Source code for pymor.analyticalproblems.burgers

# -*- coding: utf-8 -*-
# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2017 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)

import numpy as np

from pymor.analyticalproblems.elliptic import StationaryProblem
from pymor.analyticalproblems.instationary import InstationaryProblem
from pymor.domaindescriptions.basic import LineDomain, RectDomain, TorusDomain, CircleDomain
from pymor.functions.basic import ConstantFunction, ExpressionFunction
from pymor.parameters.spaces import CubicParameterSpace


[docs]def burgers_problem(v=1., circle=True, initial_data_type='sin', parameter_range=(1., 2.)): """One-dimensional Burgers-type problem. The problem is to solve :: ∂_t u(x, t, μ) + ∂_x (v * u(x, t, μ)^μ) = 0 u(x, 0, μ) = u_0(x) for u with t in [0, 0.3] and x in [0, 2]. Parameters ---------- v The velocity v. circle If `True`, impose periodic boundary conditions. Otherwise Dirichlet left, outflow right. initial_data_type Type of initial data (`'sin'` or `'bump'`). parameter_range The interval in which μ is allowed to vary. """ assert initial_data_type in ('sin', 'bump') if initial_data_type == 'sin': initial_data = ExpressionFunction('0.5 * (sin(2 * pi * x) + 1.)', 1, ()) dirichlet_data = ConstantFunction(dim_domain=1, value=0.5) else: initial_data = ExpressionFunction('(x >= 0.5) * (x <= 1) * 1.', 1, ()) dirichlet_data = ConstantFunction(dim_domain=1, value=0.) return InstationaryProblem( StationaryProblem( domain=CircleDomain([0, 2]) if circle else LineDomain([0, 2], right=None), dirichlet_data=dirichlet_data, rhs=None, nonlinear_advection=ExpressionFunction('abs(x)**exponent * v', 1, (1,), {'exponent': ()}, {'v': v}), nonlinear_advection_derivative=ExpressionFunction('exponent * abs(x)**(exponent-1) * sign(x) * v', 1, (1,), {'exponent': ()}, {'v': v}), ), T=0.3, initial_data=initial_data, parameter_space=CubicParameterSpace({'exponent': 0}, *parameter_range), name="burgers_problem({}, {}, '{}')".format(v, circle, initial_data_type) )
[docs]def burgers_problem_2d(vx=1., vy=1., torus=True, initial_data_type='sin', parameter_range=(1., 2.)): """Two-dimensional Burgers-type problem. The problem is to solve :: ∂_t u(x, t, μ) + ∇ ⋅ (v * u(x, t, μ)^μ) = 0 u(x, 0, μ) = u_0(x) for u with t in [0, 0.3], x in [0, 2] x [0, 1]. Parameters ---------- vx The x component of the velocity vector v. vy The y component of the velocity vector v. torus If `True`, impose periodic boundary conditions. Otherwise, Dirichlet left and bottom, outflow top and right. initial_data_type Type of initial data (`'sin'` or `'bump'`). parameter_range The interval in which μ is allowed to vary. """ assert initial_data_type in ('sin', 'bump') if initial_data_type == 'sin': initial_data = ExpressionFunction("0.5 * (sin(2 * pi * x[..., 0]) * sin(2 * pi * x[..., 1]) + 1.)", 2, ()) dirichlet_data = ConstantFunction(dim_domain=2, value=0.5) else: initial_data = ExpressionFunction("(x[..., 0] >= 0.5) * (x[..., 0] <= 1) * 1", 2, ()) dirichlet_data = ConstantFunction(dim_domain=2, value=0.) return InstationaryProblem( StationaryProblem( domain=TorusDomain([[0, 0], [2, 1]]) if torus else RectDomain([[0, 0], [2, 1]], right=None, top=None), dirichlet_data=dirichlet_data, rhs=None, nonlinear_advection=ExpressionFunction("abs(x)**exponent * v", 1, (2,), {'exponent': ()}, {'v': np.array([vx, vy])}), nonlinear_advection_derivative=ExpressionFunction("exponent * abs(x)**(exponent-1) * sign(x) * v", 1, (2,), {'exponent': ()}, {'v': np.array([vx, vy])}), ), initial_data=initial_data, T=0.3, parameter_space=CubicParameterSpace({'exponent': 0}, *parameter_range), name="burgers_problem_2d({}, {}, {}, '{}')".format(vx, vy, torus, initial_data_type) )