Abstract

AbstractRunge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.

Highlights

  • Runge-Kutta (RK) methods achieve high-order accuracy in time by means of combining approximations to the solution at multiple stages

  • We confirm that weak stage order (WSO) p is required for ordinary differential equation (ODE), and WSO p − 1 is required for PDE IBVPs

  • We display a diagonally-implicit Runge-Kutta (DIRK) scheme with explicit first stage (EDIRK), that is, a11 = 0, of stage order 2

Read more

Summary

Introduction

Runge-Kutta (RK) methods achieve high-order accuracy in time by means of combining approximations to the solution at multiple stages. A characteristic property of most RK schemes is that, while the non-stiff limit recovers the scheme’s order (as given by the order conditions [2, 5]), the error decays at a reduced order in the stiff limit. This phenomenon is called “order reduction” (OR) [1, 3, 7, 10, 11] and it manifests in various ways for more complex problems, including numerical boundary layers [6]. Having stage order q implies that the error decays at an order of (at least) q in the stiff regime (see [12]). We move to a weaker condition that can avoid OR in some situations for higher order in the context of DIRK schemes

Weak Stage Order
Numerical Results
Linear ODE Test Problem
Linear PDE Test Problem
Nonlinear PDE Test Problem
Stiff Nonlinear ODE
Conclusions and Outlook
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.