pith. sign in

arxiv: 2507.04024 · v3 · submitted 2025-07-05 · 🧮 math.NA · cs.NA

Exploring Exponential Runge-Kutta Methods: A Survey

Alessia and\`o , Nicol\`o Cangiotti , Mattia Sensi This is my paper

Pith reviewed 2026-05-19 06:14 UTC · model grok-4.3

classification 🧮 math.NA cs.NA MSC 65L0565L06
keywords exponential Runge-Kutta methodsexponential integratorsnumerical methods for ODEsstiff differential equationsinitial value problemshistorical surveynumerical integration
0
0 comments X

The pith

Exponential Runge-Kutta methods combine classical Runge-Kutta structure with exponential integrators for initial-value problems.

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

The survey examines exponential Runge-Kutta methods as a bridge between long-established Runge-Kutta schemes and exponential integrators first proposed in the 1960s. It traces the historical growth of these methods and supplies concrete examples to show how they operate on differential equations. A reader would care because the combination can handle stiff linear terms while retaining flexible stage computations for nonlinear parts, potentially improving accuracy and step-size choices in numerical solutions. The manuscript aims to present this material in an accessible way for people outside the immediate research community.

Core claim

Exponential Runge-Kutta methods constitute a synthesis of classical Runge-Kutta methods, which date back more than a century, and exponential integrators, which originated in the 1960s; the survey supplies a historical account of their development to the present together with illustrative examples that demonstrate their use on initial-value problems.

What carries the argument

Exponential Runge-Kutta methods, numerical schemes that embed exponential functions to treat linear stiff components exactly while applying Runge-Kutta-type stages to the remaining nonlinear terms.

If this is right

  • These methods can maintain stability for larger time steps when linear parts of the equation are stiff.
  • They reduce the need for very small steps that classical explicit Runge-Kutta methods often require on stiff problems.
  • The historical perspective clarifies how early exponential ideas were later combined with multistage Runge-Kutta frameworks.
  • Concrete examples illustrate practical implementation details that readers can adapt to their own equations.

Where Pith is reading between the lines

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

  • Researchers might test whether specific exponential Runge-Kutta variants outperform standard implicit methods on particular classes of stiff systems arising in applications.
  • The survey's accessibility focus suggests these methods could be incorporated into teaching materials for numerical analysis courses that currently emphasize only classical or exponential integrators separately.
  • Future surveys could quantify performance gains by collecting error and cost data across a standardized set of test problems.

Load-bearing premise

The selected historical developments and examples chosen for the survey are representative of the field and free from major omissions or biases.

What would settle it

Identification of a significant body of prior work on exponential Runge-Kutta methods that the survey omits or mischaracterizes in its historical account.

Figures

Figures reproduced from arXiv: 2507.04024 by Alessia and\`o, Mattia Sensi, Nicol\`o Cangiotti.

Figure 1
Figure 1. Figure 1: Relative error as a function of computational time, highlight￾ing the efficiency of each method (1a) and relative error as a function of step size, showing how accuracy evolves with different step choices (1b). The results presented in [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relative error as a function of step size, showing how accu￾racy evolves with different step choices (2a) and relative error as a func￾tion of computational time, highlighting the efficiency of each method (2b). The results presented in [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
read the original abstract

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.

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

0 major / 2 minor

Summary. The manuscript is a survey on exponential Runge-Kutta methods for the numerical integration of initial-value problems. It claims that these methods provide a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators dating back to the 1960s. The paper includes a historical analysis of the development of these methods up to the present day along with several illustrative examples intended to make the topic accessible to a broad audience.

