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Research Papers

Hyperbolic Runge–Kutta Method Using Evolutionary Algorithm

[+] Author and Article Information
A. Arun Govind Neelan

Department of Aerospace Engineering,
Indian Institute of
Space Science and Technology,
Thiruvananthapuram 695547, Kerala, India
e-mail: arungovindneelan@gmail.com

Manoj T. Nair

Department of Aerospace Engineering,
Indian Institute of
Space Science and Technology,
Thiruvananthapuram 695547, Kerala, India
e-mail: manojtnair.iist@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 29, 2017; final manuscript received June 20, 2018; published online August 1, 2018. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 13(10), 101003 (Aug 01, 2018) (7 pages) Paper No: CND-17-1527; doi: 10.1115/1.4040708 History: Received November 29, 2017; Revised June 20, 2018

A family of Runge–Kutta (RK) methods designed for better stability is proposed. Authors have optimized the stability of RK method by increasing the stability region by trading some of the higher order terms in the Taylor series. For flow involving shocks, compromising a few higher order terms will not affect convergence rate that is justified with an example. Though this kind of analysis began about three decades ago, most of the papers dealt with classical optimization and ended up in relatively nonoptimal values. Here, authors have overcome that by using evolutionary algorithm (EA), the result is refined using multisection method (MSM). The schemes designed based on this procedure have better stability than the classical RK methods, strong stability RK methods (SSPRK), and low dispersive and dissipative RK methods (LDDRK) of the same number of stages. Authors have tested the schemes on a variety of test cases and found some significant improvement.

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Figures

Grahic Jump Location
Fig. 1

Stability plots of different methods: (a) stability plots of PRK and HRK methods and (b) stability plots of SSPRK and HRK methods

Grahic Jump Location
Fig. 2

Solution of shock tube problems: (a) velocity plot of Sod problem at T = 1.7 s and (b) pressure plot of Shu-Osher problem at T = 1.8 s

Grahic Jump Location
Fig. 4

Convergence plot of spatial and temporal discretization: (a) temporal convergence (Emt) of different schemes and (b) spatial convergence (Ems) of different schemes

Grahic Jump Location
Fig. 3

Solution of supersonic flow past a diamond using HRK42: (a) density contour and (b) Mach number contour

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