Burgers::AnalyticData::Sinusoid Class Reference

Analytic data (with an "exact" solution known) that is periodic over the interval \([0,2\pi]\). More...

#include <Sinusoid.hpp>

Public Types
using	options = tmpl::list<>

Public Member Functions
	Sinusoid (const Sinusoid &)=default
Sinusoid &	operator= (const Sinusoid &)=default
	Sinusoid (Sinusoid &&)=default
Sinusoid &	operator= (Sinusoid &&)=default
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
template<typename T >
Scalar< T >	u (const tnsr::I< T, 1 > &x) const
tuples::TaggedTuple< Tags::U >	variables (const tnsr::I< DataVector, 1 > &x, tmpl::list< Tags::U >) const
void	pup (PUP::er &p) override
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr Options::String	help

Detailed Description

Analytic data (with an "exact" solution known) that is periodic over the interval \([0,2\pi]\).

The initial data is given by:

\begin{align} u(x, 0) = \sin(x) \end{align}

At future times the analytic solution can be found by solving the transcendental equation [80]

\begin{align} \label{eq:transcendental burgers periodic} \mathcal{F}=\sin\left(x-\mathcal{F}t\right) \end{align}

on the interval \(x\in(0,\pi)\). The solution from \(x\in(\pi,2\pi)\) is given by \(\mathcal{F}(x, t)=-\mathcal{F}(2\pi-x,t)\). The transcendental equation \((\ref{eq:transcendental burgers periodic})\) can be solved with a Newton-Raphson iterative scheme. Since this can be quite sensitive to the initial guess we implement this solution as analytic data. The python code below can be used to compute the analytic solution if desired.

At time \(1\) the solution develops a discontinuity at \(x=\pi\) followed by the amplitude of the solution decaying over time.

Note

We have rescaled \(x\) and \(t\) by \(\pi\) compared to [80].

import numpy as np
from scipy.optimize import newton
 
# x_grid is a np.array of positions at which to evaluate the solution
def burgers_periodic(x_grid, time):
    def opt_fun(F, x, t):
        return np.sin((x - F * t)) - F
 
    results = []
    for i in range(len(x_grid)):
        x = x_grid[i]
        greater_than_pi = False
        if x > np.pi:
            x = x - np.pi
            x = -x
            x = x + np.pi
            greater_than_pi = True
 
        guess = 0.0
        if len(results) > 0:
            if results[-1] < 0.0:
                guess = -results[-1]
            else:
                guess = results[-1]
        res = newton(lambda F: opt_fun(F, x, time), x0=guess)
 
        if greater_than_pi:
            results.append(-res)
        else:
            results.append(res)
 
    return np.asarray(results)

Member Function Documentation

get_clone()

auto Burgers::AnalyticData::Sinusoid::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

constexpr Options::String Burgers::AnalyticData::Sinusoid::help

staticconstexpr

Initial value:

{
      "A solution that is periodic over the interval [0,2pi]. The solution "
      "starts as a sinusoid: u(x,0) = sin(x) and develops a "
      "discontinuity at x=pi and t=1."}

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/AnalyticData/Burgers/Sinusoid.hpp