDE discretizations.

axioms (1)

standard math Standard assumptions of numerical analysis for hyperbolic PDEs such as consistency and stability of the spatial discretization.
The local evolution step inside each mapped tent presupposes these properties.

pith-pipeline@v0.9.0 · 5694 in / 1153 out tokens · 25367 ms · 2026-05-25T15:21:12.822682+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation (spacetime-emergence, light-cone classification) reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes... causality condition

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Abedi, B

R. Abedi, B. Petracovici, and R. B. Haber. A spacetime discontinuous Galerkin method for elastodynamics with element-wise momentum balance. Computer Methods in Applied Mechanics and Engineering, 195:3247–3273, 2006

work page 2006
[2]

Gopalakrishnan, J

J. Gopalakrishnan, J. Sch ¨oberl, and C. Wintersteiger. Mapped tent pitching schemes for hy- perbolic systems. SIAM J. Sci. Comput. 39 (2017), B1043-B1063

work page 2017
[3]

J. S. Hesthaven and T. Warburton. Nodal discontinuous Galerkin methods, volume 54 ofTexts in Applied Mathematics. Springer, New York, 2008. Algorithms, analysis, and applications

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Hochbruck; T

M. Hochbruck; T. Paur, A. Schulz, E. Thawinan, C. Wieners. Efﬁcient time integration for discontinuous Galerkin approximations of linear wave equations. ZAMM Z. Angew. Math. Mech. 95 (2015), no. 3, 237259

work page 2015
[5]

R. B. Lowrie, P. L. Roe, and B. van Leer. A space-time discontinuous Galerkin method for the time-accurate numerical solution of hyperbolic conservation laws. In Proceedings of the 12th AIAA Computational Fluid Dynamics Conference , number 95-1658, 1995

work page 1995
[6]

Monk and G

P. Monk and G. R. Richter. A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media . J. Sci. Comput., 22/23: 443–477, 2005

work page 2005
[7]

G. R. Richter. An explicit ﬁnite element method for the wave equation. Appl. Numer . Math., 16(1-2):65–80, 1994

work page 1994
[8]

L. Yin, A. Acharia, N. Sobh, R. B. Haber, and D. A. Tortorelli. A spacetime discontinuous Galerkin method for elastodynamics analysis in Discontinuous Galerkin Methods: Theory, Computation and Applications , B. Cockburn, G. Karniadakis, and C.W.Shu (eds), 459–464, 2000

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[1] [1]

Abedi, B

R. Abedi, B. Petracovici, and R. B. Haber. A spacetime discontinuous Galerkin method for elastodynamics with element-wise momentum balance. Computer Methods in Applied Mechanics and Engineering, 195:3247–3273, 2006

work page 2006

[2] [2]

Gopalakrishnan, J

J. Gopalakrishnan, J. Sch ¨oberl, and C. Wintersteiger. Mapped tent pitching schemes for hy- perbolic systems. SIAM J. Sci. Comput. 39 (2017), B1043-B1063

work page 2017

[3] [3]

J. S. Hesthaven and T. Warburton. Nodal discontinuous Galerkin methods, volume 54 ofTexts in Applied Mathematics. Springer, New York, 2008. Algorithms, analysis, and applications

work page 2008

[4] [4]

Hochbruck; T

M. Hochbruck; T. Paur, A. Schulz, E. Thawinan, C. Wieners. Efﬁcient time integration for discontinuous Galerkin approximations of linear wave equations. ZAMM Z. Angew. Math. Mech. 95 (2015), no. 3, 237259

work page 2015

[5] [5]

R. B. Lowrie, P. L. Roe, and B. van Leer. A space-time discontinuous Galerkin method for the time-accurate numerical solution of hyperbolic conservation laws. In Proceedings of the 12th AIAA Computational Fluid Dynamics Conference , number 95-1658, 1995

work page 1995

[6] [6]

Monk and G

P. Monk and G. R. Richter. A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media . J. Sci. Comput., 22/23: 443–477, 2005

work page 2005

[7] [7]

G. R. Richter. An explicit ﬁnite element method for the wave equation. Appl. Numer . Math., 16(1-2):65–80, 1994

work page 1994

[8] [8]

L. Yin, A. Acharia, N. Sobh, R. B. Haber, and D. A. Tortorelli. A spacetime discontinuous Galerkin method for elastodynamics analysis in Discontinuous Galerkin Methods: Theory, Computation and Applications , B. Cockburn, G. Karniadakis, and C.W.Shu (eds), 459–464, 2000

work page 2000