Design of IMEXRK time integration schemes via Delaunay-based derivative-free optimization with nonconvex constraints and grid-based acceleration

Abstract

This paper develops a powerful new variant, dubbed $$\varDelta $$ -DOGS( $$\varOmega _Z$$ ), of our Delaunay-based Derivative-free Optimization via Global Surrogates family of algorithms, and uses it to identify a new, low-storage, high-accuracy, Implicit/Explicit Runge–Kutta (IMEXRK) time integration scheme for the stiff ODEs arising in high performance computing applications, like the simulation of turbulence. The $$\varDelta $$ -DOGS( $$\varOmega _Z$$ ) algorithm, which we prove to be globally convergent under the appropriate assumptions, combines (a) the essential ideas of our $$\varDelta $$ -DOGS( $$\varOmega $$ ) algorithm, which is designed to efficiently optimize a nonconvex objective function f(x) within a nonconvex feasible domain $$\varOmega $$ defined by a number of constraint functions $$c_\kappa (x)$$ , with (b) our $$\varDelta $$ -DOGS(Z) algorithm, which reduces the number of function evaluations on the boundary of the search domain via the restriction that all function evaluations lie on a Cartesian grid, which is successively refined as the iterations proceed. The optimization of the parameters of low-storage IMEXRK schemes involves a complicated set of nonconvex constraints, which leads to a challenging disconnected feasible domain, and a highly nonconvex objective function; our simulations indicate significantly faster convergence using $$\varDelta $$ -DOGS( $$\varOmega _Z$$ ) as compared with the original $$\varDelta $$ -DOGS( $$\varOmega $$ ) optimization algorithm on the problem of tuning the parameters of such schemes. A low-storage third-order IMEXRK scheme with remarkably good stability and accuracy properties is ultimately identified using this approach, and is briefly tested on Burgers’ equation.