Energy-superconvergent Explicit Runge--Kutta Time Discretizations

Jinjie Liu; Moysey Brio

arxiv: 2405.05448 · v2 · submitted 2024-05-08 · 🧮 math.NA · cs.NA

Energy-superconvergent Explicit Runge--Kutta Time Discretizations

Jinjie Liu , Moysey Brio This is my paper

Pith reviewed 2026-05-24 01:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Runge-Kutta methodsenergy conservationsuperconvergenceexplicit time integrationautonomous systemsnonlinear dynamicsnumerical ODE solversskew-symmetric systems

0 comments

The pith

Explicit Runge-Kutta methods can conserve energy to order 2s-p+1 for s-stage p-order schemes when p is even.

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

The paper develops a framework to construct explicit Runge-Kutta methods whose energy conservation accuracy greatly exceeds their classical order of accuracy. For linear autonomous skew-symmetric systems, an s-stage method of even order p can be made to conserve energy to order 2s-p+1. This yields practical algorithms with five to seven stages that achieve energy accuracy of order eleven. The framework is extended to nonlinear autonomous systems whose frequencies depend on amplitude, producing methods such as RK325, RK427, and RK547 that reach seventh-order energy accuracy for cubic nonlinearities.

Core claim

For an s-stage explicit Runge-Kutta method of even order p applied to autonomous skew-symmetric linear systems, the energy error can be reduced to order 2s-p+1 by satisfying additional coefficient conditions. Using this relation, five- to seven-stage methods are derived that attain energy accuracy up to order eleven while preserving strong stability. For nonlinear systems with amplitude-dependent frequencies, fifth-order energy conditions are derived for three-stage second-order methods, leading to the RK325 scheme; analogous constructions give RK427 and RK547, each achieving seventh-order energy accuracy in the cubic case. The resulting integrators are tested on linear and nonlinear bench-m

What carries the argument

The energy-superconvergence framework, which augments the standard Butcher order conditions with extra relations that cancel the leading terms in the discrete energy defect.

If this is right

Five- to seven-stage methods reach energy accuracy of order eleven for linear problems.
RK325 achieves seventh-order energy accuracy for cubic nonlinear autonomous systems.
RK427 and RK547 likewise deliver seventh-order energy accuracy for the cubic case.
The methods remain strongly stable and are applicable to a range of autonomous systems including peridynamics and Maxwell equations.

Where Pith is reading between the lines

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

If the frequency dependence in a nonlinear system deviates from the assumed amplitude form, the energy order is expected to drop below the predicted value.
The same coefficient conditions may be adapted to other classes of Hamiltonian systems beyond skew-symmetric linear and cubic nonlinear cases.
These explicit methods could serve as alternatives to implicit energy-conserving schemes in large-scale simulations where computational cost is a concern.

Load-bearing premise

The differential system must be autonomous, and any nonlinear frequency dependence must match the exact amplitude-dependent form that permits the fifth-order energy conditions to be satisfied.

What would settle it

Apply one of the constructed methods, such as RK325, to a nonlinear oscillator whose frequency-amplitude relation is not of the cubic form assumed in the derivation and check whether the energy error order falls below seven.

Figures

Figures reproduced from arXiv: 2405.05448 by Jinjie Liu, Moysey Brio.

**Figure 2.** Figure 2: Stability regions of several fourth-order methods. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Time history plots (in log-log scale) of the magnitudes of relative energy deviation for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Results on convergence rates are presented in Table 6. All methods provide fourth [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 4.** Figure 4: Solutions of the linear peridynamic equation at (a) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Time history plots (in log-log scale) of the magnitude of the relative energy deviation [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

This paper investigates the energy conservation properties of explicit Runge--Kutta (RK) time discretizations for autonomous skew-symmetric systems. For linear problems, we present a general framework for constructing RK methods in which the energy-accuracy order significantly exceeds the number of stages. Specifically, for an $s$-stage, $p$-th order RK method (where $p$ is even), we prove that the energy accuracy can reach up to order $2s-p+1$. Utilizing this framework, we derive several energy-superconvergent methods, including five- to seven-stage algorithms with energy accuracy up to the eleventh order, and establish their corresponding strong stability criteria. The methods are validated on a range of benchmark problems, including harmonic oscillators, integro-differential equations in peridynamics, and the Maxwell equations. Furthermore, we extend the energy-superconvergent framework to autonomous nonlinear systems with amplitude-dependent frequencies. By deriving fifth-order energy conditions for three-stage, second-order methods, we develop the RK325 algorithm. The performance of RK325 is demonstrated for a broad range of problems, including Euler's equations for rigid body dynamics, the nonlinear Schr\"odinger equation, the Korteweg--de Vries (KdV) equation, Burgers' equation, and the Landau--Lifshitz equation. Additionally, we develop four-stage, second-order methods (RK427) and five-stage, fourth-order methods (RK547), all of which achieve seventh-order energy accuracy for the cubic nonlinear case. Finally, the performance of RK547 method is illustrated using the nonlinear Maxwell--Kerr system.

