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Published at November 25

An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation

math.NA

Released Date: November 25, 2024

Authors: Haifeng Wang, Jingwei Sun, Hong Zhang, Xu Qian, Songhe Song

Arxiv: http://arxiv.org/abs/2411.16271v1

Refer to caption

Stages	1	2	3	$s$
	$\Delta_{1,1}$	$\Delta_{1,1}-\frac{1}{2}\Delta_{2,1}$	$\Delta_{1,1}-\frac{1}{2}\Delta_{2,1}-\frac{1}{2}\Delta_{3,1}$	$\Delta_{1,1}-\frac{1}{2}\sum_{j=2}^{s}\Delta_{j,1}$
		$\Delta_{2,1}$	$\Delta_{2,1}$	$\Delta_{2,1}$
Energy		$\Delta_{2,2}-\frac{1}{2}\Delta_{2,1}$	$\Delta_{2,2}-\frac{1}{2}\Delta_{2,1}-\frac{1}{2}\Delta_{3,2}$	$\Delta_{2,2}-\frac{1}{2}\Delta_{2,1}-\frac{1}{2}\sum_{j=3}^{s}\Delta_{j,2}$
stability			$\Delta_{3,1}$	$\Delta_{3,1}$
matrices			$\Delta_{3,2}$	$\Delta_{3,2}$
			$\Delta_{3,3}-\frac{1}{2}\Delta_{3,1}-\frac{1}{2}\Delta_{3,2}$	$\Delta_{3,3}-\frac{1}{2}\sum_{j=1}^{2}\Delta_{3,j}$
				$\Delta_{i,j},i=1,\dots,s;j=1,\dots,i-1$
				$\Delta_{i,i}-\frac{1}{2}\sum_{j=1}^{i-1}\Delta_{i,j}-\frac{1}{2}\sum_{j=i+1}^{% s}\Delta_{j,i},i=1,\dots,s$