An Exponential Time Differencing Runge–Kutta Method ETDRK32 for Phase Field Models

The exponential time differencing Runge–Kutta method (ETDRK) developed by Cox and Matthews has proven effective in computing nonlinear partial differential equation. However, it is difficult to prove the unconditional energy stability for third-order or higher ETDRK method. In this paper, we proposed a new third-order ETDRK scheme, ETDRK3, and prove its unconditional energy stability. Combining ETDRK3 with a second-order scheme ETDRK2, we proposed an adaptive ETDRK method(ETDRK32). Furthermore, the numerical simulations for phase problems demonstrate the accuracy, stability and efficiency of adaptive step-size strategy.