A class of refined implicit-explicit Runge-Kutta methods with robust time adaptability and unconditional convergence for the Cahn-Hilliard model

TL;DR

This paper introduces a new class of refined implicit-explicit Runge-Kutta methods with robust adaptive time-stepping and proven unconditional convergence for the Cahn-Hilliard model, overcoming previous limitations related to nonlinearities.

Contribution

It develops a refined IERK class with uniform boundedness of stage solutions and unconditional convergence, enabling larger adaptive steps in phase field simulations.

Findings

Established uniform boundedness of stage solutions.

Proved unconditional energy dissipation law.

Demonstrated effectiveness through numerical tests.

Abstract

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.