Time-Accurate and highly-Stable Explicit operators for stiff differential equations

Abstract

Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. The resulting time integration methods remain the original explicit time-marching schemes, yet with nearly unconditional stability. The TASE operators can be designed to be arbitrarily high-order accurate with Richardson extrapolation such that the accuracy order of original explicit time-marching method is preserved. Theoretical analyses and stability diagrams show that the $s$-stages $s$th-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order $p \leq s$ and $p \leq 2$. On the other hand, the $s$th-order RK methods preconditioned by the TASE operators with order of $p \leq s$ and $p > 2$ are nearly unconditionally stable. The only free parameter in TASE operators can be determined a priori based on stability arguments. Unlike classical implicit methods, the TASE methodology allows for solving non-linear problems with arbitrary order without requiring solving a nonlinear system of equations. A set of benchmark problems with strong stiffness is simulated to assess the performance of the TASE method. Numerical results suggest that the proposed framework preserves the high-order accuracy of the explicit time-marching methods with very-large time steps for all the considered cases.