test_grudge.py

__copyright__ = """
Copyright (C) 2015 Andreas Kloeckner
Copyright (C) 2021 University of Illinois Board of Trustees
"""

__license__ = """
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
"""

import numpy as np
import numpy.linalg as la

from grudge.array_context import PytestPyOpenCLArrayContextFactory
from arraycontext import pytest_generate_tests_for_array_contexts
pytest_generate_tests = pytest_generate_tests_for_array_contexts(
        [PytestPyOpenCLArrayContextFactory])

from arraycontext.container.traversal import thaw

from meshmode import _acf           # noqa: F401

from meshmode.dof_array import flat_norm
import meshmode.mesh.generation as mgen

from pytools.obj_array import flat_obj_array

from grudge import DiscretizationCollection

import grudge.dof_desc as dof_desc
import grudge.op as op


import pytest

import logging

logger = logging.getLogger(__name__)


# {{{ mass operator trig integration

@pytest.mark.parametrize("ambient_dim", [1, 2, 3])
@pytest.mark.parametrize("discr_tag", [dof_desc.DISCR_TAG_BASE,
                                       dof_desc.DISCR_TAG_QUAD])
def test_mass_mat_trig(actx_factory, ambient_dim, discr_tag):
    """Check the integral of some trig functions on an interval using the mass
    matrix.
    """
    actx = actx_factory()

    nel_1d = 16
    order = 4

    a = -4.0 * np.pi
    b = +9.0 * np.pi
    true_integral = 13*np.pi/2 * (b - a)**(ambient_dim - 1)

    from meshmode.discretization.poly_element import QuadratureSimplexGroupFactory
    dd_quad = dof_desc.DOFDesc(dof_desc.DTAG_VOLUME_ALL, discr_tag)
    if discr_tag is dof_desc.DISCR_TAG_BASE:
        discr_tag_to_group_factory = {}
    else:
        discr_tag_to_group_factory = {
            discr_tag: QuadratureSimplexGroupFactory(order=2*order)
        }

    mesh = mgen.generate_regular_rect_mesh(
            a=(a,)*ambient_dim, b=(b,)*ambient_dim,
            nelements_per_axis=(nel_1d,)*ambient_dim, order=1)
    dcoll = DiscretizationCollection(
        actx, mesh, order=order,
        discr_tag_to_group_factory=discr_tag_to_group_factory
    )

    def f(x):
        return actx.np.sin(x[0])**2

    volm_disc = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
    x_volm = thaw(volm_disc.nodes(), actx)
    f_volm = f(x_volm)
    ones_volm = volm_disc.zeros(actx) + 1

    quad_disc = dcoll.discr_from_dd(dd_quad)
    x_quad = thaw(quad_disc.nodes(), actx)
    f_quad = f(x_quad)
    ones_quad = quad_disc.zeros(actx) + 1

    mop_1 = op.mass(dcoll, dd_quad, f_quad)
    num_integral_1 = op.nodal_sum(
        dcoll, dof_desc.DD_VOLUME, ones_volm * mop_1
    )

    err_1 = abs(num_integral_1 - true_integral)
    assert err_1 < 2e-9, err_1

    mop_2 = op.mass(dcoll, dd_quad, ones_quad)
    num_integral_2 = op.nodal_sum(dcoll, dof_desc.DD_VOLUME, f_volm * mop_2)

    err_2 = abs(num_integral_2 - true_integral)
    assert err_2 < 2e-9, err_2

    if discr_tag is dof_desc.DISCR_TAG_BASE:
        # NOTE: `integral` always makes a square mass matrix and
        # `QuadratureSimplexGroupFactory` does not have a `mass_matrix` method.
        num_integral_3 = op.nodal_sum(dcoll, dof_desc.DD_VOLUME, f_quad * mop_2)
        err_3 = abs(num_integral_3 - true_integral)
        assert err_3 < 5e-10, err_3

