Time Integrators

Implicit Euler

• Parameters: None
• Description:

$$\begin{array}{rl}{\dot{u}}^{t+1}& ={\dot{u}}^t+h{\ddot{u}}^{t+1}\\ u^{t+1}& =u^t+h{\dot{u}}^{t+1}\end{array}$$

where $h$ is the time step size.

• Reference: https://en.wikipedia.org/wiki/Backward_Euler_method

Implicit Newmark

• Parameters: beta, gamma
• Description:

$$\begin{array}{rl}{\dot{u}}^{t+1}& ={\dot{u}}^t+(1-\gamma )h{\ddot{u}}^t+\gamma h{\ddot{u}}^{t+1}\\ u^{t+1}& =u^t+h{\dot{u}}^t+\frac{h^2}{2}((1-2\beta ){\ddot{u}}^t+2\beta {\ddot{u}}^{t+1})\end{array}$$

where $h$ is the time step size and by default $\gamma =0.5$ and $\beta =0.25$.

• Reference: https://en.wikipedia.org/wiki/Newmark-beta_method

Note

This is equivalent to the Trapezoidal rule for $\gamma =0.5$ and $\beta =0.25$.

Backward Differentiation Formula (BDF)

• Parameters: num_steps
• Description:

$$\begin{array}{rl}{\dot{u}}^{t+1}& =(\sum _{i=0}^{n-1}{\alpha }_i{\dot{u}}^{t-i})+h\beta {\ddot{u}}^{t+1}\\ u^{t+1}& =(\sum _{i=0}^{n-1}{\alpha }_i{u}^{t-i})+h\beta {\dot{u}}^{t+1}\end{array}$$

where $h$ is the time step size, the coefficients ${\alpha }_i$ and $\beta$ are choosen to make the method $n$-th order accurate, and $n\in \{1,\dots ,6\}$ is the number of previous steps to consider (default: num_steps=1). This is equivalent to implicit Euler for $n=1$. To initialize the values for $i>0$, the method starts from $n=1$ and successively builds the history, increasing $n$ by 1 until the $n$-th step. However, we know this can result in less than order $n$ accuracy [Nishikawa 2019].

• Reference: https://en.wikipedia.org/wiki/Backward_differentiation_formula

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Last update: 2023-10-03