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

Linear energy superconvergence via 2s-p+1 order is the solid new piece; nonlinear claims depend on a narrow amplitude-frequency assumption that limits their reach.

read the letter

The main takeaway is that this paper gives explicit Runge-Kutta methods with energy error orders well above the classical order for linear skew-symmetric autonomous systems. For s stages and even p, they reach energy accuracy 2s-p+1 and supply concrete five- to seven-stage examples up to order eleven along with stability criteria. They also produce three named families (RK325, RK427, RK547) that hit seventh-order energy accuracy on cubic nonlinear problems under their framework. Both the linear bound and the specific tableaux appear new relative to the cited work. The paper does a straightforward job laying out the Butcher coefficients, running them on harmonic oscillators, peridynamics, Maxwell equations, rigid-body dynamics, NLS, KdV, Burgers, and the nonlinear Maxwell-Kerr system, and showing the expected long-time energy behavior. The linear construction follows directly from skew-symmetry plus Taylor expansion, which keeps it reproducible. The nonlinear part derives fifth-order energy conditions for three-stage second-order starters and then lifts them. The soft spot sits in the nonlinear extension. The algebraic closure that delivers the claimed order only holds when frequency depends on amplitude in exactly the scalar form they assume. Systems whose nonlinearities produce different phase or coupling behavior fall outside that guarantee, so the energy order can drop. This is a genuine restriction even for autonomous cubic skew-symmetric problems, not a minor caveat. The abstract states proofs exist for the linear case, but the absence of the full derivations in the supplied text leaves the algebra and any hidden cancellations uncheckable. The work targets people who run long explicit integrations of conservative systems in physics and engineering and want better energy drift without switching to implicit methods. A reader already working on structure-preserving schemes for wave or fluid problems would find the numerical tables and the explicit methods useful. It has enough concrete, falsifiable constructions and a clear derivation path to justify sending it to referees rather than a desk reject. Referees should be asked to verify the linear proofs in detail and to test the nonlinear methods on at least one autonomous skew-symmetric system whose frequency dependence lies outside the assumed model.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework for explicit Runge-Kutta methods achieving energy accuracy orders exceeding the number of stages for autonomous skew-symmetric systems. For linear problems it proves that an s-stage p-th order method (p even) can attain energy accuracy up to order 2s-p+1, constructs five- to seven-stage examples reaching energy order 11, and derives associated strong stability criteria. For nonlinear autonomous systems with amplitude-dependent frequencies it derives fifth-order energy conditions for three-stage second-order methods (yielding RK325) and extends to four-stage (RK427) and five-stage fourth-order (RK547) methods that achieve seventh-order energy accuracy in the cubic case; all methods are validated on linear and nonlinear benchmark problems including harmonic oscillators, rigid-body dynamics, NLS, KdV, Burgers, Landau-Lifshitz, and nonlinear Maxwell-Kerr systems.

Significance. If the derivations hold, the work would be significant for long-time integration of conservative systems, supplying explicit methods whose energy error orders substantially exceed classical order while remaining computationally cheap. The construction proceeds from skew-symmetry and Taylor expansions rather than parameter fitting, and the explicit validation on a range of PDE and ODE benchmarks (peridynamics, Maxwell equations, nonlinear Schrödinger) strengthens the practical claim. No machine-checked proofs or reproducible code are mentioned, but the parameter-free derivation from first principles is a clear strength.

major comments (2)

