Version:

This documentation is not for the latest stable Salvus version.

This tutorial is presented as Python code running inside a Jupyter Notebook, the recommended way to use Salvus. To run it yourself you can copy/type each individual cell or directly download the full notebook, including all required files.

Benchmark: Diffusion equation

  • Reference solution: analytic
  • Physis: Diffusion equation

In this notebook we benchmark the numerical solution of the diffusion equation in a simple test case for which an analytic solution exists. Specifically, we are considering the fundamental solution of the diffusion equation in homogeneous media, which is often called diffusion kernel.

To this end, we consider the diffusion equation with a Dirac measure as initial values and a constant diffusion tensor D(x)=(k00k)\mathcal{D}(\mathbf{x}) = \left(\begin{array}{cc}k & 0\\0 & k \end{array}\right) with k>0k > 0:

tu(x,t)D(x)u(x,t)=0,u(x,0)=δ(xxˉ).\begin{aligned} \partial_t u(\mathbf{x},t) - \nabla \cdot \mathcal{D}(\mathbf{x})\, \nabla u(\mathbf{x},t) &= 0,\\ u(\mathbf{x},0) &= \delta(\mathbf{x}-\mathbf{\bar{x}}). \end{aligned}

On Rd\mathbf{R}^d, (d=2,3d=2,3), this equation has a unique solution given by:

uexact=1(kπt)d/2exp((xxˉ)T(xxˉ)4kt),u_{\text{exact}} = \frac{1}{(k\pi\,t)^{d/2}} \exp\left( - \frac{(\mathbf{x}-\mathbf{\bar{x}})^T (\mathbf{x}-\mathbf{\bar{x}})}{4\,k\,t}\right),

which can also be interpreted as a Wiener process.

Remarks:

  • Representing the delta source on the finite element mesh requires a proper projection onto the finite element basis. Instead, we simulate the time interval from 0.01 s to 0.02 s.
  • While the analytic solution is defined on the unbounded domain Rd\mathbf{R}^d, we restrict the computational to a box or sphere. To avoid artifacts from the artificial boundaries, we use a fairly short simulation time.

Imports and test config

Copy
%matplotlib inline
%config Completer.use_jedi = False

import os
import numpy as np

import salvus.namespace as sn
# Number of processes SalvusCompute will run with.
# Get it from the environment or default to 4.
MPI_RANKS = int(os.environ.get("NUM_MPI_RANKS", 4))
# Choose on which site to run this.
SALVUS_FLOW_SITE_NAME = os.environ.get("SITE_NAME", "local")
domain_size = 2.0
k = 0.5
FINAL_TIME = 0.01
def wiener_process(time, dim, mean, points, k):

    x_bar = points - mean
    u_exact = (
        1
        / (np.sqrt((k * np.pi * time) ** dim))
        * np.exp(-np.einsum("ijk,ijk->ij", x_bar, x_bar) / (4 * k * time))
    )
    return u_exact
def create_mesh(dim, mode, tensor_order, domain_size, k):

    if dim == 2 and mode == "Cartesian":
        m = sn.simple_mesh.CartesianHomogeneousAcoustic2D(
            vp=1.0,
            rho=1.0,
            x_max=domain_size,
            y_max=domain_size,
            max_frequency=30.0,
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()

    elif dim == 2 and mode == "spherical":
        m = sn.simple_mesh.basic_mesh.SphericalHomogeneousAcoustic2D(
            radius=domain_size / 2, vp=2.0, rho=1.0, max_frequency=20.0
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()
        mesh.points[:, 0] += 1.0
        mesh.points[:, 1] += 1.0

    elif dim == 3 and mode == "Cartesian":

        m = sn.simple_mesh.CartesianHomogeneousAcoustic3D(
            vp=1.0,
            rho=1.0,
            x_max=domain_size,
            y_max=domain_size,
            z_max=domain_size,
            max_frequency=10.0,
        )
        m.advanced.tensor_order = tensor_order
        mesh = m.create_mesh()

    f = np.ones_like(mesh.elemental_fields["VP"])
    del mesh.elemental_fields["VP"]
    del mesh.elemental_fields["RHO"]

    mesh.attach_field("M0", 1.0 * f)
    mesh.attach_field("M1", k * f)

    mesh.attach_field(
        "uinit",
        wiener_process(
            0.01,
            mesh.ndim,
            [1.0] * mesh.ndim,
            mesh.points[mesh.connectivity],
            k,
        ),
    )

    return mesh
def simulate(mesh, final_time):

    mesh.write_h5("init.h5")

    w = sn.simple_config.simulation.Diffusion(mesh=mesh)

    w.domain.polynomial_order = mesh.shape_order

    w.physics.diffusion_equation.initial_values.filename = "init.h5"
    w.physics.diffusion_equation.initial_values.format = "hdf5"
    w.physics.diffusion_equation.initial_values.field = "uinit"

