[Paper Review] Efficient IMEX Runge-Kutta methods for nonhydrostatic dynamics

Andrew Steyer, Christopher J. Vogl+2 more|arXiv (Cornell University)|Jun 17, 2019

Meteorological Phenomena and Simulations35 references4 citations

TL;DR

This paper introduces a new family of implicit-explicit Runge-Kutta (IMEX RK) methods—IMKG2 and IMKG3—designed for efficient time integration of nonhydrostatic atmospheric models using a horizontally explicit, vertically implicit (HEVI) partitioning. The methods achieve third-order accuracy with large stability regions, demonstrating superior efficiency and stability in the HOMME-NH model compared to existing IMEX methods.

ABSTRACT

We analyze the stability and accuracy (up to third order) of a new family of implicit-explicit Runge-Kutta (IMEX RK) methods. This analysis expedites development of methods with various balances in the number of explicit stages and implicit solves. We emphasize deriving methods with large stability regions for horizontally explicit vertically implicit (HEVI) partitionings of nonhydrostatic atmosphere models. The IMKG2 and IMKG3 families of IMEX RK methods are formulated within this framework. The HOMME-NH model with a HEVI partitioning is used for testing the accuracy and stability of various IMKG2-3 methods. The efficiency of several IMKG2-3 methods is demonstrated in HOMME-NH and compared to other IMEX RK methods in the literature.

Motivation & Objective

• To develop efficient time-integration methods for nonhydrostatic atmosphere models that handle multiple time scales arising from stiff vertical dynamics.
• To address the challenge of stability and accuracy in semi-implicit time integration by designing IMEX Runge-Kutta schemes with large stability regions.
• To optimize the balance between explicit and implicit stages in IMEX methods to reduce computational cost while maintaining accuracy.
• To evaluate the performance of new IMEX methods in a realistic nonhydrostatic atmospheric model (HOMME-NH) with HEVI partitioning.
• To provide a systematic framework for deriving IMEX RK methods with customizable stage counts and stability properties.

Proposed method

• The authors formulate a new family of IMEX Runge-Kutta methods (IMKG2 and IMKG3) with arbitrary numbers of internal stages, parameterized by coefficient vectors α, β,  α̂,  β̂, and δ̂.
• The methods are designed for HEVI partitioning, where horizontal advection is treated explicitly and vertical dynamics implicitly, reducing computational cost while maintaining stability.
• Stability and accuracy are analyzed up to third order, with a focus on A-stability, stiff decay (SD), and strong stability preservation (SSP) properties.
• The methods are constructed using a Butcher tableau framework, with coefficients derived to ensure order conditions and favorable stability regions.
• The IMKG2 and IMKG3 families include multiple variants (e.g., 232a, 242b, 354a) with different stability and accuracy trade-offs, selected based on performance in numerical tests.
• The HOMME-NH nonhydrostatic atmospheric model is used as a testbed to evaluate time integration performance under HEVI partitioning.

Experimental results

Research questions

• RQ1 Can IMEX Runge-Kutta methods be systematically designed to achieve third-order accuracy while maintaining large stability regions for nonhydrostatic atmospheric models?
• RQ2 How does the choice of implicit-explicit partitioning (specifically HEVI) affect the stability and efficiency of time integration in nonhydrostatic models?
• RQ3 What is the impact of varying the number of explicit and implicit stages on the stability and accuracy of IMEX methods in nonhydrostatic dynamics?
• RQ4 How do the new IMKG2 and IMKG3 methods compare in efficiency and stability to existing IMEX RK methods in the literature when applied to the HOMME-NH model?
• RQ5 Which IMKG2-3 variants exhibit optimal trade-offs between stability, accuracy, and computational cost in nonhydrostatic atmospheric simulations?

Key findings

• The IMKG2 and IMKG3 families of IMEX Runge-Kutta methods achieve third-order accuracy with favorable stability properties, including A-stability and stiff decay (SD) for selected variants.
• The 254c variant of IMKG2 demonstrated the best balance of stability and efficiency in the HOMME-NH model, outperforming other IMEX methods in terms of computational cost per unit of accuracy.
• IMKG2-3 methods with HEVI partitioning showed superior stability compared to standard IMEX methods, particularly in resolving fast vertical gravity waves without step-size restrictions.
• The 354a variant of IMKG3 exhibited strong stability and accuracy, with a large stability region and no order reduction, making it suitable for long-time integration.
• The 243a and 253a variants showed excellent SSP properties, enhancing their robustness in simulations with sharp gradients or discontinuities.
• Overall, the IMKG2-3 methods demonstrated significant efficiency gains in HOMME-NH simulations, with reduced computational cost per time step while maintaining high-order accuracy and stability.