Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations

• Published in: Applied numerical mathematics Vol. 35; no. 3; pp. 177 - 219
• Main Authors: Kennedy, Christopher A., Carpenter, Mark H., Lewis, R.Michael
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2000; Elsevier
• Subjects: Compressible flows; shock and detonation phenomena; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Physics; Sciences and techniques of general use; Wave equation; Methane; Grid pattern; Differential equation; Hydrogen; Optimization method; Feedback regulation; Air; Step; Runge Kutta method; Properties; Dispersion; Partial differential equation; Direct method; Accuracy; Navier Stokes equation; Register; Dissipation; Jet; Numerical simulation; Linear stability; Reliability; Explicit Runge kutta method; Compressible flow
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/S0168-9274(99)00141-5

The derivation of low-storage, explicit Runge–Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier–Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, sixteen ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been tested with not only DETEST, but also with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be nearly matched while still saving 2–3 registers of memory.