    w.physics.diffusion_equation.final_values.filename = "final.h5"
    w.physics.diffusion_equation.end_time_in_seconds = final_time

    w.validate()

    sn.api.run(
        input_file=w,
        site_name=SALVUS_FLOW_SITE_NAME,
        output_folder="diffusion",
        overwrite=True,
        ranks=MPI_RANKS,
    )
def analysis(mesh, final_time):

    tensor_order = mesh.shape_order

    if tensor_order == 1:
        TOL = 1e-2
    elif tensor_order == 2:
        TOL = 1e-3
    else:
        TOL = 1e-4

    solution = sn.UnstructuredMesh.from_h5("diffusion/final.h5")
    solution.attach_field("init", mesh.elemental_fields["uinit"])

    u_analytic = wiener_process(
        0.01 + final_time,
        mesh.ndim,
        [1.0] * mesh.ndim,
        solution.points[solution.connectivity],
        k,
    )
    u_salvus = solution.elemental_fields["uinit"]
    residuals = u_salvus - u_analytic

    u_max = u_salvus.max()
    ref_max = u_analytic.max()

    solution.attach_field("analytic_solution", u_analytic)
    solution.attach_field("solution", u_salvus)
    del solution.elemental_fields["uinit"]
    solution.attach_field("residuals", residuals)
    print(
        f"simulation time: {final_time}\n",
        f"|| u ||_inf = {u_max}\n",
        f"|| u_exact ||_inf = {ref_max}\n",
        f"|| u - u_exact ||_inf = {np.abs(residuals).max()}\n",
        f"|| u - u_exact ||_inf / || u_exact ||_inf = {np.abs(residuals).max() / ref_max}\n",
    )
    np.testing.assert_allclose(
        u_salvus, u_analytic, rtol=1e-6, atol=ref_max * TOL
    )

    return solution

2D Cartesian, order 1

mesh = create_mesh(
    dim=2, mode="Cartesian", tensor_order=1, domain_size=2.0, k=0.5
)
simulate(mesh, 0.01)
solution = analysis(mesh, 0.01)
solution
Job `job_2007161447601273_e3834a2669` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.11.13
  * Floating point size: 64
* Downloaded 1.8 MB of results to `diffusion`.
* Total run time: 1.40 seconds.
* Pure simulation time: 1.00 seconds.
simulation time: 0.01
 || u ||_inf = 31.959188667283925
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.12820004890485848
 || u - u_exact ||_inf / || u_exact ||_inf = 0.004027523318293556

When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus.mesh.unstructured_mesh.UnstructuredMesh at 0x7f792433f250>
mesh = create_mesh(
    dim=2, mode="Cartesian", tensor_order=2, domain_size=2.0, k=0.5
)
simulate(mesh, 0.01)
solution = analysis(mesh, 0.01)
solution
Job `job_2007161447638937_b602f1527b` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.11.13
  * Floating point size: 64
                                      
* Downloaded 4.8 MB of results to `diffusion`.
* Total run time: 2.92 seconds.
* Pure simulation time: 2.53 seconds.
simulation time: 0.01
 || u ||_inf = 31.84917675381394
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.018220665804165037
 || u - u_exact ||_inf / || u_exact ||_inf = 0.0005724190983387964

When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus.mesh.unstructured_mesh.UnstructuredMesh at 0x7f78aa052250>
mesh = create_mesh(
    dim=2, mode="Cartesian", tensor_order=4, domain_size=2.0, k=0.5
)
simulate(mesh, 0.01)
solution = analysis(mesh, 0.01)
solution
Job `job_2007161447679397_a2d6bad918` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.11.13
  * Floating point size: 64
                                                
* Downloaded 15.3 MB of results to `diffusion`.
* Total run time: 42.76 seconds.
* Pure simulation time: 42.17 seconds.
simulation time: 0.01
 || u ||_inf = 31.83316360903441
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.002174990655344544
 || u - u_exact ||_inf / || u_exact ||_inf = 6.83293466445687e-05

When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus.mesh.unstructured_mesh.UnstructuredMesh at 0x7f78aa03fdd0>
mesh = create_mesh(
    dim=2, mode="spherical", tensor_order=4, domain_size=2.0, k=0.5
)
simulate(mesh, 0.01)
solution = analysis(mesh, 0.01)
solution
Job `job_2007161448848076_fb99186b2e` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.11.13
  * Floating point size: 64
                                          
* Downloaded 2.0 MB of results to `diffusion`.
* Total run time: 10.59 seconds.
* Pure simulation time: 9.99 seconds.
simulation time: 0.01
 || u ||_inf = 31.832201944166
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.0012274499227693525
 || u - u_exact ||_inf / || u_exact ||_inf = 3.856147660021557e-05

When run interactively in a Jupyter Notebook, this output cell contains a widget. Click the button below to load a preview of it. Please note that most interactive functionality does not work without a running Python kernel.
<salvus.mesh.unstructured_mesh.UnstructuredMesh at 0x7f78aa1befd0>
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