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Co-authored-by: Joshua Lampert <51029046+JoshuaLampert@users.noreply.github.com>

Author: SKopecz

paper/paper.md

@@ -39,7 +39,7 @@ Unfortunately, positivity is a property that almost all standard time integratio
 In particular, higher-order general linear methods cannot preserve positivity unconditionally [@bolley1978conservation].
 The only standard scheme with which unconditional positivity can be achieved is the implicit Euler method
 (assuming that the nonlinear systems are solved exactly). However, this is only first-order accurate and, in addition, the preservation of positivity within the nonlinear iteration process poses a problem. 
-Another strategy for preserving positivity used in existing open source or commercial packages (like MATLAB) is to set negative solution components that are accepted by the step size control to zero. Unfortunately, this can have a negative impact on possible conservation properties. Further approaches in the literature include projections inbetween time steps [@sandu2001positive; @nusslein2021positivity], if a negative solution was computed, or it is tried to reduce the time step size as long as a non-negative solution is calculated. Finally, strong stability preserving (SSP) methods can also be used to preserve positivity, but this is again subject to step size limitations [@gottlieb2011strong]. 
+Another strategy for preserving positivity used in existing open source or commercial packages (like MATLAB) is to set negative solution components that are accepted by the step size control to zero. Unfortunately, this can have a negative impact on possible conservation properties. Further approaches in the literature include projections in between time steps [@sandu2001positive; @nusslein2021positivity], if a negative solution was computed, or it is tried to reduce the time step size as long as a non-negative solution is calculated. Finally, strong stability preserving (SSP) methods can also be used to preserve positivity, but this is again subject to step size limitations [@gottlieb2011strong]. 
 
 Consequently, various new, unconditionally positive schemes, especially modified Patankar--Runge--Kutta (MPRK) methods, have been introduced in recent years.
 Unfortunately, these new methods are not yet available in software packages, making them inaccessible to most users and limiting their comparability within the scientific community. PositiveIntegrators.jl makes these methods available and thus usable and comparable.
@@ -64,11 +64,11 @@ In addition, all implemented MPRK methods have been extended so that non-conserv
 
 The first MPRK schemes were introduced by @burchard2003. These are the first-order scheme `MPE` and a second-order scheme based on Heun's method. To avoid the restriction to Heun's method, the second-order `MPRK22` schemes were developed by @kopeczmeister2018order2. The techniques developed therein also enabled a generalization to third-order schemes and thus the introduction of `MPRK43I` and `MPRK43II` schemes by @kopeczmeister2018order3.
 
-The aforementioned schemes were derived from the classical formulation of Runge-Kutta schemes. Using the Shu-Osher formulation instead lead to the introduction of the second-order schemes `SSPMPRK22` by @huang2019order2 and the third-order scheme `SSPMPRK43` by @huang2019order3.
+The aforementioned schemes were derived from the classical formulation of Runge-Kutta methods. Using the Shu-Osher formulation instead lead to the introduction of the second-order schemes `SSPMPRK22` by @huang2019order2 and the third-order scheme `SSPMPRK43` by @huang2019order3.
 
 Starting from a low-order scheme, the deferred correction approach can be used to increase the scheme's approximation order iteratively. @OeffnerTorlo2020 combined deferred correction with the MPRK idea to devise MPRK schemes of arbitrary order. These are implemented as `MPDeC` schemes for orders 2 up to 10.
 
-The implemented schemes were originally introduced for conservative production-destruction systems only. An extension to non-conservative production-destruction-systems was presented by @benzmeister2015. We implemented a modification of this algorithm, by treating the non-conservative production and destruction terms separately, weighting the destruction terms and leaving the production terms unweighted.
+The implemented schemes were originally introduced for conservative production-destruction systems only. An extension to non-conservative production-destruction systems was presented by @benzmeister2015. We implemented a modification of this algorithm, by treating the non-conservative production and destruction terms separately, weighting the destruction terms and leaving the production terms unweighted.
 
 Readers interested in additional theoretical background and further properties of the implemented schemes are referred to the publications of @kopeczmeister2019, @izgin2022stability1, @izgin2022stability2, @huang2023, @torlo2022, @izginoeffner2023.