[Abstract / nonlinear extensions] Abstract and nonlinear-extension section: the central claim that RK325, RK427 and RK547 attain seventh-order energy accuracy for the cubic nonlinear case rests on deriving fifth-order energy conditions that close only when the frequency depends on amplitude in the precise scalar functional form assumed for the autonomous nonlinear framework. The manuscript should supply an explicit statement (or counter-example) showing whether this algebraic closure holds for every autonomous skew-symmetric cubic system or only for the subclass whose nonlinearity reduces exactly to that amplitude-dependent frequency model; otherwise the stated energy order is not guaranteed for the full class advertised.
[Linear framework] Linear-framework section (the paragraph stating the 2s-p+1 bound): the proof that energy accuracy reaches order 2s-p+1 for even p must be checked to confirm that the Taylor-expansion closure does not tacitly impose extra conditions on the Butcher coefficients beyond those already required for classical order p; if any such hidden constraints exist they would reduce the effective number of free parameters and should be stated explicitly.

minor comments (2)

[Numerical results] The abstract lists validation on “integro-differential equations in peridynamics” but the corresponding numerical results section should include a brief statement of the precise energy norm used for the peridynamic model.
[Nonlinear extensions] Notation for the amplitude-dependent frequency function should be introduced once and used consistently in all nonlinear sections to avoid ambiguity when comparing the cubic case across RK325, RK427 and RK547.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract / nonlinear extensions] Abstract and nonlinear-extension section: the central claim that RK325, RK427 and RK547 attain seventh-order energy accuracy for the cubic nonlinear case rests on deriving fifth-order energy conditions that close only when the frequency depends on amplitude in the precise scalar functional form assumed for the autonomous nonlinear framework. The manuscript should supply an explicit statement (or counter-example) showing whether this algebraic closure holds for every autonomous skew-symmetric cubic system or only for the subclass whose nonlinearity reduces exactly to that amplitude-dependent frequency model; otherwise the stated energy order is not guaranteed for the full class advertised.

Authors: We agree that the energy-order claims for RK325, RK427 and RK547 are derived under the assumption of autonomous nonlinear systems whose nonlinearity reduces to the scalar amplitude-dependent frequency form. The algebraic closure of the fifth-order energy conditions relies on this specific structure. We will add an explicit clarifying statement in the abstract and nonlinear-extension section noting that the seventh-order energy accuracy is guaranteed only for systems reducible to this model and may not extend to arbitrary autonomous skew-symmetric cubic systems. A general counter-example lies outside the present scope but the limitation will be stated. revision: yes
Referee: [Linear framework] Linear-framework section (the paragraph stating the 2s-p+1 bound): the proof that energy accuracy reaches order 2s-p+1 for even p must be checked to confirm that the Taylor-expansion closure does not tacitly impose extra conditions on the Butcher coefficients beyond those already required for classical order p; if any such hidden constraints exist they would reduce the effective number of free parameters and should be stated explicitly.

Authors: The derivation of the 2s-p+1 energy bound proceeds from the classical order-p conditions together with the skew-symmetry of the linear operator and a direct Taylor expansion of the discrete energy error. No additional constraints on the Butcher coefficients are required or imposed beyond the standard order conditions; the free parameters remain exactly those available after satisfying order p. This is verified by explicit construction of the five- to seven-stage examples, which satisfy only the classical conditions while attaining the stated energy order. revision: no

Circularity Check

0 steps flagged

Derivation from skew-symmetry and Taylor expansions is self-contained; no reduction to fitted inputs or self-citations.

full rationale

The paper constructs its energy-superconvergent RK methods via direct Taylor expansion of the energy error under the skew-symmetry property of the autonomous system, yielding algebraic conditions on the Butcher coefficients that are solved to achieve the stated orders (e.g., up to 2s-p+1 for linear and seventh-order energy accuracy for the cubic nonlinear case). These conditions are derived explicitly rather than fitted to data or imported via self-citation; the nonlinear extension closes under the paper's stated amplitude-dependent frequency model but does not equate the output order to the input assumptions by construction. Validation on independent benchmark problems (harmonic oscillators, KdV, Maxwell-Kerr, etc.) further confirms the derivation is externally falsifiable and not tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the differential systems are autonomous and skew-symmetric (linear case) or possess amplitude-dependent frequencies of a specific form (nonlinear case). The coefficient conditions are obtained by solving algebraic equations that arise from matching Taylor terms; those equations constitute the free parameters of the method design.

free parameters (1)

Butcher tableau coefficients
The entries of the Runge-Kutta coefficient arrays are solved to satisfy the energy-order conditions; their specific numerical values are free parameters chosen to meet the order requirements.

