burgers.py¶
In this script we solve the Burgers equation on a 1D or 2D periodic domain, starting from a centered Gaussian and convecting in the positive direction of the first coordinate.
7from nutils import mesh, function, solver, export, cli, testing
8from nutils.expression_v2 import Namespace
9import numpy
10import treelog
11
12
13def main(nelems: int, ndims: int, btype: str, degree: int, timescale: float, newtontol: float, endtime: float):
14 '''
15 Burgers equation on a 1D or 2D periodic domain.
16
17 .. arguments::
18
19 nelems [20]
20 Number of elements along a single dimension.
21 ndims [1]
22 Number of spatial dimensions.
23 btype [discont]
24 Type of basis function (discont/legendre).
25 degree [1]
26 Polynomial degree for discontinuous basis functions.
27 timescale [.5]
28 Fraction of timestep and element size: timestep=timescale/nelems.
29 newtontol [1e-5]
30 Newton tolerance.
31 endtime [inf]
32 Stopping time.
33 '''
34
35 domain, geom = mesh.rectilinear([numpy.linspace(-.5, .5, nelems+1)]*ndims, periodic=range(ndims))
36
37 ns = Namespace()
38 ns.x = geom
39 ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
40 ns.basis = domain.basis(btype, degree=degree)
41 ns.u = function.dotarg('lhs', ns.basis)
42 ns.f = '.5 u^2'
43 ns.C = 1
44 ns.u0 = 'exp(-25 x_i x_i)'
45
46 res = domain.integral('-∇_0(basis_n) f dV' @ ns, degree=5)
47 res += domain.interfaces.integral('-[basis_n] n_0 ({f} - .5 C [u] n_0) dS' @ ns, degree=degree*2)
48 inertia = domain.integral('basis_n u dV' @ ns, degree=5)
49
50 sqr = domain.integral('(u - u0)^2 dV' @ ns, degree=5)
51 lhs0 = solver.optimize('lhs', sqr)
52
53 timestep = timescale/nelems
54 bezier = domain.sample('bezier', 7)
55 with treelog.iter.plain('timestep', solver.impliciteuler('lhs', res, inertia, timestep=timestep, arguments=dict(lhs=lhs0), newtontol=newtontol)) as steps:
56 for itime, lhs in enumerate(steps):
57 x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
58 export.triplot('solution.png', x, u, tri=bezier.tri, hull=bezier.hull, clim=(0, 1))
59 if itime * timestep >= endtime:
60 break
61
62 return lhs
If the script is executed (as opposed to imported), nutils.cli.run()
calls the main function with arguments provided from the command line. For
example, to simulate until 0.5 seconds run python3 burgers.py
endtime=0.5
(view log).
70if __name__ == '__main__':
71 cli.run(main)
Once a simulation is developed and tested, it is good practice to save a few
strategic return values for regression testing. The nutils.testing
module, which builds on the standard unittest
framework, facilitates
this by providing nutils.testing.TestCase.assertAlmostEqual64()
for the
embedding of desired results as compressed base64 data.
80class test(testing.TestCase):
81
82 def test_1d_p0(self):
83 lhs = main(ndims=1, nelems=10, timescale=.1, btype='discont', degree=0, endtime=.01, newtontol=1e-5)
84 self.assertAlmostEqual64(lhs, '''
85 eNrz1ttqGGOiZSZlrmbuZdZgcsEwUg8AOqwFug==''')
86
87 def test_1d_p1(self):
88 lhs = main(ndims=1, nelems=10, timescale=.1, btype='discont', degree=1, endtime=.01, newtontol=1e-5)
89 self.assertAlmostEqual64(lhs, '''
90 eNrbocann6u3yqjTyMLUwfSw2TWzKPNM8+9mH8wyTMNNZxptMirW49ffpwYAI6cOVA==''')
91
92 def test_1d_p2(self):
93 lhs = main(ndims=1, nelems=10, timescale=.1, btype='discont', degree=2, endtime=.01, newtontol=1e-5)
94 self.assertAlmostEqual64(lhs, '''
95 eNrr0c7SrtWfrD/d4JHRE6Ofxj6mnqaKZofNDpjZmQeYB5pHmL8we23mb5ZvWmjKY/LV6KPRFIMZ+o36
96 8dp92gCxZxZG''')
97
98 def test_1d_p1_legendre(self):
99 lhs = main(ndims=1, nelems=10, timescale=.1, btype='legendre', degree=1, endtime=.01, newtontol=1e-5)
100 self.assertAlmostEqual64(lhs, '''
101 eNrbpbtGt9VQyNDfxMdYzczERNZczdjYnOdsoNmc01kmE870Gj49t0c36BIAAhsO1g==''')
102
103 def test_1d_p2_legendre(self):
104 lhs = main(ndims=1, nelems=10, timescale=.1, btype='legendre', degree=2, endtime=.01, newtontol=1e-5)
105 self.assertAlmostEqual64(lhs, '''
106 eNoBPADD/8ot2y2/K4UxITFFLk00RTNNLyY2KzTTKx43QjOOzzM3Ss0pz1A2qsvhKGk0jsyXL48xzc5j
107 LswtIdLIK5SlF78=''')
108
109 def test_2d_p1(self):
110 lhs = main(ndims=2, nelems=4, timescale=.1, btype='discont', degree=1, endtime=.01, newtontol=1e-5)
111 import os
112 if os.environ.get('NUTILS_TENSORIAL'):
113 lhs = lhs.reshape(4, 2, 4, 2).transpose(0, 2, 1, 3).ravel()
114 self.assertAlmostEqual64(lhs, '''
115 eNoNyKENhEAQRuGEQsCv2SEzyQZHDbRACdsDJNsBjqBxSBxBHIgJ9xsqQJ1Drro1L1/eYBZceGz8njrR
116 yacm8UQLBvPYCw1airpyUVYSJLhKijK4IC01WDnqqxvX8OTl427aU73sctPGr3qqceBnRzOjo0xy9JpJ
117 R73m6R6YMZo/Q+FCLQ==''')