Significance. Should the historical analysis prove accurate and the examples representative without major omissions, this survey could serve as an accessible entry point into the field of exponential integrators and their connection to traditional Runge-Kutta schemes. It does not introduce new mathematical results but organizes and presents existing literature, which can be significant for educational purposes and for guiding future research in numerical methods for differential equations. The survey format allows for a broad overview that might not be found in individual research papers.

minor comments (2)
  1. The abstract states that the methods 'offer a valuable synthesis', but this positioning could be elaborated in the introduction with a brief comparison of key features from both parent classes to strengthen the central framing.
  2. The manuscript would benefit from a dedicated subsection on open problems or future directions in exponential Runge-Kutta methods to provide forward-looking value beyond the historical survey.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our survey manuscript on exponential Runge-Kutta methods. We appreciate the recognition of its potential value as an accessible entry point into the field and its educational significance. The recommendation for minor revision is noted, and we will incorporate any necessary clarifications or corrections in the revised version.

Circularity Check

0 steps flagged

No significant circularity in this literature survey

full rationale

This manuscript is a survey paper that provides historical analysis and illustrative examples of exponential Runge-Kutta methods without presenting any original derivations, theorems, predictions, fitted parameters, or new mathematical results. The central positioning statement frames the methods as a synthesis of classical Runge-Kutta and exponential integrators, but this is purely descriptive and relies on citations to external prior literature rather than any internal chain that reduces to self-definition or self-citation. No equations, proofs, or data-fitting steps exist that could create circularity by construction, making the work self-contained against external benchmarks with no load-bearing internal reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey the paper introduces no new free parameters, axioms, or invented entities; it relies entirely on established literature in numerical analysis.

pith-pipeline@v0.9.0 · 5602 in / 958 out tokens · 96041 ms · 2026-05-19T06:14:08.996837+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

87 extracted references · 87 canonical work pages

  1. [1]

    A. H. Al-Mohy and N. J. Higham. A new scaling and squaring algorithm for the matrix expo- nential. SIAM Journal on Matrix Analysis and Applications , 31(3):970–989, 2010

    work page 2010

  2. [2]

    A. H. Al-Mohy and N. J. Higham. Computing the action of the matrix exponential, with an ap- plication to exponential integrators. SIAM journal on scientific computing , 33(2):488–511, 2011

    work page 2011

  3. [3]

    And` o and R

    A. And` o and R. Vermiglio. Exponential Runge-Kutta methods fo r delay equations in the sun-star abstract framework. Discrete and Continuous Dynamical Systems-B , 30(6):1842–1858, 2025

    work page 2025

  4. [4]

    And` o and R

    A. And` o and R. Vermiglio. Exponential time integration for delay d ifferential equations via pseu- dospectral discretization. IF AC-PapersOnLine, 58(27):190–195, 2024. 18th IF AC Workshop on Time Delay Systems TDS 2024

    work page 2024

  5. [5]

    Bellen and M

    A. Bellen and M. Zennaro. Numerical methods for delay differential equations . Numerical Math- emathics and Scientifing Computing series. Oxford University Press , 2003

    work page 2003

  6. [6]

    Berland, B

    H. Berland, B. Owren, and B. Skaflestad. B-series and order co nditions for exponential integrators. SIAM Journal on Numerical Analysis , 43(4):1715–1727, 2005

    work page 2005

  7. [7]

    Berland, B

    H. Berland, B. Owren, and B. Skaflestad. Solving the nonlinear Sc hr¨ odinger equation using ex- ponential integrators. Modeling, Identification And Control , 27(4):201–217, 2006

    work page 2006

  8. [8]

    Berland, B

    H. Berland, B. Skaflestad, and W. M. Wright. EXPINT—A MATLAB p ackage for exponential integrators. ACM Trans. Math. Software , 33(1), 2007

    work page 2007

  9. [9]

    Beylkin, J

    G. Beylkin, J. M. Keiser, and L. Vozovoi. A new class of time discret ization schemes for the solution of nonlinear PDEs. Journal of computational physics , 147(2):362–387, 1998

    work page 1998

  10. [10]

    Breda, O

    D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, and R. Verm iglio. Pseudospectral dis- cretization of nonlinear delay equations: new prospects for numer ical bifurcation analysis. SIAM J. Appl. Dyn. Sys. , 15(1):1–23, 2016

    work page 2016

  11. [11]

