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Abstract
Abstract In this work, modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity-preserving for any time step size, we impose the same requirements on the corresponding dense output formula. In particular, we discover that there is an explicit first-order formula. However, to develop a boot-strapping technique, we propose to use implicit formulae which naturally fit into the framework of MPRK schemes. In particular, if lower-order MPRK schemes are used to construct methods of higher order, the same can be done with the dense output formulae we propose in this work. We explicitly construct formulae up to order three and demonstrate how to generalize this approach as long as the underlying Runge-Kutta method possesses a dense output formula of appropriate accuracy. We also note that even though linear systems have to be solved to compute an approximation for intermediate points in time using these higher-order dense output formulae, the overall computational effort to reach a given number of approximations is reduced compared to using the scheme with a smaller step size. We support this fact and our theoretical findings by means of numerical experiments.
Highlights
The first modified Patankar-Runge-Kutta (MPRK) method was introduced in 2003 based on the explicit Euler method [4]
We have developed a boot-strapping technique to equip MPRK methods with a dense output formula of appropriate accuracy
There, the first-order dense output formula is explicit while the remaining ones are linearly implicit
Summary
The first modified Patankar-Runge-Kutta (MPRK) method was introduced in 2003 based on the explicit Euler method [4]. Communications on Applied Mathematics and Computation that the sum of all constituents of the numerical approximation in any time step and for any t > 0 equals the sum of the constituents of the initial data While these two properties are guaranteed by the implicit Euler method, the advantage of the MPE scheme is that it only requires the solution of a linear system of equations at each time step, even for nonlinear differential equations. Designing the first dense output formulae for all MPRK schemes up to order four, and developing such a boot-strapping process for higher-order MPRK methods is the purpose of the present work. We present the boot-strapping process together with formulae for schemes up to fourth order and validate our theoretical findings by performing various numerical experiments before we conclude this work with a summary and an outlook
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