solver module

Numerical solver of ordinary differential equations.

Solves the initial value problem for systems of first order ordinary differential equations.

Date

2015-09-21

e1(f, x0, t, *p, verbose=False)

Explicit first-order method / (standard, or forward) Euler method / Runge-Kutta 1st order method.

de: Euler’sche Polygonzugverfahren / explizite Euler-Verfahren / Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• verbose (bool) – print information (default = False)

Approximate the solution of the initial value problem

\[\begin{split}\dot{x} &= f(t,x) \\ x(t_0) &= x_0\end{split}\]

Choose a value h for the size of every step and set

\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]

One step \(h\) of the Euler method from \(t_i\) to \(t_{i+1}\) is

\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\ x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]

Example 1:

\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\ \ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]

with

\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]

becomes

\[\begin{split}\dot{x}_1 &= x_2 \\ \dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]

or

\[\begin{split}\dot{x} &= f(t,x) \\ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_1 \end{bmatrix} &= \begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1) \end{bmatrix} \\ &= \begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]

Example 2:

\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\ \ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]

with

\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\ x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]

becomes

\[\begin{split}\dot{x}_1 &= x_2 \\ \dot{x}_2 &= m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]

or

\[\begin{split}\dot{x} &= f(t,x) \\ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= \begin{bmatrix} x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \end{bmatrix} \\ &= \begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2) \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]

The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.

e2(f, x0, t, *p, verbose=False)

Explicit second-order method / Runge-Kutta 2nd order method.

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• verbose (bool) – print information (default = False)

e4(f, x0, t, *p, verbose=False)

Explicit fourth-order method / Runge-Kutta 4th order method.

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• verbose (bool) – print information (default = False)

i1(f, x0, t, *p, max_iterations=1000, tol=1e-09, verbose=False)

Implicite first-order method / backward Euler method.

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• max_iterations (int) – maximum number of iterations

• tol (float) – tolerance against residuum (default = 1e-9)

• verbose (bool) – print information (default = False)

The backward Euler method has order one and is A-stable.

newmark_newtonraphson(f, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, max_iterations=1000, tol=1e-09, verbose=False)

Newmark method.

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• xp0 (list) – initial condition

• xpp0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• gamma (float) – newmark parameter for velocity (default = 0.5)

• beta (float) – newmark parameter for displacement (default = 0.25)

• max_iterations (int) – maximum number of iterations

• tol (float) – tolerance against residuum (default = 1e-9)

• verbose (bool) – print information (default = False)

newmark_newtonraphson_rdk(fnm, x0, xp0, xpp0, t, *p, gamma=0.5, beta=0.25, maxIterations=1000, tol=1e-09, verbose=False)

Newmark method.

Parameters

• f (function) – the function to solve

• x0 (list) – initial condition

• xp0 (list) – initial condition

• xpp0 (list) – initial condition

• t (list) – time

• *p – parameters of the function (thickness, diameter, …)

• gamma (float) – newmark parameter for velocity (default = 0.5)

• beta (float) – newmark parameter for displacement (default = 0.25)

• max_iterations (int) – maximum number of iterations

• tol (float) – tolerance against residuum (default = 1e-9)

• verbose (bool) – print information (default = False)