    J. C. Butcher. Implicit Runge-Kutta processes. Mathematics of Computation , 18:50–64, 1964

    work page 1964

  12. [12]

    J. C. Butcher. A history of Runge-Kutta methods. Applied numerical mathematics , 20(3):247– 260, 1996

    work page 1996

  13. [13]

    Caliari, F

    M. Caliari, F. Cassini, L. Einkemmer, and A. Ostermann. Accelera ting exponential integrators to efficiently solve semilinear advection-diffusion-reaction equations . SIAM Journal on Scientific Computing, 46(2):A906–A928, 2024

    work page 2024

  14. [14]

    Caliari and A

    M. Caliari and A. Ostermann. Implementation of exponential Ro senbrock-type integrators. Ap- plied Numerical Mathematics , 59(3-4):568–581, 2009

    work page 2009

  15. [15]

    Caliari, M

    M. Caliari, M. Vianello, and L. Bergamaschi. Interpolating discret e advection–diffusion prop- agators at Leja sequences. Journal of Computational and Applied Mathematics , 172(1):79–99, 2004

    work page 2004

  16. [16]

    Certaine

    J. Certaine. The solution of ordinary differential equations with large time constants. Mathemat- ical methods for digital computers , 1:128–132, 1960. 28 A. AND `O, N. CANGIOTTI, AND M. SENSI

    work page 1960

  17. [17]

    S. C. Chu and M. Berman. An exponential method for the solutio n of systems of ordinary differ- ential equations. Communications of the ACM , 17(12):699–702, 1974

    work page 1974

  18. [18]

    Costabile

    F. Costabile. A survey of pseudo Runge-Kutta methods. Rend. Mat. Roma , 23(VII):217–234, 2003

    work page 2003

  19. [19]

    S. M. Cox and P. C. Matthews. Exponential time differencing for stiff systems. Journal of Com- putational Physics , 176(2):430–455, 2002

    work page 2002

  20. [20]

    C. F. Curtiss and J. O. Hirschfelder. Integration of stiff equat ions. Proceedings of the national academy of sciences , 38(3):235–243, 1952

    work page 1952

  21. [21]

    A special stability problem for linear multistep method s

    G Dahlquist. A special stability problem for linear multistep method s. BIT Numerical Mathe- matics, 3:27–43, 1963

    work page 1963

  22. [22]

    Dahlquist

    G. Dahlquist. Problems related to the numerical treatment of s tiff differential equations. In G¨ unther, E., et al. (eds.) International Computing Sympos ium, North Holland, Amsterdam, 1973

    work page 1973

  23. [23]

    T. T. Dang and T. H. Hoang. How to avoid order reduction in third -order exponential Runge– Kutta methods for problems with non-commutative operators? Preprint available on arXiv: 2412.11920, 2024

    work page arXiv 2024

  24. [24]

    P. I. Davies and N. J. Higham. A Schur-Parlett algorithm for com puting matrix functions. SIAM Journal on Matrix Analysis and Applications , 25(2):464–485, 2003

    work page 2003

  25. [25]

    Diekmann, P

    O. Diekmann, P. Getto, and M. Gyllenberg. Stability and bifurcat ion analysis of Volterra func- tional equations in the light of suns and stars. SIAM J. Math. Anal. , 39(4):1023–1069, 2008

    work page 2008

  26. [26]

    Diekmann, S

    O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walthe r. Delay Equations – Functional, Complex and Nonlinear Analysis . Number 110 in Applied Mathematical Sciences. Springer Verlag, New York, 1995

    work page 1995

  27. [27]

    Dimarco and L

    G. Dimarco and L. Pareschi. Exponential Runge–Kutta method s for stiff kinetic equations. SIAM Journal on Numerical Analysis , 49(5):2057–2077, 2011

    work page 2057

  28. [28]