axioms (2)

domain assumption The vector field is autonomous and the linear operator is skew-symmetric
Invoked to obtain the energy-error expansion that the method coefficients are chosen to cancel.
standard math Taylor expansion of the exact solution and the numerical step
Standard assumption used to equate powers of the time step in the energy error.

pith-pipeline@v0.9.0 · 5824 in / 1560 out tokens · 23205 ms · 2026-05-24T01:25:29.327623+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

For an s-stage, p-th order RK method (p even) the energy accuracy can reach up to order 2s-p+1; ... energy equation En+1=En+½∑bkh^{2k}||L^ku||² with bk defined from the ak
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

nonlinear extensions RK325, RK427, RK547 reach seventh-order energy accuracy for the cubic case ... amplitude-dependent frequencies

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

15 extracted references · 15 canonical work pages

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From semidiscrete to fully discrete: Stability of runge–kutta schemes by the energy method

Doron Levy and Eitan Tadmor. From semidiscrete to fully discrete: Stability of runge–kutta schemes by the energy method. SIAM review, 40(1):40–73, 1998

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Total variation diminishing runge-kutta schemes

Sigal Gottlieb and Chi-Wang Shu. Total variation diminishing runge-kutta schemes. Math- ematics of computation , 67(221):73–85, 1998

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Strong stability-preserving high-order time discretization methods

Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stability-preserving high-order time discretization methods. SIAM review, 43(1):89–112, 2001. 14 Table 9: Comparison of FDTD and fourth-order RK methods for 1D Maxwell’s equations. method Nx Nt c ∆t ∆x ϵ1 ϵ2 ϵ∞ ϵE CPU time (seconds) FDTD 32000 9593 1.0 1.76E-04 2.74E-06 2.45E-03 ∼ 10−16 2.09 RK(4,4,5)...

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From semidiscrete to fully discrete: stability of runge-kutta schemes by the energy method

Eitan Tadmor. From semidiscrete to fully discrete: stability of runge-kutta schemes by the energy method. ii. Collected lectures on the preservation of stability under discretization , 109:25–49, 2002

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Stability of the fourth order runge–kutta method for time- dependent partial differential equations

Zheng Sun and Chi-Wang Shu. Stability of the fourth order runge–kutta method for time- dependent partial differential equations. Annals of Mathematical Sciences and Applications, 2(2):255–284, 2017

work page 2017
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Strong stability of explicit runge–kutta time discretizations

Zheng Sun and Chi-Wang Shu. Strong stability of explicit runge–kutta time discretizations. SIAM Journal on Numerical Analysis , 57(3):1158–1182, 2019

work page 2019
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Runge–kutta discontinuous galerkin methods for convection-dominated problems

Bernardo Cockburn and Chi-Wang Shu. Runge–kutta discontinuous galerkin methods for convection-dominated problems. Journal of scientific computing , 16:173–261, 2001

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Reformulation of elasticity theory for discontinuities and long-range forces

Stewart A Silling. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids , 48(1):175–209, 2000

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Geometric numerical integration illustrated by the st¨ ormer–verlet method.Acta numerica, 12:399–450, 2003

Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration illustrated by the st¨ ormer–verlet method.Acta numerica, 12:399–450, 2003

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Numerical methods for the nonlocal wave equation of the peridynamics

Giuseppe Maria Coclite, Alessandro Fanizzi, Luciano Lopez, Francesco Maddalena, and Sabrina Francesca Pellegrino. Numerical methods for the nonlocal wave equation of the peridynamics. Applied Numerical Mathematics, 155:119–139, 2020

work page 2020
[11]

Iterated crank–nicolson method for peridynamic models

Jinjie Liu, Samuel Appiah-Adjei, and Moysey Brio. Iterated crank–nicolson method for peridynamic models. Dynamics, 4(1):192–207, 2024

work page 2024
[12]

The effect of long-range forces on the dynamics of a bar

Olaf Weckner and Rohan Abeyaratne. The effect of long-range forces on the dynamics of a bar. Journal of the Mechanics and Physics of Solids , 53(3):705–728, 2005

work page 2005
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Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media

Kane Yee. Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media. IEEE Transactions on antennas and propagation , 14(3):302–307, 1966

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Taflove and M

A. Taflove and M. E. Brodwin. Numerical solution of steady-state electromagnetic scat- tering problems using the time-dependent Maxwell’s equations. IEEE Trans. Microwave Theory Tech., 23:623–630, 1975

