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Unconditionally positive and conservative third order modified Patankar–Runge–Kutta discretizations of production–destruction systems

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Abstract

Modified Patankar–Runge–Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production–destruction systems. They adapt explicit Runge–Kutta schemes to ensure positivity and conservation irrespective of the time step size. The first two members of this class, the first order MPE scheme and the second order MPRK22(1) scheme, have been successfully applied in a large number of applications. Recently, a general definition of MPRK schemes was introduced and necessary as well as sufficient conditions to obtain first and second order MPRK schemes were presented. In this paper we derive necessary and sufficient conditions for third order MPRK schemes and introduce the first family of such schemes. The theoretical results are confirmed by numerical experiments considering linear and nonlinear as well as nonstiff and stiff systems of differential equations.

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Notes

  1. The theorem can be deduced from the binomial series \(\sum _{k=0}^\infty \left( {\begin{array}{c}s\\ k\end{array}}\right) x^k=(1+x)^s\), which is convergent for \(|x|<1\). See for instance [16].

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Correspondence to Stefan Kopecz.

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Communicated by Christian Lubich.

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Kopecz, S., Meister, A. Unconditionally positive and conservative third order modified Patankar–Runge–Kutta discretizations of production–destruction systems. Bit Numer Math 58, 691–728 (2018). https://doi.org/10.1007/s10543-018-0705-1

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  • DOI: https://doi.org/10.1007/s10543-018-0705-1

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