Taylor series Explicit Euler Implicit Euler

1 Taylor series

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}\left(a\right)}{n!}{(x-a)}^n\\ f\left(a\right)+\frac{f^{\prime }\left(a\right)}{1!}(x-a)+\frac{f^{\prime \prime }\left(a\right)}{2!}(x-a{)}^2+\frac{f^{\prime \prime \prime }\left(a\right)}{3!}(x-a{)}^3+\dots \end{array}$$

$$y(t+\mathrm{\Delta }t)=y(t)+\mathrm{\Delta }t{y}^{\prime }(t)+\frac{1}{2}\mathrm{\Delta }{t}^2{y}^{″}(t)+\frac{1}{3!}\mathrm{\Delta }{t}^3{y}^{‴}(t)+...$$

2 Explicit Euler

$$\mathrm{\Delta }t=t_1-t_0$$

$$\begin{array}{rl}{\int }_{t^{[0]}}^{t^{[1]}}\mathbf{v}(t)dt& =\mathrm{\Delta }t\mathbf{v}(t^{[0]})+\frac{\mathrm{\Delta }{t}^2}{2}{\mathbf{v}}^{\prime }(t^{[0]})+\cdots \\ & =\mathrm{\Delta }t\mathbf{v}(t^{[0]})+O(\mathrm{\Delta }{t}^2)\end{array}$$

$$\mathbf{x}(t^{[1]})-\mathbf{x}(t^{[0]})=[\text{Taylor at }\mathbf{x}(t^{[1]})\text{ based on }\mathbf{x}(t^{[0]})]-\mathbf{x}(t^{[0]})$$

3 Implicit Euler

$$\mathrm{\Delta }t=t_1-t_0$$

$$\begin{array}{rl}{\int }_{t^{[0]}}^{t^{[1]}}\mathbf{v}(t)dt& =\mathrm{\Delta }t\mathbf{v}(t^{[1]})-\frac{\mathrm{\Delta }{t}^2}{2}{\mathbf{v}}^{\prime }(t^{[1]})+\cdots \\ & =\mathrm{\Delta }t\mathbf{v}(t^{[1]})+O(\mathrm{\Delta }{t}^2)\end{array}$$

$$\mathbf{x}(t^{[1]})-\mathbf{x}(t^{[0]})=\mathbf{x}(t^{[1]})-[\text{Taylor at }\mathbf{x}(t^{[0]})\text{ based on }\mathbf{x}(t^{[1]})]$$

X ref

1. Euler's Method, Taylor Series Method, Runge Kutta ... Brown University
2. Mathew Mariani Midpoint Integration
3. GAMES103: Intro to Physics-Based Animation by Rigid Body Dynamics