[Paper Review] High-order low-storage explicit Runge-Kutta schemes for equations with quadratic nonlinearities

Pablo D. Mininni, Mininni, Pablo D.+2 more|ArXiv.org|Aug 13, 2008

Computational Fluid Dynamics and Aerodynamics21 references3 citations

TL;DR

This paper proposes high-order, low-storage explicit Runge-Kutta schemes for equations with quadratic nonlinearities by extending the JST algorithm with correction terms, enabling third- and fourth-order accuracy while maintaining minimal memory usage. The method is validated on Burgers' equation and the incompressible Euler equation, showing improved temporal accuracy and stability with only a modest increase in computational cost.

ABSTRACT

We show in this paper that third- and fourth-order low storage Runge-Kutta algorithms can be built specifically for quadratic nonlinear operators, at the expense of roughly doubling the time needed for evaluating the temporal derivatives. The resulting algorithms are especially well suited for computational fluid dynamics. Examples are given for the Hénon-Heiles Hamiltonian system and, in one and two space dimensions, for the Burgers equation using both a pseudo-spectral code and a spectral element code, respectively. The scheme is also shown to be practical in three space solving the incompressible Euler equation using a fully parallelized pseudo-spectral code.

Motivation & Objective

• To address the limitation of the JST algorithm, which is only second-order accurate for nonlinear quadratic systems, despite being high-order for linear problems.
• To develop correction terms for the JST algorithm that restore third- and fourth-order accuracy when the right-hand side contains quadratic nonlinearities.
• To preserve the low-storage property of the original JST scheme, ensuring applicability in large-scale simulations with limited memory.
• To validate the new schemes on representative problems including the Burgers equation and incompressible Euler equations in multiple dimensions.
• To demonstrate that higher-order time integration is essential in high-resolution simulations where temporal errors can dominate spatial errors.

Proposed method

• The method begins with the JST algorithm, a low-storage Runge-Kutta scheme, and modifies it by adding correction terms derived from Taylor series expansion of the exact solution.
• The correction terms are derived specifically for systems where the right-hand side F(u) = L(u) + N(u,u), with L linear and N quadratic and symmetric in its arguments.
• The correction terms are applied in a way that maintains the original two-level storage structure of the JST algorithm, ensuring low memory footprint.
• The scheme is implemented in both pseudo-spectral and spectral element codes, using existing time-stepping frameworks with minimal modifications.
• The method is applied to the Hénon-Heiles system, Burgers' equation in 1D and 2D, and the 3D incompressible Euler equation with a fully parallelized pseudo-spectral code.
• The 3+ scheme, which includes third-order correction terms, is identified as the most cost-effective implementation, requiring four JST iterations.

Experimental results

Research questions

• RQ1 Can the JST algorithm be corrected to achieve third- and fourth-order accuracy for quadratic nonlinear systems without increasing storage requirements?
• RQ2 What are the specific correction terms needed to extend the JST algorithm beyond second order for nonlinear quadratic operators?
• RQ3 How does the performance of the corrected scheme compare to standard low-storage Runge-Kutta methods in terms of accuracy and stability for high-resolution simulations?
• RQ4 Is the increased temporal accuracy of the new scheme practically beneficial in large-scale simulations, such as direct numerical simulation of turbulence?
• RQ5 Can the corrected scheme be efficiently implemented in existing high-order spectral and spectral element codes with minimal code changes?

Key findings

• The JST algorithm, while exact for linear problems, introduces third-order errors when applied to quadratic nonlinear systems, necessitating correction terms for higher-order accuracy.
• The derived correction terms restore third- and fourth-order accuracy for quadratic nonlinearities while preserving the original two-level storage structure of the JST scheme.
• The 3+ scheme, which includes third-order corrections, is found to be the most cost-effective, requiring only four JST iterations and providing significantly lower global errors than standard third-order schemes.
• Numerical results for Burgers' equation show that truncation errors decrease with increasing order of the scheme, and the 3+ scheme outperforms standard third-order methods.
• For the 3D incompressible Euler equation, the new scheme enables accurate long-time integration in high-resolution simulations, preventing spurious conservation properties seen with low-order time integration.
• The method is practical and efficient in large-scale simulations, including 3D pseudo-spectral codes, and is suitable for petascale direct numerical simulations of turbulence.