时间推进格式 Runge-Kutta 方法

显式时间推进格式 Runge-Kutta 基本公式

Runge-Kutta

A general m-stage Runge–Kutta method is written in the form

\begin{align} & \bold{U}^{(0)} = \bold{U}^{n}, \quad \bold{U}^{n+1} = \bold{U}^{(m)} \\ & \bold{U}^{(i)} = \sum_{k=0}^{i-1} \left( \alpha_{i,k} \bold{U}^{(k)} + \Delta t \beta_{i,k} \bold{R}^{(k)} \right) \end{align}

\begin{align} \bold{U}^{(i)} = \bold{U}^{(0)} - \alpha_i \Delta t \bold{R}^{(i-1)} \end{align}

stage	3	4	5
CFL $\sigma$	1.5	2.0	1.5
$\alpha_1$	0.1481	0.0833	0.0533
$\alpha_2$	0.4000	0.2069	0.1263
$\alpha_3$	1.0000	0.4265	0.2375
$\alpha_4$		1.0000	0.4414
$\alpha_5$			1.0000

stage	3	4	5
CFL $\sigma$	0.69	0.92	1.15
$\alpha_1$	0.1918	0.1084	0.0695
$\alpha_2$	0.4929	0.2602	0.1602
$\alpha_3$	1.0000	0.5052	0.2898
$\alpha_4$		1.0000	0.5060
$\alpha_5$			1.0000

\begin{align} \bold{U}^{(i)} = \alpha_{i,0} \bold{U}^{(0)} + \alpha_{i,i-1} \bold{U}^{(i-1)} + \beta_i \Delta t \bold{R}^{(i-1)} \end{align}

\begin{align} d\bold{U}^{(i)} = \alpha_i (\bold{U}^{(0)} - \bold{U}^{(i-1)}) + \beta_i \Delta t c \bold{R}^{(i-1)} \end{align}