# }}}


# {{{ mass operator on surface

def _ellipse_surface_area(radius, aspect_ratio):
    # https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipe.html
    eccentricity = 1.0 - (1/aspect_ratio)**2

    if abs(aspect_ratio - 2.0) < 1.0e-14:
        # NOTE: hardcoded value so we don't need scipy for the test
        ellip_e = 1.2110560275684594
    else:
        from scipy.special import ellipe        # pylint: disable=no-name-in-module
        ellip_e = ellipe(eccentricity)

    return 4.0 * radius * ellip_e


def _spheroid_surface_area(radius, aspect_ratio):
    # https://en.wikipedia.org/wiki/Ellipsoid#Surface_area
    a = 1.0
    c = aspect_ratio

    if a < c:
        e = np.sqrt(1.0 - (a/c)**2)
        return 2.0 * np.pi * radius**2 * (1.0 + (c/a) / e * np.arcsin(e))
    else:
        e = np.sqrt(1.0 - (c/a)**2)
        return 2.0 * np.pi * radius**2 * (1 + (c/a)**2 / e * np.arctanh(e))


@pytest.mark.parametrize("name", [
    "2-1-ellipse", "spheroid", "box2d", "box3d"
    ])
def test_mass_surface_area(actx_factory, name):
    actx = actx_factory()

    # {{{ cases

    if name == "2-1-ellipse":
        from mesh_data import EllipseMeshBuilder
        builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
        surface_area = _ellipse_surface_area(builder.radius, builder.aspect_ratio)
    elif name == "spheroid":
        from mesh_data import SpheroidMeshBuilder
        builder = SpheroidMeshBuilder()
        surface_area = _spheroid_surface_area(builder.radius, builder.aspect_ratio)
    elif name == "box2d":
        from mesh_data import BoxMeshBuilder
        builder = BoxMeshBuilder(ambient_dim=2)
        surface_area = 1.0
    elif name == "box3d":
        from mesh_data import BoxMeshBuilder
        builder = BoxMeshBuilder(ambient_dim=3)
        surface_area = 1.0
    else:
        raise ValueError("unknown geometry name: %s" % name)

    # }}}

    # {{{ convergence

    from pytools.convergence import EOCRecorder
    eoc = EOCRecorder()

    for resolution in builder.resolutions:
        mesh = builder.get_mesh(resolution, builder.mesh_order)
        dcoll = DiscretizationCollection(actx, mesh, order=builder.order)
        volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)

        logger.info("ndofs:     %d", volume_discr.ndofs)
        logger.info("nelements: %d", volume_discr.mesh.nelements)

        # {{{ compute surface area

        dd = dof_desc.DD_VOLUME
        ones_volm = volume_discr.zeros(actx) + 1
        approx_surface_area = actx.to_numpy(op.integral(dcoll, dd, ones_volm))

        logger.info("surface: got {:.5e} / expected {:.5e}".format(
            approx_surface_area, surface_area))
        area_error = abs(approx_surface_area - surface_area) / abs(surface_area)

        # }}}

        # compute max element size
        from grudge.dt_utils import h_max_from_volume

        h_max = h_max_from_volume(dcoll)

        eoc.add_data_point(actx.to_numpy(h_max), area_error)

    # }}}

    logger.info("surface area error\n%s", str(eoc))

    assert eoc.max_error() < 3e-13 or eoc.order_estimate() > builder.order

# }}}


# {{{ mass inverse

@pytest.mark.parametrize("name", [
    "2-1-ellipse",
    "spheroid",
    "warped_rect2",
    "warped_rect3",
    ])
def test_mass_operator_inverse(actx_factory, name):
    actx = actx_factory()

    # {{{ cases

    import mesh_data
    if name == "2-1-ellipse":
        # curve
        builder = mesh_data.EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
    elif name == "spheroid":
        # surface
        builder = mesh_data.SpheroidMeshBuilder()
    elif name.startswith("warped_rect"):
        builder = mesh_data.WarpedRectMeshBuilder(dim=int(name[-1]))

    else:
        raise ValueError("unknown geometry name: %s" % name)

    # }}}

    # {{{ inv(m) @ m == id

    from pytools.convergence import EOCRecorder
    eoc = EOCRecorder()

    for resolution in builder.resolutions:
        mesh = builder.get_mesh(resolution, builder.mesh_order)
        dcoll = DiscretizationCollection(actx, mesh, order=builder.order)
        volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)

        logger.info("ndofs:     %d", volume_discr.ndofs)
        logger.info("nelements: %d", volume_discr.mesh.nelements)

