MPI Numerical Differentiation¶

The MPI differentiation functions provide distributed numerical derivatives for data decomposed across MPI ranks. These functions are designed for large datasets where the computational domain is split between processes.

The implementation supports:

Distributed finite-difference differentiation
Periodic and non-periodic boundaries
1D, 2D, and 3D scalar fields

Overview¶

The MPI differentiation module contains:

mpi_dfdx — MPI derivative along the x-axis
mpi_dfdy — MPI derivative along the y-axis
mpi_dfdz — MPI derivative along the z-axis

The x-axis derivative performs explicit MPI communication because the data domain is decomposed along the x direction. The y and z derivatives are performed locally on each MPI slab.

MPI Domain Decomposition¶

The data is assumed to be distributed across MPI ranks along the x-axis.

For example, a global 3D grid:

Global grid:
+-----------------------------+
| Rank 0 | Rank 1 | Rank 2    |
+-----------------------------+

Each MPI rank owns a slab of the x-domain.

To compute derivatives near slab boundaries, neighboring values are exchanged between MPI ranks using halo communication.

mpi_dfdx¶

Overview¶

mpi_dfdx computes the derivative along the x-axis for distributed data.

Because the x-axis is distributed across MPI processes, edge cells are communicated between neighboring ranks before differentiation.

Periodic Boundaries¶

For periodic domains:

First and last MPI ranks communicate with each other
Derivatives wrap continuously across the domain

For non-periodic domains:

Edge ranks use one-sided finite differences

Example: 1D Distributed Derivative¶

# Assuming correct initialisation of MPI object, see tutorial.
import numpy as np
import fiesta

# Local grid
x_local = np.linspace(0, 1, 32)

# Local field
f_local = np.sin(2 * np.pi * x_local)

# Distributed derivative
dfdx_local = fiesta.maths.mpi_dfdx(
    x_local,
    f_local,
    MPI,
    periodic=True
)

Example: 3D Scalar Field¶

# Assuming correct initialisation of MPI object, see tutorial.
import numpy as np
import fiesta

nx, ny, nz = 32, 64, 64

x_local = np.linspace(0, 1, nx)

field = np.random.random((nx, ny, nz))

dfdx = fiesta.maths.mpi_dfdx(
    x_local,
    field,
    MPI,
    periodic=True
)

mpi_dfdy¶

Overview¶

mpi_dfdy computes derivatives along the y-axis.

Since the domain decomposition is along x, all y-values required for the derivative already exist locally. No MPI communication is necessary.

Example¶

# Assuming correct initialisation of MPI object, see tutorial.
import fiesta

dfdy = fiesta.maths.mpi_dfdy(
    ygrid,
    field,
    MPI,
    periodic=True
)

mpi_dfdz¶

Overview¶

mpi_dfdz computes derivatives along the z-axis.

As with mpi_dfdy, no MPI communication is required because the z-axis is fully local to each rank.

Example¶

# Assuming correct initialisation of MPI object, see tutorial.
import fiesta

dfdz = fiesta.maths.mpi_dfdz(
    zgrid,
    field,
    MPI,
    periodic=True
)

Finite Difference Scheme¶

The derivatives use:

Symmetric two-point finite differences
Three-point one-sided differences near boundaries

For interior points:

\[\frac{df}{dx} \approx \frac{f(x+\Delta x) - f(x-\Delta x)} {2\Delta x}\]

Boundary points use one-sided estimators.

Periodic Domains¶

When periodic=True:

Boundary values wrap around
MPI edge ranks exchange data cyclically
Derivatives remain continuous across domain edges

Example:

# Assuming correct initialisation of MPI object, see tutorial.
import fiesta

dfdx = fiesta.maths.mpi_dfdx(
    xgrid,
    field,
    MPI,
    periodic=True
)

Non-Periodic Domains¶

When periodic=False:

One-sided derivatives are applied near edges
No wrapping occurs
Physical domain boundaries are preserved

Example:

# Assuming correct initialisation of MPI object, see tutorial.
import fiesta

dfdx = fiesta.maths.mpi_dfdx(
    xgrid,
    field,
    MPI,
    periodic=False
)

API References¶

fiesta.maths.mpi_dfdx(xgrid: ndarray, f: ndarray, MPI: object, periodic: bool = False) → ndarray

Differentiate using a symmetric two-point derivative and the non-symmetric three point derivative estimator.

Parameters:

xgrid (array) – X-axis.
f (array) – Function values at x.
MPI (object) – shift.mpiutils MPI object.
periodic (bool, optional) – Sets periodic boundary conditions.

Returns:

dfdx – Numerical differentiation values for f evaluated at points x.

Return type:

array

fiesta.maths.mpi_dfdy(ygrid: ndarray, f: ndarray, MPI: object, periodic: bool = False) → ndarray

Differentiate using a symmetric two-point derivative and the non-symmetric three point derivative estimator.

Parameters:

ygrid (array) – Y-axis.
f (array) – Function values at y.
MPI (object) – shift.mpiutils MPI object.
periodic (bool, optional) – Sets periodic boundary conditions.

Returns:

dfdy – Numerical differentiation values for f evaluated at points y.

Return type:

array

fiesta.maths.mpi_dfdz(zgrid: ndarray, f: ndarray, MPI: object, periodic: bool = False) → ndarray

Differentiate using a symmetric two-point derivative and the non-symmetric three point derivative estimator.

Parameters:

zgrid (array) – Z-axis.
f (array) – Function values at z.
MPI (object) – shift.mpiutils MPI object.
periodic (bool, optional) – Sets periodic boundary conditions.

Returns:

dfdz – Numerical differentiation values for f evaluated at points z.

Return type:

array