    Dujardin

    G. Dujardin. Exponential Runge–Kutta methods for the Schr ¨ odinger equation.Applied numerical mathematics, 59(8):1839–1857, 2009

    work page 2009

  29. [29]

    Einkemmer, T.-H

    L. Einkemmer, T.-H. Hoang, and A. Ostermann. Should exponen tial integrators be used for advection-dominated problems? Preprint available on arXiv: 2410.12765, 2024

    work page arXiv 2024

  30. [30]

    W. H. Enright, D. J. Higham, B. Owren, and P. W. Sharp. A surve y of the explicit Runge-Kutta method

    work page

  31. [31]

    Fang and R

    J. Fang and R. Zhan. High order explicit exponential Runge–Kut ta methods for semilinear delay differential equations. Journal of Computational and Applied Mathematics , 388:113279, 2021

    work page 2021

  32. [32]

    Gallopoulos and Y

    E. Gallopoulos and Y. Saad. On the parallel solution of parabolic eq uations. In Proceedings of the 3rd international conference on Supercomputing , pages 17–28, 1989

    work page 1989

  33. [33]

    Gallopoulos and Y

    E. Gallopoulos and Y. Saad. Efficient solution of parabolic equation s by Krylov approximation methods. SIAM journal on scientific and statistical computing , 13(5):1236–1264, 1992

    work page 1992

  34. [34]

    Gonz´ alez, A

    C. Gonz´ alez, A. Ostermann, and M. Thalhammer. A second-or der magnus-type integrator for nonautonomous parabolic problems. Journal of Computational and Applied Mathematics , 189(1):142–156, 2006. Proceedings of The 11th International C ongress on Computational and Applied Mathematics

    work page 2006

  35. [35]

    K. G. Guderley and C.-C. Hsu. A predictor-corrector method f or a certain class of stiff differential equations. Mathematics of Computation , 26(117):51–69, 1972

    work page 1972

  36. [36]

    Solving ordinary differential equations II: stiff and differe ntial-algebraic problems

    E Hairer and G Wanner. Solving ordinary differential equations II: stiff and differe ntial-algebraic problems. Number 14 in Computational Mathematics. Springer-Verlag, Berlin , 1991

    work page 1991

  37. [37]

    J. Hersch. Contribution ` a la m´ ethode des ´ equations aux diff´erences. Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP , 9:129–180, 1958

    work page 1958

  38. [38]

    N. J. Higham. The scaling and squaring method for the matrix exp onential revisited. SIAM Journal on Matrix Analysis and Applications , 26(4):1179–1193, 2005

    work page 2005

  39. [39]

    N. J. Higham. Functions of Matrices: Theory and Computation, 2008

    work page 2008

  40. [40]

    N. J. Higham and A. H. Al-Mohy. Computing matrix functions. Acta Numerica , 19:159–208, 2010

    work page 2010

  41. [41]

    T. H. Hoang. Order Reduction of Exponential Runge–Kutta Me thods: Non-Commuting Opera- tors. Preprint available on arXiv: 2410.00470, 2024. EXPLORING EXPONENTIAL RUNGE-KUTTA METHODS 29

    work page arXiv 2024

  42. [42]

    Hochbruck and C

    M. Hochbruck and C. Lubich. On Krylov subspace approximation s to the matrix exponential operator. SIAM Journal on Numerical Analysis , 34(5):1911–1925, 1997

    work page 1911

  43. [43]

    Hochbruck, C

    M. Hochbruck, C. Lubich, and H. Selhofer. Exponential integr ators for large systems of differential equations. SIAM Journal on Scientific Computing , 19(5):1552–1574, 1998

    work page 1998

  44. [44]

    Hochbruck and A

    M. Hochbruck and A. Ostermann. Explicit exponential Runge–K utta methods for semilinear parabolic problems. SIAM Journal on Numerical Analysis , 43(3):1069–1090, 2005

    work page 2005

  45. [45]