        # {{{ compute inverse mass

        def f(x):
            return actx.np.cos(4.0 * x[0])

        dd = dof_desc.DD_VOLUME
        x_volm = thaw(volume_discr.nodes(), actx)
        f_volm = f(x_volm)
        f_inv = op.inverse_mass(
            dcoll, op.mass(dcoll, dd, f_volm)
        )

        inv_error = actx.to_numpy(
                op.norm(dcoll, f_volm - f_inv, 2) / op.norm(dcoll, f_volm, 2))

        # }}}

        # compute max element size
        from grudge.dt_utils import h_max_from_volume

        h_max = h_max_from_volume(dcoll)

        eoc.add_data_point(actx.to_numpy(h_max), inv_error)

    logger.info("inverse mass error\n%s", str(eoc))

    # NOTE: both cases give 1.0e-16-ish at the moment, but just to be on the
    # safe side, choose a slightly larger tolerance
    assert eoc.max_error() < 1.0e-14

    # }}}

# }}}


# {{{ surface face normal orthogonality

@pytest.mark.parametrize("mesh_name", ["2-1-ellipse", "spheroid"])
def test_face_normal_surface(actx_factory, mesh_name):
    """Check that face normals are orthogonal to the surface normal"""
    actx = actx_factory()

    # {{{ geometry

    if mesh_name == "2-1-ellipse":
        from mesh_data import EllipseMeshBuilder
        builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
    elif mesh_name == "spheroid":
        from mesh_data import SpheroidMeshBuilder
        builder = SpheroidMeshBuilder()
    else:
        raise ValueError("unknown mesh name: %s" % mesh_name)

    mesh = builder.get_mesh(builder.resolutions[0], builder.mesh_order)
    dcoll = DiscretizationCollection(actx, mesh, order=builder.order)

    volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
    logger.info("ndofs:    %d", volume_discr.ndofs)
    logger.info("nelements: %d", volume_discr.mesh.nelements)

    # }}}

    # {{{ Compute surface and face normals
    from meshmode.discretization.connection import FACE_RESTR_INTERIOR
    from grudge.geometry import normal

    dv = dof_desc.DD_VOLUME
    df = dof_desc.as_dofdesc(FACE_RESTR_INTERIOR)

    ambient_dim = mesh.ambient_dim

    surf_normal = op.project(
        dcoll, dv, df,
        normal(actx, dcoll, dd=dv)
    )
    surf_normal = surf_normal / actx.np.sqrt(sum(surf_normal**2))

    face_normal_i = thaw(dcoll.normal(df), actx)
    face_normal_e = dcoll.opposite_face_connection()(face_normal_i)

    if mesh.ambient_dim == 3:
        from grudge.geometry import pseudoscalar, area_element
        # NOTE: there's only one face tangent in 3d
        face_tangent = (
            pseudoscalar(actx, dcoll, dd=df) / area_element(actx, dcoll, dd=df)
        ).as_vector(dtype=object)

    # }}}

    # {{{ checks

    def _eval_error(x):
        return op.norm(dcoll, x, np.inf, dd=df)

    rtol = 1.0e-14

    # check interpolated surface normal is orthogonal to face normal
    error = _eval_error(surf_normal.dot(face_normal_i))
    logger.info("error[n_dot_i]:    %.5e", error)
    assert error < rtol

    # check angle between two neighboring elements
    error = _eval_error(face_normal_i.dot(face_normal_e) + 1.0)
    logger.info("error[i_dot_e]:    %.5e", error)
    assert error > rtol

    # check orthogonality with face tangent
    if ambient_dim == 3:
        error = _eval_error(face_tangent.dot(face_normal_i))
        logger.info("error[t_dot_i]:  %.5e", error)
        assert error < 5 * rtol

    # }}}

# }}}


# {{{ diff operator

@pytest.mark.parametrize("dim", [1, 2, 3])
def test_tri_diff_mat(actx_factory, dim, order=4):
    """Check differentiation matrix along the coordinate axes on a disk

