implementations.problem_classes.nonlinear_ODE_1 module

class nonlinear_ODE_1(u0=0.0, newton_maxiter=200, newton_tol=5e-11, stop_at_nan=True)

Bases: Problem

This class implements a simple nonlinear ODE with a singularity in the derivative, taken from https://www.osti.gov/servlets/purl/6111421 (Problem E-4). For \(0 \leq t \leq 5\), the problem is given by

\[\frac{du(t)}{dt} = \sqrt{1 - u(t)}\]

with initial condition \(u(0) = 0\). The exact solution is

\[u(t) = t - \frac{t^2}{4}.\]

Parameters:

• u0 (float, optional) – Initial condition.

• newton_maxiter (float, optional) – Maximum number of iterations for Newton’s method.

• newton_tol (float, optional) – Tolerance for Newton’s method to terminate.

• stop_at_nan (bool, optional) – Indicates that Newton solver has to stop if nan values arise.

newton_itercount

Counts the Newton iterations.

Type:

int

newton_ncalls

Counts calls of Newton method.

Type:

int

dtype_f

alias of mesh

dtype_u

alias of mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time of the numerical solution is computed (not used here).

Returns:

f – The right-hand side of the problem (one component).

Return type:

dtype_f

solve_system(rhs, dt, u0, t)

Simple Newton solver for the nonlinear equation

Parameters:

• rhs (dtype_f) – Right-hand side for the nonlinear system.

• dt (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver.

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

u – The solution as mesh.

Return type:

dtype_u

u_exact(t)

Routine to compute the exact solution at time \(t\).

Parameters:

t (float) – Time of the exact solution.

Returns:

me – The exact solution.

Return type:

dtype_u