Thermalized LB: test 4 rel time model velocity and pressure fluctuations

This changes the fluctuating LB test to match as closely as possible what we are going to use. Model with 4 relaxation times for shear/bulk/odd/even modes. Tests:

• mass conservation
• momentum conservation
• <m v^2> = k_B T
• Maxwell velocity distribution
• average stress tensor
• stress tensor fluctuations accodring to Duenweg, Schiller, Ladd

lbmpy_tests/test_fluctuating_lb.py

-"""
-This tests that for the thermalized LB (MRT with 15 equal relaxation times),
-the correct temperature is obtained and the velocity distribution matches
-the Maxwell-Boltzmann distribution
-"""
+"""Tests velocity and stress fluctuations for thermalized LB"""
 
 
 import pystencils as ps
-from lbmpy.lbstep import LatticeBoltzmannStep
-from lbmpy.creationfunctions import create_lb_collision_rule
-from lbmpy.relaxationrates import relaxation_rate_from_lattice_viscosity, relaxation_rate_from_magic_number
+from lbmpy.creationfunctions import *
+from lbmpy.macroscopic_value_kernels import macroscopic_values_setter
 import numpy as np
-import pickle
-import gzip
-from time import time
+from lbmpy.moments import is_bulk_moment, is_shear_moment, get_order
 