    Uses sines as the function to differentiate.
    """

    actx = actx_factory()

    from pytools.convergence import EOCRecorder
    axis_eoc_recs = [EOCRecorder() for axis in range(dim)]

    def f(x, axis):
        return actx.np.sin(3*x[axis])

    def df(x, axis):
        return 3*actx.np.cos(3*x[axis])

    for n in [4, 8, 16]:
        mesh = mgen.generate_regular_rect_mesh(a=(-0.5,)*dim, b=(0.5,)*dim,
                nelements_per_axis=(n,)*dim, order=4)

        dcoll = DiscretizationCollection(actx, mesh, order=4)
        volume_discr = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
        x = thaw(volume_discr.nodes(), actx)

        for axis in range(dim):
            df_num = op.local_grad(dcoll, f(x, axis))[axis]
            df_volm = df(x, axis)

            linf_error = flat_norm(df_num - df_volm, ord=np.inf)
            axis_eoc_recs[axis].add_data_point(1/n, actx.to_numpy(linf_error))

    for axis, eoc_rec in enumerate(axis_eoc_recs):
        logger.info("axis %d\n%s", axis, eoc_rec)
        assert eoc_rec.order_estimate() > order - 0.25

# }}}


# {{{ divergence theorem

def test_2d_gauss_theorem(actx_factory):
    """Verify Gauss's theorem explicitly on a mesh"""

    pytest.importorskip("meshpy")

    from meshpy.geometry import make_circle, GeometryBuilder
    from meshpy.triangle import MeshInfo, build

    geob = GeometryBuilder()
    geob.add_geometry(*make_circle(1))
    mesh_info = MeshInfo()
    geob.set(mesh_info)

    mesh_info = build(mesh_info)

    from meshmode.mesh.io import from_meshpy
    from meshmode.mesh import BTAG_ALL

    mesh = from_meshpy(mesh_info, order=1)

    actx = actx_factory()

    dcoll = DiscretizationCollection(actx, mesh, order=2)
    volm_disc = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
    x_volm = thaw(volm_disc.nodes(), actx)

    def f(x):
        return flat_obj_array(
            actx.np.sin(3*x[0]) + actx.np.cos(3*x[1]),
            actx.np.sin(2*x[0]) + actx.np.cos(x[1])
        )

    f_volm = f(x_volm)
    int_1 = op.integral(dcoll, "vol", op.local_div(dcoll, f_volm))

    prj_f = op.project(dcoll, "vol", BTAG_ALL, f_volm)
    normal = thaw(dcoll.normal(BTAG_ALL), actx)
    int_2 = op.integral(dcoll, BTAG_ALL, prj_f.dot(normal))

    assert abs(int_1 - int_2) < 1e-13


@pytest.mark.parametrize("mesh_name", ["2-1-ellipse", "2-1-spheroid"])
def test_surface_divergence_theorem(actx_factory, mesh_name, visualize=False):
    r"""Check the surface divergence theorem.

        .. math::

            \int_Sigma \phi \nabla_i f_i =
            \int_\Sigma \nabla_i \phi f_i +
            \int_\Sigma \kappa \phi f_i n_i +
            \int_{\partial \Sigma} \phi f_i m_i

        where :math:`n_i` is the surface normal and :class:`m_i` is the
        face normal (which should be orthogonal to both the surface normal
        and the face tangent).
    """
    actx = actx_factory()

    # {{{ cases

    if mesh_name == "2-1-ellipse":
        from mesh_data import EllipseMeshBuilder
        builder = EllipseMeshBuilder(radius=3.1, aspect_ratio=2.0)
    elif mesh_name == "2-1-spheroid":
        from mesh_data import SpheroidMeshBuilder
        builder = SpheroidMeshBuilder(radius=3.1, aspect_ratio=2.0)
    elif mesh_name == "circle":
        from mesh_data import EllipseMeshBuilder
        builder = EllipseMeshBuilder(radius=1.0, aspect_ratio=1.0)
    elif mesh_name == "starfish":
        from mesh_data import StarfishMeshBuilder
        builder = StarfishMeshBuilder()
    elif mesh_name == "sphere":
        from mesh_data import SphereMeshBuilder
        builder = SphereMeshBuilder(radius=1.0, mesh_order=16)
    else:
        raise ValueError("unknown mesh name: %s" % mesh_name)