    Hochbruck and A

    M. Hochbruck and A. Ostermann. Exponential Runge–Kutta m ethods for parabolic problems. Applied Numerical Mathematics , 53(2-4):323–339, 2005

    work page 2005

  46. [46]

    Hochbruck and A

    M. Hochbruck and A. Ostermann. Exponential integrators. Acta Numerica, 19:209–286, 2010

    work page 2010

  47. [47]

    Hochbruck and A

    M. Hochbruck and A. Ostermann. Exponential multistep metho ds of Adams-type. BIT, 51(4):889–908, December 2011

    work page 2011

  48. [48]

    Hochbruck, A

    M. Hochbruck, A. Ostermann, and J. Schweitzer. Exponentia l Rosenbrock-type methods. SIAM Journal on Numerical Analysis , 47(1):786–803, 2009

    work page 2009

  49. [49]

    Huang, A

    Q. Huang, A. Ostermann, and G. Zhong. Exponential Runge-K utta methods of collocation type for parabolic equations with time-dependent delay. Preprint available on arXiv: 2503.04674, 2025

    work page arXiv 2025

  50. [50]

    K. J. In ’t Hout. A new interpolation procedure for adapting Run ge-Kutta methods to delay differential equations. BIT, 32(4):634–649, 1992

    work page 1992

  51. [51]

    Kassam and L

    A.-K. Kassam and L. N. Trefethen. Fourth-order time-stepp ing for stiff PDEs. SIAM Journal on Scientific Computing , 26(4):1214–1233, 2005

    work page 2005

  52. [52]

    S. Koikari. Rooted tree analysis of Runge–Kutta methods with e xact treatment of linear terms. Journal of computational and applied mathematics , 177(2):427–453, 2005

    work page 2005

  53. [53]

    Kovacic and M

    I. Kovacic and M. J. Brennan, editors. The Duffing Equation: Nonlinear Oscillators and their Behaviour. Wiley, 2011

    work page 2011

  54. [54]

    W. Kutta. Beitrag zur n¨ aherungsweisen integration totaler d ifferentialgleichungen. Zeitschrift f¨ ur Mathematik und Physik , 46:435–453, 1901

    work page 1901

  55. [55]

    J.D. Lambert. Computational Methods in Ordinary Differential Equations . Introductory mathe- matics for scientists and engineers. Wiley, 1973

    work page 1973

  56. [56]

    J.D. Lambert. Numerical Methods for Ordinary Differential Systems: The In itial Value Problem . Wiley, 1991

    work page 1991

  57. [57]

    J. D. Lawson. Generalized Runge-Kutta processes for stable systems with large Lipschitz con- stants. SIAM Journal on Numerical Analysis , 4(3):372–380, 1967

    work page 1967

  58. [58]

    Y. Y. Lu. Computing a matrix function for exponential integrat ors. Journal of computational and applied mathematics, 161(1):203–216, 2003

    work page 2003

  59. [59]

    V. T. Luan. Fourth-order two-stage explicit exponential inte grators for time-dependent PDEs. Applied Numerical Mathematics , 112:91–103, 2017

    work page 2017

  60. [60]

    V. T. Luan. Efficient exponential Runge–Kutta methods of high order: construction and imple- mentation. BIT Numerical Mathematics , 61(2):535–560, 2021

    work page 2021

  61. [61]

    V. T. Luan and T. Alhsmy. Sixth-order exponential Runge–Kut ta methods for stiff systems. Applied Mathematics Letters , 153:109036, 2024

    work page 2024

  62. [62]

    V. T. Luan, R. Chinomona, and D. R. Reynolds. A new class of high -order methods for multirate differential equations. SIAM Journal on Scientific Computing , 42(2):A1245–A1268, 2020

    work page 2020

  63. [63]

    V. T. Luan and A. Ostermann. Exponential B-series: The stiff c ase. SIAM Journal on Numerical Analysis, 51(6):3431–3445, 2013

    work page 2013

  64. [64]

