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

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

## 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,
)
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
)
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,
)
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
Job job_2010191444055063_21d5336616 running on local_f64 with 4 rank(s).
Site information:
* Salvus version: 0.11.19
* Floating point size: 64

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.
* Downloaded 1.8 MB of results to diffusion.
* Total run time: 1.16 seconds.
* Pure simulation time: 0.64 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 0x7f2ff0190b10>

### 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
Job job_2010191444243769_d6c6c864d2 running on local_f64 with 4 rank(s).
Site information:
* Salvus version: 0.11.19
* Floating point size: 64

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.
* Downloaded 4.8 MB of results to diffusion.
* Total run time: 2.32 seconds.
* Pure simulation time: 1.90 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 0x7f2fe8c72850>

### 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
Job job_2010191444167197_bf0ca25184 running on local_f64 with 4 rank(s).
Site information:
* Salvus version: 0.11.19
* Floating point size: 64


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.
* Downloaded 15.3 MB of results to diffusion.
* Total run time: 10.92 seconds.
* Pure simulation time: 10.09 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 0x7f2fee87b790>

### 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
Job job_2010191444108538_ffcbcfba59 running on local_f64 with 4 rank(s).
Site information:
* Salvus version: 0.11.19
* Floating point size: 64


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.
* Downloaded 2.0 MB of results to diffusion.
* Total run time: 2.44 seconds.
* Pure simulation time: 2.08 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 0x7f2fee098850>
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