[Paper Review] Partially implicit Runge-Kutta methods for wave-like equations in spherical-type coordinates

I. Cordero-Carrión, P. Cerdá–Durán|arXiv (Cornell University)|Nov 26, 2012

Numerical methods for differential equations13 references4 citations

TL;DR

This paper proposes partially implicit Runge-Kutta methods to stabilize time evolution of wave-like partial differential equations in spherical-type coordinates, addressing numerical instabilities caused by stiff terms or coordinate-induced factors. The approach improves stability over explicit Runge-Kutta methods while maintaining high-order accuracy for long-term simulations.

ABSTRACT

Partially implicit Runge-Kutta methods are presented in this work in order to numerically evolve in time a set of partial differential equations. These methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations due to potential numerical unstable terms in the sources, such as stiff terms or the presence of factors as a result of a partic- ular chosen system of coordinates. In this article, partially implicit Runge-Kutta methods for several convergence orders have been derived and stability properties have been analyzed. These methods are shown to be appropriated to avoid the de- velopment of numerical instabilities in the evolution in time of wave-like equations in spherical-type coordinates, in contrast to the explicit Runge-Kutta methods.

Motivation & Objective

• Address numerical instabilities arising in time evolution of wave-like equations due to stiff source terms or coordinate system choices.
• Develop high-order partially implicit Runge-Kutta schemes tailored for spherical-type coordinate systems.
• Ensure stability and accuracy in long-term simulations where explicit methods fail.
• Analyze the stability properties of the proposed methods to validate their robustness.

Proposed method

• Derive partially implicit Runge-Kutta methods with multiple convergence orders for time integration of wave-like PDEs.
• Separate stiff or unstable terms in the source into implicit components to stabilize the time evolution.
• Apply the method to systems of partial differential equations in spherical-type coordinates, where coordinate singularities or factors can induce instability.
• Use Butcher tableau representation to structure the coefficients of the Runge-Kutta schemes.
• Perform von Neumann stability analysis to evaluate the stability properties of the derived methods.
• Compare the performance and stability of the partially implicit schemes against standard explicit Runge-Kutta methods.

Experimental results

Research questions

• RQ1 How can numerical instabilities caused by stiff terms or coordinate-induced factors be mitigated in time integration of wave-like equations?
• RQ2 What is the stability behavior of partially implicit Runge-Kutta methods when applied to wave-like PDEs in spherical-type coordinates?
• RQ3 Can high-order accuracy be preserved while achieving improved stability compared to explicit Runge-Kutta methods?
• RQ4 What are the optimal coefficients for partially implicit Runge-Kutta schemes to ensure stability in stiff systems?
• RQ5 How do the stability regions of the proposed methods compare to those of explicit Runge-Kutta methods in the context of spherical coordinates?

Key findings

• The proposed partially implicit Runge-Kutta methods effectively suppress numerical instabilities that arise in time evolution due to stiff or coordinate-induced terms.
• Stability analysis confirms that the methods maintain stability over long integration times, unlike explicit Runge-Kutta methods which diverge under similar conditions.
• High-order convergence is preserved in the derived schemes, enabling accurate long-term simulations.
• The methods are specifically designed to handle the mathematical challenges introduced by spherical-type coordinate systems, such as singularities or radial factors.
• The partially implicit formulation allows for larger stable time steps compared to explicit methods, improving computational efficiency.
• The stability regions of the proposed schemes are significantly larger than those of explicit Runge-Kutta methods, particularly for stiff components in the source terms.