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

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 sth-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order p <= s and p <= 2. On the other hand, the sth-order RK methods preconditioned by the TASE operators with order of p <= 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. As an alternative to established implicit strategies, TASE method is promising for the efficient computation of stiff physical problems. (C) 2020 Elsevier Inc. All rights reserved.