# Citing

To credit the JuliaDiffEq software, please star the repositories that you would like to support. If you use JuliaDiffEq software as part of your research, teaching, or other activities, we would be grateful if you could cite our work. Since JuliaDiffEq is a collection of individual modules, in order to give proper credit please cite each related component.

To give proper academic credit, all software should be cited. See this link for more information on citing software. Listed below are relevant publications which should be cited upon usage. For software which do not have publications, recommended citations are also included. If you have any questions about citations, please feel free to file an issue at the Github repository or ask in the Gitter channel. If any of this information needs to be updated, please open an issue or pull request at the website repository.

#### DifferentialEquations.jl

• Rackauckas, C. & Nie, Q., (2017). DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software. 5(1), p.15. DOI: http://doi.org/10.5334/jors.151

#### StochasticDiffEq.jl

• Rackauckas C., Nie Q., (2016). Adaptive Methods for Stochastic Differential Equations via Natural Embeddings and Rejection Sampling with Memory. Discrete and Continuous Dynamical Systems - Series B, 22(7), pp. 2731-2761. doi:10.3934/dcdsb.2017133

• P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems, SIAM J. Sci. Comp., 15 (1994), pp. 1467-1488.

• P. N. Brown, A. C. Hindmarsh, and L. R. Petzold, Consistent Initial Condition Calculation for Differential-Algebraic Systems, SIAM J. Sci. Comp. 19 (1998), pp. 1495-1512.

#### DASSL.jl

• DASSL.jl. Date of access. Current version. https://github.com/JuliaDiffEq/DASSL.jl.

#### DelayDiffEq.jl

• DelayDiffEq.jl. Date of access. Current version. https://github.com/JuliaDiffEq/DelayDiffEq.jl.

#### LSODA.jl

• Alan Hindmarsh, ODEPACK, a Systematized Collection of ODE Solvers, in Scientific Computing, edited by Robert Stepleman, Elsevier, 1983, ISBN13: 978-0444866073, LC: Q172.S35.

• K Radhakrishnan, Alan Hindmarsh, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations, Technical report UCRL-ID-113855, Lawrence Livermore National Laboratory, December 1993.

• Linda Petzold, Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations, SIAM J. Sci. and Stat. Comput., 4(1), 136–148.

• LSODA.jl. Date of access. Current version. https://github.com/rveltz/LSODA.jl.

#### ODE.jl

• ODE.jl. Date of access. Current version. https://github.com/JuliaDiffEq/ODE.jl.

#### OrdinaryDiffEq.jl

• OrdinaryDiffEq.jl. Date of access. Current version. https://github.com/JuliaDiffEq/OrdinaryDiffEq.jl.

#### ParameterizedFunctions.jl

• ParameterizedFunctions.jl. Date of access. Current version. https://github.com/JuliaDiffEq/ParameterizedFunctions.jl.

#### StochasticDiffEq.jl

• StochasticDiffEq.jl. Date of access. Current version. https://github.com/JuliaDiffEq/StochasticDiffEq.jl.

#### Sundials.jl

• A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,” ACM Transactions on Mathematical Software, 31(3), pp. 363-396, 2005. Also available as LLNL technical report UCRL-JP-200037.

#### SymEngine.jl (if the ode_def macro is used)

• SymEngine. Date of access. Current version. https://github.com/symengine/symengine.

## Algorithm Citations

Many of the algorithms which are included as part of this ecosystem of software packages originated as part of academic research. If you know which algorithms were used in your work, please use this as a reference for determining additional citations.

### Ordinary Differential Equations

#### BS3, ode23

• Bogacki, Przemyslaw; Shampine, Lawrence F. (1989), “A 3(2) pair of Runge–Kutta formulas”, Applied Mathematics Letters, 2 (4): 321–325, doi:10.1016/0893-9659(89)90079-7

#### Tsit5

• Tsitouras Ch., “Runge–Kutta pairs of order 5(4) satisfying only the first column simplifying assumption”, Computers & Mathematics with Applications, 62 (2): 770-775, dx.doi.org/10.1016/j.camwa.2011.06.002

#### DP5, dopri5, ode45

• Dormand, J. R.; Prince, P. J. (1980), “A family of embedded Runge-Kutta formulae”, Journal of Computational and Applied Mathematics, 6 (1): 19–26, doi:10.1016/0771-050X(80)90013-3