    V. T. Luan and A. Ostermann. Explicit exponential Runge–Kutt a methods of high order for parabolic problems. Journal of Computational and Applied Mathematics , 256:168–179, 2014

    work page 2014

  65. [65]

    V. T. Luan and A. Ostermann. Exponential Rosenbrock metho ds of order five—construction, analysis and numerical comparisons. Journal of Computational and Applied Mathematics , 255:417–431, 2014

    work page 2014

  66. [66]

    V. T. Luan and A. Ostermann. Stiff order conditions for expone ntial Runge–Kutta methods of order five. In Modeling, Simulation and Optimization of Complex Processe s-HPSC 2012: Proceed- ings of the Fifth International Conference on High Performa nce Scientific Computing, March 5-9, 2012, Hanoi, Vietnam , pages 133–143. Springer, 2014. 30 A. AND `O, N. CANGI...

    work page 2012

  67. [67]

    V. T. Luan and A. Ostermann. Parallel exponential Rosenbroc k methods. Computers & Mathe- matics with Applications , 71(5):1137–1150, 2016

    work page 2016

  68. [68]

    V. T. Luan, N. Van Hoang, and J. O. Ehigie. Efficient exponential methods for genetic regulatory systems. Journal of Computational and Applied Mathematics , 436:115424, 2024

    work page 2024

  69. [69]

    Maset and M

    S. Maset and M. Zennaro. Unconditional stability of explicit expo nential Runge–Kutta methods for semi-linear ordinary differential equations. Mathematics of computation , 78(266):957–967, 2009

    work page 2009

  70. [70]

    Maset and M

    S. Maset and M. Zennaro. Stability properties of explicit expone ntial Runge–Kutta methods. IMA Journal of Numerical Analysis , 33(1):111–135, 2013

    work page 2013

  71. [71]

    D. L. Michels, G. A. Sobottka, and A. G. Weber. Exponential int egrators for stiff elastodynamic problems. ACM Transactions on Graphics (TOG) , 33(1):1–20, 2014

    work page 2014

  72. [72]

    B. V. Minchev. Exponential Integrators for Semilinear Problems. Phd thesis, University of Bergen, Bergen, NO, August 2004. Available at https://www.ii.uib.no/~borko/pub/PhD_thesis.pdf

    work page 2004

  73. [73]

    B. V. Minchev and W. Wright. A review of exponential integrator s for first order semi-linear problems. Technical Report Preprint numerics 2/2005, Norwegian University Of Science And Technology, Trondheim, Norway, 2005

    work page 2005

  74. [74]

    W. L. Miranker. Numerical methods for stiff equations and singular perturba tion problems. Math- ematics and its applications ; 5. D. Reidel, Dordrecht, Holland, 1981

    work page 1981

  75. [75]

    Moler and C

    C. Moler and C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM review , 45(1):3–49, 2003

    work page 2003

  76. [76]

    Montanelli and N

    H. Montanelli and N. Bootland. Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators. Mathematics and Computers in Simulation , 178:307–327, 2020

    work page 2020

  77. [77]

    S. P. Nørsett. An A-stable modification of Adams-Bashforth’s method for he numerical integra- tion of ordinary stiff differential equations. Preprint series: Pure mathematics http://urn. nb. no/URN: NBN: no-8076 , 1969

    work page 1969

  78. [78]

    B. N. Parlett. Computation of Functions of Triangular Matrices . Memorandum (University of California, Berkeley, Electronics Research Laboratory). Defens e Technical Information Center, 1974

    work page 1974

  79. [79]

    D. A. Pope. An exponential method of numerical integration of ordinary differential equations. Communications of the ACM , 6(8):491–493, 1963

    work page 1963

  80. [80]

    C. Runge. ¨Uber die numerische aufl¨ osung von differentialgleichungen. Mathematische Annalen , 46:167–178, 1895

    work page

Showing first 80 references.