Hyperbolic Runge–Kutta Method Using Evolutionary Algorithm

Abstract

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.