# Source code for devito.finite_differences.finite_difference

```
from sympy import finite_diff_weights
from devito.finite_differences.tools import (form_side, symbolic_weights, left, right,
generate_indices, centered, check_input,
check_symbolic, direct, transpose)
__all__ = ['first_derivative', 'second_derivative', 'cross_derivative',
'generic_derivative', 'left', 'right', 'centered', 'transpose',
'generate_indices', 'form_side']
# Number of digits for FD coefficients to avoid roundup errors and non-deterministic
# code generation
_PRECISION = 9
[docs]@check_input
@check_symbolic
def first_derivative(expr, dim, fd_order=None, side=centered, matvec=direct,
symbolic=False):
"""
First-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the first-order derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int, optional
Coefficient discretization order. Note: this impacts the width of
the resulting stencil. Defaults to ``expr.space_order``
side : Side, optional
Side of the finite difference location, centered (at x), left (at x - 1)
or right (at x +1). Defaults to ``centered``.
matvec : Transpose, optional
Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to ``direct``.
Returns
-------
expr-like
First-order derivative of ``expr``.
Examples
--------
>>> from devito import Function, Grid, first_derivative, transpose
>>> grid = Grid(shape=(4, 4))
>>> x, _ = grid.dimensions
>>> f = Function(name='f', grid=grid)
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x)
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
Semantically, this is equivalent to
>>> (f*g).dx
Derivative(f(x, y)*g(x, y), x)
The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via ``evaluate``
>>> (f*g).dx.evaluate
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
For the adjoint mode of the first derivative, pass ``matvec=transpose``
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x, matvec=transpose)
-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x
This is also accessible via the .T shortcut
>>> (f*g).dx.T.evaluate
-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x
"""
side = side
diff = dim.spacing
order = fd_order or expr.space_order
# Stencil positions for non-symmetric cross-derivatives with symmetric averaging
ind = generate_indices(expr, dim, diff, order, side=side)[0]
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, 1, ind, dim)
else:
c = finite_diff_weights(1, ind, dim)[-1][-1]
# Loop through positions
deriv = 0
all_dims = tuple(set((dim,) + tuple([i for i in expr.indices if i.root == dim])))
for i in range(len(ind)):
subs = dict([(d, ind[i].subs({dim: d, diff: matvec.val*diff})) for d in all_dims])
deriv += expr.subs(subs) * c[i]
# Evaluate up to _PRECISION digits
deriv = deriv.evalf(_PRECISION)
return deriv
[docs]@check_input
@check_symbolic
def second_derivative(expr, dim, fd_order, stagger=None, **kwargs):
"""
Second-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int
Coefficient discretization order. Note: this impacts the width of
the resulting stencil.
stagger : Side, optional
Shift of the finite-difference approximation.
Returns
-------
expr-like
Second-order derivative of ``expr``.
Examples
--------
>>> from devito import Function, Grid, second_derivative
>>> grid = Grid(shape=(4, 4))
>>> x, _ = grid.dimensions
>>> f = Function(name='f', grid=grid, space_order=2)
>>> g = Function(name='g', grid=grid, space_order=2)
>>> second_derivative(f*g, dim=x, fd_order=2)
-2.0*f(x, y)*g(x, y)/h_x**2 + f(x - h_x, y)*g(x - h_x, y)/h_x**2 +\
f(x + h_x, y)*g(x + h_x, y)/h_x**2
Semantically, this is equivalent to
>>> (f*g).dx2
Derivative(f(x, y)*g(x, y), (x, 2))
The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via ``evaluate``
>>> (f*g).dx2.evaluate
-2.0*f(x, y)*g(x, y)/h_x**2 + f(x - h_x, y)*g(x - h_x, y)/h_x**2 +\
f(x + h_x, y)*g(x + h_x, y)/h_x**2
"""
return generic_derivative(expr, dim, fd_order, 2, stagger=None, **kwargs)
[docs]@check_input
@check_symbolic
def cross_derivative(expr, dims, fd_order, deriv_order, stagger=None, **kwargs):
"""
Arbitrary-order cross derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the cross derivative is produced.
dims : tuple of Dimension
Dimensions w.r.t. which to differentiate.
fd_order : tuple of ints
Coefficient discretization order. Note: this impacts the width of
the resulting stencil.
deriv_order : tuple of ints
Derivative order, e.g. 2 for a second-order derivative.
stagger : tuple of Side, optional
Shift of the finite-difference approximation.
Returns
-------
expr-like
Cross-derivative of ``expr``.
Examples
--------
>>> from devito import Function, Grid, second_derivative
>>> grid = Grid(shape=(4, 4))
>>> x, y = grid.dimensions
>>> f = Function(name='f', grid=grid, space_order=2)
>>> g = Function(name='g', grid=grid, space_order=2)
>>> cross_derivative(f*g, dims=(x, y), fd_order=(2, 2), deriv_order=(1, 1))
-0.5*(-0.5*f(x - h_x, y - h_y)*g(x - h_x, y - h_y)/h_x +\
0.5*f(x + h_x, y - h_y)*g(x + h_x, y - h_y)/h_x)/h_y +\
0.5*(-0.5*f(x - h_x, y + h_y)*g(x - h_x, y + h_y)/h_x +\
0.5*f(x + h_x, y + h_y)*g(x + h_x, y + h_y)/h_x)/h_y
Semantically, this is equivalent to
>>> (f*g).dxdy
Derivative(f(x, y)*g(x, y), x, y)
The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via ``evaluate``
>>> (f*g).dxdy.evaluate
-0.5*(-0.5*f(x - h_x, y - h_y)*g(x - h_x, y - h_y)/h_x +\
0.5*f(x + h_x, y - h_y)*g(x + h_x, y - h_y)/h_x)/h_y +\
0.5*(-0.5*f(x - h_x, y + h_y)*g(x - h_x, y + h_y)/h_x +\
0.5*f(x + h_x, y + h_y)*g(x + h_x, y + h_y)/h_x)/h_y
"""
stagger = stagger or [None]*len(dims)
for d, fd, dim, s in zip(deriv_order, fd_order, dims, stagger):
expr = generic_derivative(expr, dim=dim, fd_order=fd, deriv_order=d, stagger=s)
return expr
[docs]@check_input
@check_symbolic
def generic_derivative(expr, dim, fd_order, deriv_order, stagger=None, symbolic=False,
matvec=direct):
"""
Arbitrary-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int
Coefficient discretization order. Note: this impacts the width of
the resulting stencil.
deriv_order : int
Derivative order, e.g. 2 for a second-order derivative.
stagger : Side, optional
Shift of the finite-difference approximation.
Returns
-------
expr-like
``deriv-order`` derivative of ``expr``.
"""
diff = dim.spacing
# Stencil positions
indices, x0 = generate_indices(expr, dim, diff, fd_order, stagger=stagger)
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, deriv_order, indices, x0)
else:
c = finite_diff_weights(deriv_order, indices, x0)[-1][-1]
# Loop through positions
deriv = 0
all_dims = tuple(set((dim,) + tuple(i for i in expr.indices if i.root == dim)))
for i in range(len(indices)):
subs = dict((d, indices[i].subs({dim: d, diff: matvec.val*diff}))
for d in all_dims)
deriv += expr.subs(subs) * c[i]
# Evaluate up to _PRECISION digits
deriv = deriv.evalf(_PRECISION)
return deriv
```