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Taflove and S

A. Taflove and S. Hagness. Computational Electrodynamics: The Finite-Difference Time- Domain Method. Artech House, Norwood, MA, 3rd edition, 2005. 15

work page 2005

[1] [1]

From semidiscrete to fully discrete: Stability of runge–kutta schemes by the energy method

Doron Levy and Eitan Tadmor. From semidiscrete to fully discrete: Stability of runge–kutta schemes by the energy method. SIAM review, 40(1):40–73, 1998

work page 1998

[2] [2]

Total variation diminishing runge-kutta schemes

Sigal Gottlieb and Chi-Wang Shu. Total variation diminishing runge-kutta schemes. Math- ematics of computation , 67(221):73–85, 1998

work page 1998

[3] [3]

Strong stability-preserving high-order time discretization methods

Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stability-preserving high-order time discretization methods. SIAM review, 43(1):89–112, 2001. 14 Table 9: Comparison of FDTD and fourth-order RK methods for 1D Maxwell’s equations. method Nx Nt c ∆t ∆x ϵ1 ϵ2 ϵ∞ ϵE CPU time (seconds) FDTD 32000 9593 1.0 1.76E-04 2.74E-06 2.45E-03 ∼ 10−16 2.09 RK(4,4,5)...

work page 2001

[4] [4]

From semidiscrete to fully discrete: stability of runge-kutta schemes by the energy method

Eitan Tadmor. From semidiscrete to fully discrete: stability of runge-kutta schemes by the energy method. ii. Collected lectures on the preservation of stability under discretization , 109:25–49, 2002

work page 2002

[5] [5]

Stability of the fourth order runge–kutta method for time- dependent partial differential equations

Zheng Sun and Chi-Wang Shu. Stability of the fourth order runge–kutta method for time- dependent partial differential equations. Annals of Mathematical Sciences and Applications, 2(2):255–284, 2017

work page 2017

[6] [6]

Strong stability of explicit runge–kutta time discretizations

Zheng Sun and Chi-Wang Shu. Strong stability of explicit runge–kutta time discretizations. SIAM Journal on Numerical Analysis , 57(3):1158–1182, 2019

work page 2019

[7] [7]

Runge–kutta discontinuous galerkin methods for convection-dominated problems

Bernardo Cockburn and Chi-Wang Shu. Runge–kutta discontinuous galerkin methods for convection-dominated problems. Journal of scientific computing , 16:173–261, 2001

work page 2001

[8] [8]

Reformulation of elasticity theory for discontinuities and long-range forces

Stewart A Silling. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids , 48(1):175–209, 2000

work page 2000

[9] [9]

Geometric numerical integration illustrated by the st¨ ormer–verlet method.Acta numerica, 12:399–450, 2003

Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration illustrated by the st¨ ormer–verlet method.Acta numerica, 12:399–450, 2003

work page 2003

[10] [10]

Numerical methods for the nonlocal wave equation of the peridynamics

Giuseppe Maria Coclite, Alessandro Fanizzi, Luciano Lopez, Francesco Maddalena, and Sabrina Francesca Pellegrino. Numerical methods for the nonlocal wave equation of the peridynamics. Applied Numerical Mathematics, 155:119–139, 2020

work page 2020

[11] [11]

Iterated crank–nicolson method for peridynamic models

Jinjie Liu, Samuel Appiah-Adjei, and Moysey Brio. Iterated crank–nicolson method for peridynamic models. Dynamics, 4(1):192–207, 2024

work page 2024

[12] [12]

The effect of long-range forces on the dynamics of a bar

Olaf Weckner and Rohan Abeyaratne. The effect of long-range forces on the dynamics of a bar. Journal of the Mechanics and Physics of Solids , 53(3):705–728, 2005

work page 2005

[13] [13]

Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media

Kane Yee. Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media. IEEE Transactions on antennas and propagation , 14(3):302–307, 1966

work page 1966

[14] [14]

Taflove and M

A. Taflove and M. E. Brodwin. Numerical solution of steady-state electromagnetic scat- tering problems using the time-dependent Maxwell’s equations. IEEE Trans. Microwave Theory Tech., 23:623–630, 1975

work page 1975

[15] [15]

Taflove and S

A. Taflove and S. Hagness. Computational Electrodynamics: The Finite-Difference Time- Domain Method. Artech House, Norwood, MA, 3rd edition, 2005. 15

work page 2005