One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity assumption of nonlinear bulk. With the help of discrete orthogonal convolution kernels, an updated time-space splitting technique is developed to establish the uniform boundedness of stage solutions for a refined class of IERK methods in which the associated differentiation matrices and the average dissipation rates are always independent of the time-space discretization meshes. This makes the refined IERK methods highly advantageous in self-adaptive time-stepping procedures as some larger adaptive step sizes in actual simulations become possible. From the perspective of optimizing the average dissipation rate, we construct some parameterized refined IERK methods up to third-order accuracy, in which the involved diagonally implicit Runge-Kutta methods for the implicit part have an explicit first stage and allow a stage-order of two such that they are not necessarily algebraically stable.

Then we are able to establish, for the first time, the original energy dissipation law and the unconditional L2 norm convergence. Extensive numerical tests are presented to support our theory.

Publication:

MATHEMATICS OF COMPUTATION

https://doi.org/10.1090/mcom/4090

Author:

HONG-LIN LIAO

School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China; and Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, People’s Republic of China

Email address: liaohl@nuaa.edu.cn; liaohl@csrc.ac.cn

TAO TANG

School of Mathematics and Statistics, Guangzhou Nanfang College, Guangzhou

510970, People’s Republic of China

Email address: ttang@nfu.edu.cn

XUPING WANG

School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing

211106, People’s Republic of China

Email address: wangxp@nuaa.edu.cn

TAO ZHOU

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Email address: tzhou@lsec.cc.ac.cn