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

Thomas Izgin, David I. Ketcheson, Andreas Meister

TL;DR

This paper develops a unified NB-series framework for non-standard additive Runge–Kutta (NSARK) methods whose coefficients depend on the solution, enabling order conditions to be read off from labeled N-trees for arbitrary order. It applies the theory to GeCo and MPRK schemes, deriving comprehensive order conditions up to 3rd order for GeCo and up to 4th order for MPRK, with several results recast into explicit, reduced forms. A new 4th-order MPRK method (MPRKord4) is constructed and its convergence is numerically validated, illustrating the practical impact of the theory. The framework facilitates systematic development of higher-order, positivity-preserving integrators and sets the stage for extensions to stiff problems and other Patankar-type schemes.

Abstract

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. We also present a new 4th order MPRK method within this framework and numerically confirm its convergence rate.

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

TL;DR

This paper develops a unified NB-series framework for non-standard additive Runge–Kutta (NSARK) methods whose coefficients depend on the solution, enabling order conditions to be read off from labeled N-trees for arbitrary order. It applies the theory to GeCo and MPRK schemes, deriving comprehensive order conditions up to 3rd order for GeCo and up to 4th order for MPRK, with several results recast into explicit, reduced forms. A new 4th-order MPRK method (MPRKord4) is constructed and its convergence is numerically validated, illustrating the practical impact of the theory. The framework facilitates systematic development of higher-order, positivity-preserving integrators and sets the stage for extensions to stiff problems and other Patankar-type schemes.

Abstract

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. We also present a new 4th order MPRK method within this framework and numerically confirm its convergence rate.
Paper Structure (11 sections, 19 theorems, 112 equations, 1 figure, 1 table)

This paper contains 11 sections, 19 theorems, 112 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Order Conditions for NSARK methods
  3. Preliminaries
  4. Main result
  5. Application to GeCo Schemes
  6. Application to MPRK Schemes
  7. Order Conditions for MPRK Schemes from the Literature
  8. Reduced Order Conditions for 4th Order MPRK Methods
  9. Summary and Outlook
  10. Results for the Derivation of the Main Theorem
  11. Results for Reducing Order Conditions of NSARK methods

Key Result

Theorem 2

Let the functions $\mathbf{F}^{[\mu]}$ from ivp satisfy $\mathbf{F}^{[\mu]}\in \mathcal{C}^{k+1}$ for $\mu=1,\dotsc,N$. If $\mathbf{y}$ solves ivp, then where $\gamma$ is the density defined in eq:sigmagamma.

Figures (1)

  • Figure 1: Error plot of MPRKord4 applied to various problems. The error was computed at the respective final times using ode45 as a reference solution.

Theorems & Definitions (42)

  • Example 1
  • Theorem 2: Ntrees
  • Remark 3
  • Example 4
  • Theorem 5
  • proof
  • Corollary 6
  • Corollary 7
  • proof
  • Remark 8
  • ...and 32 more