    # }}}

    # {{{ convergence

    def f(x):
        return flat_obj_array(
            actx.np.sin(3*x[1]) + actx.np.cos(3*x[0]) + 1.0,
            actx.np.sin(2*x[0]) + actx.np.cos(x[1]),
            3.0 * actx.np.cos(x[0] / 2) + actx.np.cos(x[1]),
        )[:ambient_dim]

    from pytools.convergence import EOCRecorder
    eoc_global = EOCRecorder()
    eoc_local = EOCRecorder()

    theta = np.pi / 3.33
    ambient_dim = builder.ambient_dim
    if ambient_dim == 2:
        mesh_rotation = np.array([
            [np.cos(theta), -np.sin(theta)],
            [np.sin(theta), np.cos(theta)],
            ])
    else:
        mesh_rotation = np.array([
            [1.0, 0.0, 0.0],
            [0.0, np.cos(theta), -np.sin(theta)],
            [0.0, np.sin(theta), np.cos(theta)],
            ])

    mesh_offset = np.array([0.33, -0.21, 0.0])[:ambient_dim]

    for i, resolution in enumerate(builder.resolutions):
        from meshmode.mesh.processing import affine_map
        from meshmode.discretization.connection import FACE_RESTR_ALL

        mesh = builder.get_mesh(resolution, builder.mesh_order)
        mesh = affine_map(mesh, A=mesh_rotation, b=mesh_offset)

        from meshmode.discretization.poly_element import \
                QuadratureSimplexGroupFactory

        qtag = dof_desc.DISCR_TAG_QUAD
        dcoll = DiscretizationCollection(
            actx, mesh, order=builder.order,
            discr_tag_to_group_factory={
                qtag: QuadratureSimplexGroupFactory(2 * builder.order)
            }
        )

        volume = dcoll.discr_from_dd(dof_desc.DD_VOLUME)
        logger.info("ndofs:     %d", volume.ndofs)
        logger.info("nelements: %d", volume.mesh.nelements)

        dd = dof_desc.DD_VOLUME
        dq = dd.with_discr_tag(qtag)
        df = dof_desc.as_dofdesc(FACE_RESTR_ALL)
        ambient_dim = dcoll.ambient_dim

        # variables
        f_num = f(thaw(dcoll.nodes(dd=dd), actx))
        f_quad_num = f(thaw(dcoll.nodes(dd=dq), actx))

        from grudge.geometry import normal, summed_curvature

        kappa = summed_curvature(actx, dcoll, dd=dq)
        normal = normal(actx, dcoll, dd=dq)
        face_normal = thaw(dcoll.normal(df), actx)
        face_f = op.project(dcoll, dd, df, f_num)

        # operators
        stiff = op.mass(dcoll, sum(op.local_d_dx(dcoll, i, f_num_i)
                                   for i, f_num_i in enumerate(f_num)))
        stiff_t = sum(op.weak_local_d_dx(dcoll, i, f_num_i)
                      for i, f_num_i in enumerate(f_num))
        kterm = op.mass(dcoll, dq, kappa * f_quad_num.dot(normal))
        flux = op.face_mass(dcoll, face_f.dot(face_normal))

        # sum everything up
        op_global = op.nodal_sum(dcoll, dd, stiff - (stiff_t + kterm))
        op_local = op.elementwise_sum(dcoll, dd, stiff - (stiff_t + kterm + flux))

        # compute max element size
        from grudge.dt_utils import h_max_from_volume

        h_max = actx.to_numpy(h_max_from_volume(dcoll))
        err_global = actx.to_numpy(abs(op_global))
        err_local = actx.to_numpy(op.norm(dcoll, op_local, np.inf))
        logger.info("errors: h_max %.5e global %.5e local %.5e",
                h_max, err_global, err_local)

        eoc_global.add_data_point(h_max, err_global)
        eoc_local.add_data_point(h_max, err_local)

        if visualize:
            from grudge.shortcuts import make_visualizer
            vis = make_visualizer(dcoll)

            filename = f"surface_divergence_theorem_{mesh_name}_{i:04d}.vtu"
            vis.write_vtk_file(filename, [
                ("r", actx.np.log10(op_local))
                ], overwrite=True)

