bte.utils.ERK#

Module Contents#

Classes#

RK

Explicit Runge-Kutta Method

RK1

1st order Runge-Kutta

RK2

2nd order Runge-Kutta

RK3_SSP

Explicit Runge-Kutta Method

RK4

4nd order Runge-Kutta

RK4_2

Explicit Runge-Kutta Method
class bte.utils.ERK.RK(c, b, A)[source]#

Explicit Runge-Kutta Method

Create a runge-kutta method to integrate \(u_t=f(t,u)\)

\[\begin{split}\begin{aligned} y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\ \ \vdots \\k_{s}&=f(t_{n}+c_{s}h,y_{n}+h(a_{s1}k_{1}+a_{s2}k_{2}+\cdots +a_{s,s-1}k_{s-1})). \end{aligned}\end{split}\]
Parameters:

  • c (list) – runge-kutta coef

  • b (list) – runge-kutta coef

  • A (list) – runge-kutta coef

Butcher tableau

\[\begin{split}\begin{array}{c|c} c & A \\ \hline & b \end{array}\end{split}\]
__call__(f, t0, u0, h, f0=None)[source]#

single step of runge-kutta

u_t = f(u,t)

f_0=f(u0,t0)

Parameters:

  • f (Callable) – [description]

  • t0 (float) – [description]

  • u0 (float) – [description]

  • h (float) – [description]

  • f0 ([type], optional) – [description]. Defaults to None.

Returns:

[description]

Return type:

[type]

class bte.utils.ERK.RK1[source]#

Bases: RK

1st order Runge-Kutta

Butcher tableau:

\[\begin{split}\begin{array}{c|c} 0 & 0 \\ \hline & 1 \end{array}\end{split}\]
class bte.utils.ERK.RK2(alpha=1 / 2)[source]#

Bases: RK

2nd order Runge-Kutta

Butcher tableau:

\[\begin{split}\begin{array}{c|cc} 0 & 0 & 0\\ \alpha & \alpha & 0 \\ \hline & (1 - \frac{1}{2\alpha}) & \frac{1}{2\alpha} \end{array}\end{split}\]
Parameters:

alpha (float, optional) – Defaults to 1/2.

class bte.utils.ERK.RK3_SSP[source]#

Bases: RK

Explicit Runge-Kutta Method

Create a runge-kutta method to integrate \(u_t=f(t,u)\)

\[\begin{split}\begin{aligned} y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\ \ \vdots \\k_{s}&=f(t_{n}+c_{s}h,y_{n}+h(a_{s1}k_{1}+a_{s2}k_{2}+\cdots +a_{s,s-1}k_{s-1})). \end{aligned}\end{split}\]
Parameters:

  • c (list) – runge-kutta coef

  • b (list) – runge-kutta coef

  • A (list) – runge-kutta coef

Butcher tableau

\[\begin{split}\begin{array}{c|c} c & A \\ \hline & b \end{array}\end{split}\]
class bte.utils.ERK.RK4[source]#

Bases: RK

4nd order Runge-Kutta

Butcher tableau:

\[\begin{split}\begin{array}{c|cc} 0 & & & &\\ \frac{1}{2} & \frac{1}{2} & & & \\ \frac{1}{2} & 0 & \frac{1}{2} & & \\ 1 & 0 & 0 & 1 & \\ \hline & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} & \frac{1}{6} \end{array}\end{split}\]
class bte.utils.ERK.RK4_2[source]#

Bases: RK

Explicit Runge-Kutta Method

Create a runge-kutta method to integrate \(u_t=f(t,u)\)

\[\begin{split}\begin{aligned} y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\ \ \vdots \\k_{s}&=f(t_{n}+c_{s}h,y_{n}+h(a_{s1}k_{1}+a_{s2}k_{2}+\cdots +a_{s,s-1}k_{s-1})). \end{aligned}\end{split}\]
Parameters:

  • c (list) – runge-kutta coef

  • b (list) – runge-kutta coef

  • A (list) – runge-kutta coef

Butcher tableau

\[\begin{split}\begin{array}{c|c} c & A \\ \hline & b \end{array}\end{split}\]