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@@ -36,7 +36,7 @@ Many systems of ordinary differential equations that model real-life application
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For some of these systems unconditionally positivity-preserving time integration methods are helpful or even necessary to obtain meaningful solutions.
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Unfortunately, positivity is a property that almost all standard time integration schemes, such as Runge–Kutta methods, Rosenbrock methods, or linear multistep methods, do not preserve.
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The only standard scheme with which unconditional positivity can be achieved is the implicit Euler method. However, this is only first-order accurate and, in addition, the preservation of positivity within the nonlinear iteration process poses a problem.
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Another strategy for preserving positivity used in existing open source or commercial packages (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, 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, SSP Runge-Kutta methods can also be used to presere positivity, but this is again subject to step size limitations.
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Another strategy for preserving positivity used in existing open source or commercial packages (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, 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, SSP Runge-Kutta methods can also be used to preserve positivity, but this is again subject to step size limitations.
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Consequently, various new, unconditionally positive schemes, especially modified Patankar--Runge--Kutta (MPRK) methods, have been introduced in recent years.
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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.

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