    # }}}

    order = min(builder.order, builder.mesh_order) - 0.5
    logger.info("eoc_global:\n%s", str(eoc_global))
    logger.info("eoc_local:\n%s", str(eoc_local))

    assert eoc_global.max_error() < 1.0e-12 \
            or eoc_global.order_estimate() > order - 0.5

    assert eoc_local.max_error() < 1.0e-12 \
            or eoc_local.order_estimate() > order - 0.5

# }}}


# {{{ models: advection

@pytest.mark.parametrize(("mesh_name", "mesh_pars"), [
    ("segment", [8, 16, 32]),
    ("disk", [0.07, 0.02, 0.01]),
    ("rect2", [4, 8]),
    ("rect3", [4, 6]),
    ("warped2", [4, 8]),
    ])
@pytest.mark.parametrize("op_type", ["strong", "weak"])
@pytest.mark.parametrize("flux_type", ["central"])
@pytest.mark.parametrize("order", [3, 4, 5])
# test: 'test_convergence_advec(cl._csc, "disk", [0.1, 0.05], "strong", "upwind", 3)'
def test_convergence_advec(actx_factory, mesh_name, mesh_pars, op_type, flux_type,
        order, visualize=False):
    """Test whether 2D advection actually converges"""

    actx = actx_factory()

    from pytools.convergence import EOCRecorder
    eoc_rec = EOCRecorder()

    for mesh_par in mesh_pars:
        if mesh_name == "segment":
            mesh = mgen.generate_box_mesh(
                [np.linspace(-1.0, 1.0, mesh_par)],
                order=order)

            dim = 1
            dt_factor = 1.0
        elif mesh_name == "disk":
            pytest.importorskip("meshpy")

            from meshpy.geometry import make_circle, GeometryBuilder
            from meshpy.triangle import MeshInfo, build

            geob = GeometryBuilder()
            geob.add_geometry(*make_circle(1))
            mesh_info = MeshInfo()
            geob.set(mesh_info)

            mesh_info = build(mesh_info, max_volume=mesh_par)

            from meshmode.mesh.io import from_meshpy
            mesh = from_meshpy(mesh_info, order=1)
            dim = 2
            dt_factor = 4
        elif mesh_name.startswith("rect"):
            dim = int(mesh_name[-1:])
            mesh = mgen.generate_regular_rect_mesh(a=(-0.5,)*dim, b=(0.5,)*dim,
                    nelements_per_axis=(mesh_par,)*dim, order=4)

            if dim == 2:
                dt_factor = 4
            elif dim == 3:
                dt_factor = 2
            else:
                raise ValueError("dt_factor not known for %dd" % dim)
        elif mesh_name.startswith("warped"):
            dim = int(mesh_name[-1:])
            mesh = mgen.generate_warped_rect_mesh(dim, order=order,
                    nelements_side=mesh_par)

            if dim == 2:
                dt_factor = 4
            elif dim == 3:
                dt_factor = 2
            else:
                raise ValueError("dt_factor not known for %dd" % dim)
        else:
            raise ValueError("invalid mesh name: " + mesh_name)

        v = np.array([0.27, 0.31, 0.1])[:dim]
        norm_v = la.norm(v)

        def f(x):
            return actx.np.sin(10*x)

        def u_analytic(x, t=0):
            return f(-v.dot(x)/norm_v + t*norm_v)

        from grudge.models.advection import (
            StrongAdvectionOperator, WeakAdvectionOperator
        )
        from meshmode.mesh import BTAG_ALL

        dcoll = DiscretizationCollection(actx, mesh, order=order)
        op_class = {"strong": StrongAdvectionOperator,
                    "weak": WeakAdvectionOperator}[op_type]
        adv_operator = op_class(dcoll, v,
                                inflow_u=lambda t: u_analytic(
                                    thaw(dcoll.nodes(dd=BTAG_ALL), actx),
                                    t=t
                                ),
                                flux_type=flux_type)

        nodes = thaw(dcoll.nodes(), actx)
        u = u_analytic(nodes, t=0)

        def rhs(t, u):
            return adv_operator.operator(t, u)

        compiled_rhs = actx.compile(rhs)

        if dim == 3:
            final_time = 0.1
        else:
            final_time = 0.2

