First thoughts on realted software

Author: SKopecz

paper/paper.bib

@@ -190,4 +190,20 @@ @article {OeffnerTorlo2020
   MRNUMBER = {4064785},
   DOI = {10.1016/j.apnum.2020.01.025},
   URL = {https://doi.org/10.1016/j.apnum.2020.01.025},
+}
+
+@article {Shampine2005,
+  AUTHOR = {Shampine, L. F. and Thompson, S. and Kierzenka, J. A. and
+              Byrne, G. D.},
+  TITLE = {Non-negative solutions of {ODE}s},
+  JOURNAL = {Applied Mathematics and Computation},
+  VOLUME = {170},
+  YEAR = {2005},
+  NUMBER = {1},
+  PAGES = {556--569},
+  ISSN = {0096-3003,1873-5649},
+  MRCLASS = {34C11},
+  MRNUMBER = {2177561},
+  DOI = {10.1016/j.amc.2004.12.011},
+  URL = {https://doi.org/10.1016/j.amc.2004.12.011},
 }

paper/paper.md

@@ -54,6 +54,15 @@ All implemented schemes were originally introduced for conservative production-d
 
 Readers interested in additional theoretical background and further properties of the implemented schemes are referred to the following papers: @kopeczmeister2019, @izgin2022stability1, @izgin2022stability2, @huang2023, @torlo2022, @izginoeffner2023
 
+Existing software libraries do not have a strong focus on unconditional positivity and, to the authors' knowledge, there is no other software library which offers MPRK schemes. 
+
+A common strategy to obtain nonnegative solutions used in the `PositiveDomain` callback of `Differentialequtions'jl` or the commercial package `Matlab` is described by @Shampine2005. In this approach negative components of approximate solutions that have been accepted by the adaptive time stepping algorithm are set to zero.
+
+Another possibility is to reduce the chosen time step size beyond accuracy considerations until a non-negative approximation is calculated. This can be achieved in `DifferentialEquations.jl` using the solver option `isoutofdomain`.
+
+Some papers on MPRK schemes offer supplementary codes. However, these are mainly small scripts for the reproduction of results shown in the papers and are not intended as software libraries.
+
+
 TODO
 
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