Order conditions for Runge–Kutta-like methods with solution-dependent coefficients

05/23/2023
by   Thomas Izgin, et al.
0

In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge–Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent factors. These include the classes of modified Patankar–Runge–Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, the analysis of accuracy and stability of these methods is more complicated. In this work, we provide a comprehensive and unifying theory of order conditions for such RK-like methods, which differ from original RK schemes in that their coefficients are solution-dependent. The resulting order conditions are themselves solution-dependent and obtained using the theory of NB-series, and thus, can easily be read off from labeled N-trees. We present for the first time order conditions for MPRK and GeCo schemes of arbitrary order; For MPRK schemes, the order conditions are given implicitly in terms of the stages. From these results, we recover as particular cases all known order conditions from the literature for first- and second-order GeCo as well as first-, second- and third-order MPRK methods. Additionally, we derive sufficient and necessary conditions in an explicit form for 3rd and 4th order GeCo schemes as well as 4th order MPRK methods.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/02/2022

On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar–Runge–Kutta Schemes

Since almost twenty years, modified Patankar–Runge–Kutta (MPRK) methods ...
research
03/01/2023

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

The approximate solution of the Cauchy problem for second-order evolutio...
research
03/06/2020

General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes

Recently, an approach known as relaxation has been developed for preserv...
research
08/10/2023

Necessary and sufficient conditions for strong stability of explicit Runge-Kutta methods

Strong stability is a property of time integration schemes for ODEs that...
research
04/07/2022

Algebraic Structure of the Weak Stage Order Conditions for Runge-Kutta Methods

Runge-Kutta (RK) methods may exhibit order reduction when applied to sti...
research
12/09/2019

Two-derivative error inhibiting schemes with post-processing

High order methods are often desired for the evolution of ordinary diffe...
research
10/05/2021

Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints

We investigate a family of bilevel imaging learning problems where the l...

Please sign up or login with your details

Forgot password? Click here to reset