        from grudge.dt_utils import h_max_from_volume

        h_max = h_max_from_volume(dcoll, dim=dcoll.ambient_dim)
        dt = actx.to_numpy(dt_factor * h_max/order**2)
        nsteps = (final_time // dt) + 1
        tol = 1e-14
        dt = final_time/nsteps + tol

        from grudge.shortcuts import make_visualizer, compiled_lsrk45_step
        vis = make_visualizer(dcoll)

        step = 0
        t = 0

        while t < final_time - tol:
            step += 1
            logger.debug("[%04d] t = %.5f", step, t)

            u = compiled_lsrk45_step(actx, u, t, dt, compiled_rhs)

            if visualize:
                vis.write_vtk_file(
                    "fld-%s-%04d.vtu" % (mesh_par, step),
                    [("u", u)]
                )

            t += dt

            if t + dt >= final_time - tol:
                dt = final_time-t

        error_l2 = op.norm(
            dcoll,
            u - u_analytic(nodes, t=t),
            2
        )
        logger.info("h_max %.5e error %.5e", actx.to_numpy(h_max), error_l2)
        eoc_rec.add_data_point(actx.to_numpy(h_max), actx.to_numpy(error_l2))

    logger.info("\n%s", eoc_rec.pretty_print(
        abscissa_label="h",
        error_label="L2 Error"))

    if mesh_name.startswith("warped"):
        # NOTE: curvilinear meshes are hard
        assert eoc_rec.order_estimate() > order - 0.5
    else:
        assert eoc_rec.order_estimate() > order

# }}}


# {{{ models: maxwell

@pytest.mark.parametrize("order", [3, 4, 5])
def test_convergence_maxwell(actx_factory,  order):
    """Test whether 3D Maxwell's actually converges"""

    actx = actx_factory()

    from pytools.convergence import EOCRecorder
    eoc_rec = EOCRecorder()

    dims = 3
    ns = [4, 6, 8]
    for n in ns:
        mesh = mgen.generate_regular_rect_mesh(
                a=(0.0,)*dims,
                b=(1.0,)*dims,
                nelements_per_axis=(n,)*dims)

        dcoll = DiscretizationCollection(actx, mesh, order=order)

        epsilon = 1
        mu = 1

        from grudge.models.em import get_rectangular_cavity_mode

        def analytic_sol(x, t=0):
            return get_rectangular_cavity_mode(actx, x, t, 1, (1, 2, 2))

        nodes = thaw(dcoll.nodes(), actx)
        fields = analytic_sol(nodes, t=0)

        from grudge.models.em import MaxwellOperator

        maxwell_operator = MaxwellOperator(
            dcoll,
            epsilon,
            mu,
            flux_type=0.5,
            dimensions=dims
        )
        maxwell_operator.check_bc_coverage(mesh)

        def rhs(t, w):
            return maxwell_operator.operator(t, w)

        dt = actx.to_numpy(maxwell_operator.estimate_rk4_timestep(actx, dcoll))
        final_t = dt * 5
        nsteps = int(final_t/dt)

        from grudge.shortcuts import set_up_rk4
        dt_stepper = set_up_rk4("w", dt, fields, rhs)

        logger.info("dt %.5e nsteps %5d", dt, nsteps)

        step = 0
        for event in dt_stepper.run(t_end=final_t):
            if isinstance(event, dt_stepper.StateComputed):
                assert event.component_id == "w"
                esc = event.state_component

                step += 1
                logger.debug("[%04d] t = %.5e", step, event.t)

        sol = analytic_sol(nodes, t=step * dt)
        total_error = op.norm(dcoll, esc - sol, 2)
        eoc_rec.add_data_point(1.0/n, actx.to_numpy(total_error))

    logger.info("\n%s", eoc_rec.pretty_print(
        abscissa_label="h",
        error_label="L2 Error"))

    assert eoc_rec.order_estimate() > order

# }}}


# {{{ models: variable coefficient advection oversampling

@pytest.mark.parametrize("order", [2, 3, 4])
def test_improvement_quadrature(actx_factory, order):
    """Test whether quadrature improves things and converges"""
    from grudge.models.advection import VariableCoefficientAdvectionOperator
    from pytools.convergence import EOCRecorder
    from meshmode.discretization.poly_element import QuadratureSimplexGroupFactory
    from meshmode.mesh import BTAG_ALL

