A review on the time-accurate and highly-stable explicit (TASE) scheme for solving stiff differential equations

The stiff problem within ordinary differential equations and partial differential equations presents numerous challenges for the stability and convergence of numerical methodologies due to their significant differences in scale. While explicit time-marching schemes have their advantages, their need for extremely small time steps significantly sacrifices computational efficiency. Implicit time-marching schemes, on the other hand, allow for larger time steps with better stability properties, where traditional schemes such as the diagonally implicit Runge-Kutta method and the implicit-explicit Runge-Kutta scheme are already widely used. However, when it comes to nonlinear problems, we still need to solve the nonlinear implicit equation, which is fundamentally difficult at high-order accuracy. To tackle this, the time-accurate and highly-stable explicit (TASE) operators were proposed. Differing from the traditional implicit time-marching schemes, TASE operators are preconditioners for existing explicit time-marching schemes, such as the explicit Runge-Kutta (RK) schemes, where their combination enables RK schemes to solve stiff problems with larger time steps and enhances stability. Furthermore, TASE operators are linear in nature, avoiding the need to solve non-linear problems, where the accuracy of TASE operators theoretically can also be of an arbitrarily high order through Richardson extrapolation. These inherent advantages have led to the rapid growth of the family of TASE-schemes recently, including theoretical analysis and algorithmic improvements. In this review, the TASE operators and their variants are summarised, highlighting their stability properties, parameter settings, comparisons with traditional implicit time-marching schemes, and promising future directions of the TASE family of operators.