 
-def single_component_maxwell(x1, x2, kT):
+def single_component_maxwell(x1, x2, kT, mass):
     """Integrate the probability density from x1 to x2 using the trapezoidal rule"""
     x = np.linspace(x1, x2, 1000)
-    return np.trapz(np.exp(-x**2 / (2. * kT)), x) / np.sqrt(2. * np.pi * kT)
-
-
-def run_scenario(scenario, steps):
-    scenario.pre_run()
-    for t in range(scenario.time_steps_run, scenario.time_steps_run + steps):
-        scenario.kernel_params['time_step'] = t
-        scenario.time_step()
-    scenario.post_run()
-    scenario.time_steps_run += steps
-
-
-def create_scenario(domain_size, temperature=None, viscosity=None, seed=2, target='cpu', openmp=4, num_rel_rates=None):
-    rr = [relaxation_rate_from_lattice_viscosity(viscosity)]
-    rr = rr*num_rel_rates
-    cr = create_lb_collision_rule(
-        stencil='D3Q19', compressible=True,
-        method='mrt', weighted=True, relaxation_rates=rr,
-        fluctuating={'temperature': temperature, 'seed': seed},
-        optimization={'cse_global': True, 'split': False,
-                      'cse_pdfs': True, 'vectorization': True}
+    return np.trapz(np.exp(-mass * x**2 / (2. * kT)), x) / np.sqrt(2. * np.pi * kT/mass)
+
+
+def rr_getter(moment_group):
+    """Mapos moments to 4 relaxation rates for shear, bulk, odd, even modes"""
+    is_shear = [is_shear_moment(m, 3) for m in moment_group]
+    is_bulk = [is_bulk_moment(m, 3) for m in moment_group]
+    order = [get_order(m) for m in moment_group]
+    assert min(order) == max(order)
+    order = order[0]
+
+    if order < 2:
+        return 0
+    elif any(is_bulk):
+        assert all(is_bulk)
+        return sp.Symbol("omega_bulk")
+    elif any(is_shear):
+        assert all(is_shear)
+        return sp.Symbol("omega_shear")
+    elif order % 2 == 0:
+        assert order > 2
+        return sp.Symbol("omega_even")
+    else:
+        return sp.Symbol("omega_odd")
+
+
+def second_order_moment_tensor_assignments(function_values, stencil, output_field):
+    """Assignments for calculating the pressure tensor"""
+    assert len(function_values) == len(stencil)
+    dim = len(stencil[0])
+    return [ps.Assignment(output_field(i, j),
+                          sum(c[i] * c[j] * f for f, c in zip(function_values, stencil)))
+            for i in range(dim) for j in range(dim)]
+
+
+def add_pressure_output_to_collision_rule(collision_rule, pressure_field):
+    pressure_ouput = second_order_moment_tensor_assignments(collision_rule.method.pre_collision_pdf_symbols,
+                                                            collision_rule.method.stencil, pressure_field)
+    collision_rule.main_assignments = collision_rule.main_assignments + pressure_ouput
+
+
+def test():
+
+    # Parameters
+    stencil = get_stencil('D3Q19')
+    L = [60]*3
+    kT = 4E-4
+    rho_0 = 0.8
+    omega_shear = 0.8
+    omega_bulk = 0.3
+    omega_even = 0.5
+    omega_odd = 0.4
+    target = 'cpu'
+
+    # Setup data handling
+    dh = ps.create_data_handling(L, periodicity=True, default_target=target)
+    src = dh.add_array('src', values_per_cell=len(stencil), layout='f')
+    dst = dh.add_array_like('dst', 'src')
+    rho = dh.add_array('rho', layout='f', latex_name='\\rho')
+    u = dh.add_array('u', values_per_cell=dh.dim, layout='f')
+    pressure_field = dh.add_array('pressure', values_per_cell=(
+        3, 3), layout='f', gpu=target == 'gpu')
+    force_field = dh.add_array(
+        'force', values_per_cell=3, layout='f', gpu=target == 'gpu')
+
+    # Method setup
+    method = create_mrt_orthogonal(
+        stencil=get_stencil('D3Q19'),
+        compressible=True,
+        weighted=True,
+        relaxation_rate_getter=rr_getter,
+        force_model=force_model_from_string('luo', force_field.center_vector))
+    collision_rule = create_lb_collision_rule(
+        method,
+        fluctuating={
+            'temperature': kT
+        },
+        optimization={'cse_global': True}
     )
-    return LatticeBoltzmannStep(periodicity=(True, True, True), domain_size=domain_size, compressible=True, stencil='D3Q19', collision_rule=cr, optimization={'target': target, 'openmp': openmp})
-
-
-def test_fluctuating_mrt():
-    # Unit conversions (MD to lattice) for parameters known to work with Espresso
-    agrid = 1.
-    m = 1.  # mass per node
-    tau = 0.01  # time step
-    temperature = 4. / (m * agrid**2/tau**2)
-    viscosity = 3. * tau / agrid**2
-    n = 8
-    sc = create_scenario((n, n, n), viscosity=viscosity, temperature=temperature,
-                         target='cpu', openmp=4, num_rel_rates=15)
-    assert np.average(sc.velocity[:, :, :]) == 0.
-
-    # Warmup
-    run_scenario(sc, steps=500)
-    # sampling:
-    steps = 20000
-    v = np.zeros((steps, n, n, n, 3))
-    for i in range(steps):
-        run_scenario(sc, steps=2)
-        v[i, :, :, :, :] = np.copy(sc.velocity[:, :, :, :])
-
-    v = v.reshape((steps*n*n*n, 3))
-    np.testing.assert_allclose(np.mean(v, axis=0), [0, 0, 0], atol=6E-7)
-    np.testing.assert_allclose(
-        np.var(v, axis=0), [temperature, temperature, temperature], rtol=1E-2)
-    v_hist, v_bins = np.histogram(v, bins=11, range=(-.08, .08), density=True)
-
-    # Calculate expected values from single
-    v_expected = []
-    for i in range(len(v_hist)):
-        # Maxwell distribution
-        res = np.exp(-v_bins[i]**2/(2.*temperature)) / \
-            np.sqrt(2*np.pi*temperature)
-        res = 1./(v_bins[i+1]-v_bins[i]) * \
-            single_component_maxwell(v_bins[i], v_bins[i+1], temperature)
-        v_expected.append(res)
-    v_expected = np.array(v_expected)
-
-    # 8% accuracy on the entire histogram
-    np.testing.assert_allclose(v_hist, v_expected, rtol=0.08)
-    # 0.5% accuracy on the middle part
-    remove = 3
-    np.testing.assert_allclose(
-        v_hist[remove:-remove], v_expected[remove:-remove], rtol=0.005)
 