    actx = actx_factory()

    dims = 2

    def gaussian_mode(x):
        source_width = 0.1
        return actx.np.exp(-np.dot(x, x) / source_width**2)

    def conv_test(descr, use_quad):
        logger.info("-" * 75)
        logger.info(descr)
        logger.info("-" * 75)
        eoc_rec = EOCRecorder()

        if use_quad:
            qtag = dof_desc.DISCR_TAG_QUAD
        else:
            qtag = None

        ns = [20, 25]
        for n in ns:
            mesh = mgen.generate_regular_rect_mesh(
                a=(-0.5,)*dims,
                b=(0.5,)*dims,
                nelements_per_axis=(n,)*dims,
                order=order)

            if use_quad:
                discr_tag_to_group_factory = {
                    qtag: QuadratureSimplexGroupFactory(order=4*order)
                }
            else:
                discr_tag_to_group_factory = {}

            dcoll = DiscretizationCollection(
                actx, mesh, order=order,
                discr_tag_to_group_factory=discr_tag_to_group_factory
            )

            nodes = thaw(dcoll.nodes(), actx)

            def zero_inflow(dtag, t=0):
                dd = dof_desc.DOFDesc(dtag, qtag)
                return dcoll.discr_from_dd(dd).zeros(actx)

            adv_op = VariableCoefficientAdvectionOperator(
                dcoll,
                flat_obj_array(-1*nodes[1], nodes[0]),
                inflow_u=lambda t: zero_inflow(BTAG_ALL, t=t),
                flux_type="upwind",
                quad_tag=qtag
            )

            total_error = op.norm(
                dcoll, adv_op.operator(0, gaussian_mode(nodes)), 2
            )
            eoc_rec.add_data_point(1.0/n, actx.to_numpy(total_error))

        logger.info("\n%s", eoc_rec.pretty_print(
            abscissa_label="h",
            error_label="L2 Error"))

        return eoc_rec.order_estimate(), np.array([x[1] for x in eoc_rec.history])

    eoc, errs = conv_test("no quadrature", False)
    q_eoc, q_errs = conv_test("with quadrature", True)

    assert q_eoc > eoc
    assert (q_errs < errs).all()
    assert q_eoc > order - 0.1

# }}}


# {{{ bessel

@pytest.mark.xfail
def test_bessel(actx_factory):
    actx = actx_factory()

    dims = 2

    mesh = mgen.generate_regular_rect_mesh(
            a=(0.1,)*dims,
            b=(1.0,)*dims,
            nelements_per_axis=(8,)*dims)

    dcoll = DiscretizationCollection(actx, mesh, order=3)

    nodes = thaw(dcoll.nodes(), actx)
    r = actx.np.sqrt(nodes[0]**2 + nodes[1]**2)

    # FIXME: Bessel functions need to brought out of the symbolic
    # layer. Related issue: https://github.com/inducer/grudge/issues/93
    def bessel_j(actx, n, r):
        from grudge import sym, bind  # pylint: disable=no-name-in-module
        return bind(dcoll, sym.bessel_j(n, sym.var("r")))(actx, r=r)

    # https://dlmf.nist.gov/10.6.1
    n = 3
    bessel_zero = (bessel_j(actx, n+1, r)
                   + bessel_j(actx, n-1, r)
                   - 2*n/r * bessel_j(actx, n, r))

    z = op.norm(dcoll, bessel_zero, 2)

    assert z < 1e-15

# }}}


@pytest.mark.parametrize("p", [2, np.inf])
def test_norm_real(actx_factory, p):
    actx = actx_factory()

    dim = 2
    mesh = mgen.generate_regular_rect_mesh(
            a=(0,)*dim, b=(1,)*dim,
            nelements_per_axis=(8,)*dim, order=1)
    dcoll = DiscretizationCollection(actx, mesh, order=4)
    nodes = thaw(dcoll.nodes(), actx)

    norm = op.norm(dcoll, nodes[0], p)
    if p == 2:
        ref_norm = (1/3)**0.5
    elif p == np.inf:
        ref_norm = 1