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 $L^2$ norm convergence. Extensive numerical tests are presented to support our theory.

翻译：验证隐式-显式Runge-Kutta方法对相场方程能量耗散律的主要障碍之一，是在非线性体项不具备全局Lipschitz连续性的假设下，建立阶段解的一致有界性。借助离散正交卷积核，本文发展了一种更新的时空分裂技术，为一类精细化IERK方法建立了阶段解的一致有界性，其中关联的微分矩阵与平均耗散率始终独立于时空离散网格。这使得精细化IERK方法在自适应时间步进过程中具有显著优势，因为在实际模拟中可以采用更大的自适应步长。从优化平均耗散率的角度出发，我们构造了若干参数化的精细化IERK方法，其精度可达三阶，其中隐式部分所涉及的对角隐式Runge-Kutta方法具有显式首级，并允许达到二阶阶段精度，因此不要求代数稳定性。在此基础上，我们首次建立了原始能量耗散律与无条件$L^2$范数收敛性。大量数值实验验证了理论结果。