This tutorial is not for the latest stable Salvus version (viewing 2024.1.3).

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 $\mathcal{D}(\mathbf{x}) = \left(\begin{array}{cc}k & 0\\0 & k \end{array}\right)$ with $k > 0$ :

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

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 $\mathbf{R}^d$ , 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

%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

Initial values and exact solution

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

Simulation

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,
    )

Error analysis

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

Scenarios

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

SalvusJob `job_2504152113232952_30f0894ec5` running on `local` with 4 rank(s).
Site information:
  * Salvus version: 2024.1.3
  * Floating point size: 32

* Downloaded 1.6 MB of results to `diffusion`.
* Total run time: 2.72 seconds.
* Pure simulation time: 2.33 seconds.
simulation time: 0.01
 || u ||_inf = 31.959190368652344
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.12820175027327707
 || u - u_exact ||_inf / || u_exact ||_inf = 0.004027576768358805

<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x778a69db5cd0>

2D Cartesian, order 2

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

SalvusJob `job_2504152113904303_85f2e43a2c` running on `local` with 4 rank(s).
Site information:
  * Salvus version: 2024.1.3
  * Floating point size: 32

* Downloaded 4.3 MB of results to `diffusion`.
* Total run time: 2.89 seconds.
* Pure simulation time: 2.46 seconds.
simulation time: 0.01
 || u ||_inf = 31.849178314208984
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.01822223726619754
 || u - u_exact ||_inf / || u_exact ||_inf = 0.0005724684672745635

<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x778a671e9290>

2D Cartesian, order 4

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

SalvusJob `job_2504152113629459_bb1bb38658` running on `local` with 4 rank(s).
Site information:
  * Salvus version: 2024.1.3
  * Floating point size: 32

* Downloaded 14.0 MB of results to `diffusion`.
* Total run time: 10.58 seconds.
* Pure simulation time: 10.17 seconds.
simulation time: 0.01
 || u ||_inf = 31.83320426940918
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.0022156510301130083
 || u - u_exact ||_inf / || u_exact ||_inf = 6.960672999121685e-05

<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x778b14949ad0>

2D spherical, order 4

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

SalvusJob `job_2504152114355949_8fbdc91172` running on `local` with 4 rank(s).
Site information:
  * Salvus version: 2024.1.3
  * Floating point size: 32

* Downloaded 2.5 MB of results to `diffusion`.
* Total run time: 4.90 seconds.
* Pure simulation time: 4.39 seconds.
simulation time: 0.01
 || u ||_inf = 31.83287811279297
 || u_exact ||_inf = 31.830988618379067
 || u - u_exact ||_inf = 0.0018894944139020708
 || u - u_exact ||_inf / || u_exact ||_inf = 5.936021769713698e-05

<salvus.mesh.data_structures.unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x778a671e8410>