DeepAI AI Chat
Log In Sign Up

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

by   Abhijit Biswas, et al.
King Abdullah University of Science and Technology
Temple University
New Jersey Institute of Technology

Runge-Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This paper outlines the first algebraic theory of WSO, and establishes general order barriers that relate the WSO of a RK scheme to its order and number of stages for both fully-implicit and DIRK schemes. It is shown in several scenarios that the constructed bounds are sharp. The theory characterizes WSO in terms of orthogonal invariant subspaces and associated minimal polynomials. The resulting necessary conditions on the structure of RK methods with WSO are then shown to be of practical use for the construction of such schemes.


page 1

page 2

page 3

page 4


Design of DIRK Schemes with High Weak Stage Order

Runge-Kutta (RK) methods may exhibit order reduction when applied to cer...

Isomeric trees and the order of Runge–Kutta methods

The conditions for a Runge–Kutta method to be of order p with p≥ 5 for a...

Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods

When applied to stiff, linear differential equations with time-dependent...

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

In recent years, many positivity-preserving schemes for initial value pr...

Syzygies among reduction operators

We introduce the notion of syzygy for a set of reduction operators and r...

Discrete Adjoint Implicit Peer Methods in Optimal Control

It is well known that in the first-discretize-then-optimize approach in ...

Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling

B-series and generalizations are a powerful tool for the analysis of num...