#### BS5

• Bogacki P. and Shampine L.F., (1996), “An Efficient Runge-Kutta (4,5) Pair”, Computers and Mathematics with Applications, 32 (6): 15-28

#### Verner Methods (Vern6, Vern7, Vern8, Vern9)

• J.H. Verner, Numerically optimal Runge–Kutta pairs with interpolants. Numerical Algorithms, 53, (2010) pp. 383–396. 10.1007/s11075-009-9290-3

#### TanYam7

• Tanaka M., Muramatsu S., Yamashita S., (1992), “On the Optimization of Some Nine-Stage Seventh-order Runge-Kutta Method”, Information Processing Society of Japan, 33 (12), pp. 1512-1526.

#### DP8, dop853, odex, seulex, rodas,

• E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

#### RK4 Residual Control

• L. F. Shampine. 2005. Solving ODEs and DDEs with residual control. Appl. Numer. Math. 52, 1 (January 2005), 113-127. DOI=http://dx.doi.org/10.1016/j.apnum.2004.07.003

#### OwrenZen3, OwrenZen4, OwrenZen5

• Brynjulf Owren and Marino Zennaro. 1992. Derivation of efficient, continuous, explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput. 13, 6 (November 1992), 1488-1501. DOI=http://dx.doi.org/10.1137/0913084

• E. Hairer and G. Wanner, (1999) Stiff differential equations solved by Radau methods, Journal of Computational and Applied Mathematics, 111 (1-2), pp. 93-111.

#### Rosenbrock23, Rosenbrock32, ode23s, ModifiedRosenbrockIntegrator

• Shampine L.F. and Reichelt M., (1997) The MATLAB ODE Suite, SIAM Journal of Scientific Computing, 18 (1), pp. 1-22.

#### ROS3P

• Lang, J. & Verwer, ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems J. BIT Numerical Mathematics (2001) 41: 731. doi:10.1023/A:1021900219772

#### Rodas3, Ros4LStab, Rodas4, Rodas42

• E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

#### RosShamp4

• L. F. Shampine, Implementation of Rosenbrock Methods, ACM Transactions on Mathematical Software (TOMS), 8: 2, 93-113. doi:10.1145/355993.355994

#### Veldd4, Velds4

• van Veldhuizen, D-stability and Kaps-Rentrop-methods, M. Computing (1984) 32: 229. doi:10.1007/BF02243574

#### GRK4T, GRK4A

• Kaps, P. & Rentrop, Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. P. Numer. Math. (1979) 33: 55. doi:10.1007/BF01396495

#### Rodas4P

• Steinebach G. Order-reduction of ROW-methods for DAEs and method of lines applications. Preprint-Nr. 1741, FB Mathematik, TH Darmstadt; 1995.

#### Rodas5

• Di Marzo G. RODAS5(4) – Méthodes de Rosenbrock d’ordre 5(4) adaptées aux problemes différentiels-algébriques. MSc mathematics thesis, Faculty of Science, University of Geneva, Switzerland.

• Andre Vladimirescu. 1994. The Spice Book. John Wiley & Sons, Inc., New York, NY, USA.

#### TRBDF2

• M.E. Hosea, L.F. Shampine, Analysis and implementation of TR-BDF2, Applied Numerical Mathematics, Volume 20, Issue 1, 1996, Pages 21-37, ISSN 0168-9274, http://dx.doi.org/10.1016/0168-9274(95)00115-8.

#### SDIRK2, Cash4

• A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,” ACM Transactions on Mathematical Software, 31(3), pp. 363-396,
1. Also available as LLNL technical report UCRL-JP-200037.

#### Kvaerno3, Kvaerno4, Kvaerno5

• Kværnø, A., Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage, BIT Numerical Mathematics (2004) 44: 489. https://doi.org/10.1023/B:BITN.0000046811.70614.38

#### Hairer4, Hairer42

• E. Hairer, G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Computational mathematics (2nd revised ed.), Springer (1996)

#### KenCarp3, KenCarp4, KenCarp5

• Christopher A. Kennedy and Mark H. Carpenter. 2003. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44, 1-2 (January 2003), 139-181. DOI=http://dx.doi.org/10.1016/S0168-9274(02)00138-1

#### Nystrom4, Nystrom4VelocityIndependent, Nystrom5VelocityIndependent

• E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

#### ERKN4

• M. A. Demba, N. Senu, F. Ismail “An embedded 4(3) pair of explicit trigonometrically-fitted Runge-Kutta-Nystrom method for solving periodic initial value problems” Applied Mathematical Sciences, Vol. 11, 2017, no. 17, 819-838, https://doi.org/10.12988/ams.2017.7262

#### ERKN5

• Demba, Musa & Senu, Norazak & Ismail, Fudziah. (2016). A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems. Mathematical and Computational Applications. 21. . 10.3390/mca21040046.