+    add_pressure_output_to_collision_rule(collision_rule, pressure_field)
+
+    collision = create_lb_update_rule(collision_rule=collision_rule,
+                                      stencil=stencil,
+                                      method=method,
+                                      compressible=True,
+                                      kernel_type='collide_only',
+                                      optimization={'symbolic_field': src})
+    stream = create_stream_pull_with_output_kernel(collision.method, src, dst,
+                                                   {'density': rho, 'velocity': u})
+
+    opts = {'cpu_openmp': True,
+            'cpu_vectorize_info': None,
+            'target': dh.default_target}
+
+    # Compile kernels
+    stream_kernel = ps.create_kernel(stream, **opts).compile()
+    collision_kernel = ps.create_kernel(collision, **opts).compile()
+
+    sync_pdfs = dh.synchronization_function([src.name])
+
+    # Initialization
+    init = macroscopic_values_setter(collision.method, velocity=(0,)*dh.dim,
+                                     pdfs=src.center_vector, density=rho.center)
+    init_kernel = ps.create_kernel(init, ghost_layers=0).compile()
+
+    dh.fill(rho.name, rho_0)
+    dh.fill(u.name, np.nan, ghost_layers=True, inner_ghost_layers=True)
+    dh.fill(u.name, 0)
+    dh.run_kernel(init_kernel)
+
+    # time loop
+    def time_loop(start, steps):
+        dh.all_to_gpu()
+        for i in range(start, start+steps):
+            dh.run_kernel(collision_kernel, omega_shear=omega_shear, omega_bulk=omega_bulk,
+                          omega_odd=omega_odd, omega_even=omega_even, seed=42, time_step=i)
+
+            sync_pdfs()
+            dh.run_kernel(stream_kernel)
+
+            dh.swap(src.name, dst.name)
+        return start+steps
+
+    # Test
+    t = 0
+    # warm up
+    t = time_loop(t, 10)
+
+    # Measurement
+    for i in range(10):
+        t = time_loop(t, 5)
+
+        res_u = dh.gather_array("u").reshape((-1, 3))
+        res_rho = dh.gather_array("rho").reshape((-1,))
+
+        # mass conservation
+        np.testing.assert_allclose(np.mean(res_rho), rho_0, atol=3E-12)
+
+        # momentum conservation
+        momentum = np.dot(res_rho, res_u)
+        np.testing.assert_allclose(momentum, [0, 0, 0], atol=1E-10)
+
+        # temperature
+        kinetic_energy = 1/2*np.dot(res_rho, res_u*res_u)/np.product(L)
+        np.testing.assert_allclose(
+            kinetic_energy, [kT/2]*3, atol=kT*0.01)
+
+        # velocity distribution
+        v_hist, v_bins = np.histogram(
+            res_u, bins=11, range=(-.075, .075), density=True)
+
+        # Calculate expected values from single
+        v_expected = []
+        for j in range(len(v_hist)):
+            # Maxwell distribution
+            res = 1./(v_bins[j+1]-v_bins[j]) * \
+                single_component_maxwell(
+                    v_bins[j], v_bins[j+1], kT, rho_0)
+            v_expected.append(res)
+        v_expected = np.array(v_expected)
+
+        # 10% accuracy on the entire histogram
+        np.testing.assert_allclose(v_hist, v_expected, rtol=0.1)
+        # 1% accuracy on the middle part
+        remove = 3
+        np.testing.assert_allclose(
+            v_hist[remove:-remove], v_expected[remove:-remove], rtol=0.01)
+
+        # pressure tensor against expressions from
+        # Duenweg, Schiller, Ladd, https://arxiv.org/abs/0707.1581
+
+        res_pressure = dh.gather_array("pressure").reshape((-1, 3, 3))
+
+        c_s = np.sqrt(1/3)  # speed of sound
+
+        # average of pressure tensor
+        # Diagonal elements are rho c_s^22 +<u,u>. When the fluid is
+        # thermalized, the expectation value of <u,u> = kT due to the
+        # equi-partition theorem.
+        p_av_expected = np.diag([rho_0*c_s**2 + kT]*3)
+        np.testing.assert_allclose(
+            np.mean(res_pressure, axis=0), p_av_expected, atol=c_s**2/2000)