An Exponential Time-Integrator Scheme for Steady and Unsteady Inviscid Flows

TL;DR

This paper introduces a second-order exponential time-integrator scheme, PCEXP, for fluid dynamics equations, demonstrating improved efficiency and accuracy for steady and unsteady inviscid flows through comparisons with existing methods.

Contribution

The paper develops and applies a novel second-order exponential integrator scheme, PCEXP, optimized with Krylov methods for fluid flow simulations, enhancing stability and efficiency over traditional schemes.

Findings

PCEXP achieves higher accuracy than BDF2 for unsteady flows.

PCEXP offers computational efficiency comparable to implicit schemes for steady flows.

The scheme demonstrates stability and effectiveness in multi-dimensional fluid simulations.

Abstract

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.