#### IRKN3, IRKN4

• Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method, International Science Index, Mathematical and Computational Sciences Vol:6, No:9, 2012 waset.org/Publication/1175

#### DPRKN6

• J.R. Dormand, P.J. Prince, Runge-Kutta-Nystrom triples, Computers & Mathematics with Applications, Volume 13, Issue 12, 1987, Pages 937-949, ISSN 0898-1221, http://dx.doi.org/10.1016/0898-1221(87)90066-6.

#### DPRKN8, DPRKN12

• J. R. DORMAND, M. E. A. EL-MIKKAWY, P. J. PRINCE; High-Order Embedded Runge-Kutta-Nystrom Formulae, IMA Journal of Numerical Analysis, Volume 7, Issue 4, 1 October 1987, Pages 423–430, https://doi.org/10.1093/imanum/7.4.423

#### VelocityVerlet, VerletLeapfrog, PseudoVerletLeapfrog

• Verlet, Loup (1967). “Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard−Jones Molecules”. Physical Review. 159: 98–103. doi:10.1103/PhysRev.159.98

• Etienne Forest, Ronald D. Ruth, Fourth-order symplectic integration, Physica D: Nonlinear Phenomena, Volume 43, Issue 1, 1990, Pages 105-117, ISSN 0167-2789, http://dx.doi.org/10.1016/0167-2789(90)90019-L.

#### Ruth3

• Ruth, Ronald D. (August 1983). “A Canonical Integration Technique”. Nuclear Science, IEEE Trans. on. NS-30 (4): 2669–2671. Bibcode:1983ITNS…30.2669R. doi:10.1109/TNS.1983.4332919

• Etienne Forest, Ronald D. Ruth, Fourth-order symplectic integration, Physica D: Nonlinear Phenomena, Volume 43, Issue 1, 1990, Pages 105-117, ISSN 0167-2789, http://dx.doi.org/10.1016/0167-2789(90)90019-L.

#### McAte2, McAte3, McAte4, McAte42, McAte5

• R. I. McLachlan and P. Atela, The accuracy of symplectic integrators, Nonlinearity 5 (1992), 541-562.

• Stephen K. Gray, Donald W. Noid and Bobby G. Sumpter, Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods The Journal of Chemical Physics 101, 4062 (1994); doi: http://dx.doi.org/10.1063/1.467523

#### CandyRoz4

• Candy, J.; Rozmus, W (1991). “A Symplectic Integration Algorithm for Separable Hamiltonian Functions”. J. Comput. Phys. 92: 230. Bibcode:1991JCoPh..92..230C. doi:10.1016/0021-9991(91)90299-Z

#### CalvoSanz4

• Stephen K. Gray, Donald W. Noid and Bobby G. Sumpter, Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods The Journal of Chemical Physics 101, 4062 (1994); doi: http://dx.doi.org/10.1063/1.467523

• M. P. Calvo & J. M. Sanz-Serna, Symplectic numerical methods for Hamiltonian problems, Int. J. Mod. Phys. C 4(1993), 617-634.

#### KahanLi6, KahanLi8

• Kahan, W. & Li, Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Mathematics of Computation. 66, 219, p. 1089-1099 11 p.

#### Yoshida6

• Yoshida, H. (1990). “Construction of higher order symplectic integrators”. Phys. Lett. A. 150 (5–7): 262. Bibcode:1990PhLA..150..262Y. doi:10.1016/0375-9601(90)90092-3

#### McAte8

• R. I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comp. 16, (1995), 151-168.
##### SofSpa10
• Mark Sofroniou & Giulia Spaletta, Derivation of symmetric composition constants for symmetric integrators, Optimization Methods and Software Vol. 20 , 4-5,2005

#### Feagin10, Feagin12, Feagin14

• Feagin, T., “High-order Explicit Runge-Kutta Methods Using M-Symmetry,” Neural, Parallel & Scientific Computations, Vol. 20, No. 4, December 2012, pp. 437-458

• Feagin, T., “An Explicit Runge-Kutta Method of Order Fourteen,” Numerical Algorithms, 2009

#### CarpenterKennedy2N54

• M.H. Carpenter, C.A. Kennedy, Fourth-Order Kutta Schemes, NASA Langley Research Center, Hampton, Virginia 23681-0001, 1994.

#### Strong Stability Preserving (SSP) Runge-Kutta Methods: General Information, SSPRK432, SSPRK932

• Gottlieb, Sigal, David I. Ketcheson, and Chi-Wang Shu. Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific, 2011.

#### SSPRK22, SSPRK33

• Shu, Chi-Wang, and Stanley Osher. “Efficient implementation of essentially non-oscillatory shock-capturing schemes.” Journal of Computational Physics 77.2 (1988): 439-471.

#### SSPRK53, SSPRK63, SSPRK73, SSPRK83, SSPRK54

• Ruuth, Steven. “Global optimization of explicit strong-stability-preserving Runge-Kutta methods.” Mathematics of Computation 75.253 (2006): 183-207.

#### SSPRK104

• Ketcheson, David I. “Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations.” SIAM Journal on Scientific Computing 30.4 (2008): 2113-2136.

#### SSP Dense Output

• Ketcheson, David I., et al. “Dense output for strong stability preserving Runge–Kutta methods.” Journal of Scientific Computing 71.3 (2017): 944-958.

#### LawsonEuler, NorsettEuler

• Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–86. doi:10.1017/S0962492910000048.

#### GenericIIF1, GenericIIF2

• Q. Nie, Y. Zhang and R. Zhao. Efficient Semi-implicit Schemes for Stiff Systems. Journal of Computational Physics, 214, pp 521-537, 2006.

### Delay Differential Equations

#### State-Dependent Delays

• S. P. Corwin, D. Sarafyan and S. Thompson in “DKLAG6: a code based on continuously imbedded sixth-order Runge-Kutta methods for the solution of state-dependent functional differential equations”, Applied Numerical Mathematics, 1997.

### Stochastic Differential Equations

#### RK-Mil

• Kloeden, P.E., Platen, E., Numerical Solution of Stochastic Differential Equations. Springer. Berlin Heidelberg (2011)

#### SRI, SRIW1, SRA, SRA1

• Rößler A., Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations, SIAM J. Numer. Anal., 48 (3), pp. 922–952. DOI:10.1137/09076636X

• Rackauckas C. Nie Q., (2016) Adaptive Methods for Stochastic Differential Equations via Natural Embeddings and Rejection Sampling with Memory. Discrete and Continuous Dynamical Systems - Series B. Accepted December 2016.

### Bifurcation Analysis

• Clewley R (2012) Hybrid Models and Biological Model Reduction with PyDSTool. PLoS Comput Biol 8(8): e1002628. doi:10.1371/journal.pcbi.1002628

### ProbInts (Uncertainty Quantification)

• Conrad P., Girolami M., Särkkä S., Stuart A., Zygalakis. K, Probability Measures for Numerical Solutions of Differential Equations, arXiv:1506.04592

### Manifold Projection Callback

• Ernst Hairer, Christian Lubich, Gerhard Wanner. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Berlin ; New York :Springer, 2002.

### PositiveDomain Callback

• Shampine, Lawrence F., Skip Thompson, Jacek Kierzenka and G. D. Byrne. “Non-negative solutions of ODEs.” Applied Mathematics and Computation 170 (2005): 556-569.

### Constant Rate Jump Aggregators

#### Direct

• Gillespie, Daniel T. (1976). A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics. 22 (4): 403–434. doi:10.1016/0021-9991(76)90041-3.

### Variable Rate Jumps

• Salis H., Kaznessis Y., Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions, Journal of Chemical Physics, 122 (5), DOI:10.1063/1.1835951

### Split Coupling

• David F. Anderson, Masanori Koyama; An asymptotic relationship between coupling methods for stochastically modeled population processes. IMA J Numer Anal 2015; 35 (4): 1757-1778. doi: 10.1093/imanum/dru044