Finitedifference approximations¶

devito.finite_differences.finite_difference.
first_derivative
(expr, dim, fd_order=None, side=centered[0], matvec=direct[1], symbolic=False, x0=None)[source]¶ Firstorder derivative of a given expression.
 Parameters
expr (exprlike) – Expression for which the firstorder 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
.x0 (dict, optional) – Origin of the finitedifference scheme as a map dim: origin_dim.
 Returns
Firstorder derivative of
expr
. Return type
exprlike
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
Finally the x0 argument allows to choose the origin of the finitedifference
>>> first_derivative(f, dim=x, x0={x: 1}) f(1, y)/h_x + f(h_x + 1, y)/h_x

devito.finite_differences.finite_difference.
second_derivative
(expr, dim, fd_order, **kwargs)[source]¶ Secondorder derivative of a given expression.
 Parameters
expr (exprlike) – 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 finitedifference approximation.
x0 (dict, optional) – Origin of the finitedifference scheme as a map dim: origin_dim.
 Returns
Secondorder derivative of
expr
. Return type
exprlike
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
Finally the x0 argument allows to choose the origin of the finitedifference
>>> second_derivative(f, dim=x, fd_order=2, x0={x: 1}) 2.0*f(1, y)/h_x**2 + f(1  h_x, y)/h_x**2 + f(h_x + 1, y)/h_x**2

devito.finite_differences.finite_difference.
cross_derivative
(expr, dims, fd_order, deriv_order, **kwargs)[source]¶ Arbitraryorder cross derivative of a given expression.
 Parameters
expr (exprlike) – 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 secondorder derivative.
stagger (tuple of Side, optional) – Shift of the finitedifference approximation.
x0 (dict, optional) – Origin of the finitedifference scheme as a map dim: origin_dim.
 Returns
Crossderivative of
expr
. Return type
exprlike
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)) (f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x)/h_y + (f(x, y + h_y)*g(x, y + h_y)/h_x + 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 (f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x)/h_y + (f(x, y + h_y)*g(x, y + h_y)/h_x + f(x + h_x, y + h_y)*g(x + h_x, y + h_y)/h_x)/h_y
Finally the x0 argument allows to choose the origin of the finitedifference
>>> cross_derivative(f*g, dims=(x, y), fd_order=(2, 2), deriv_order=(1, 1), x0={x: 1, y: 2}) (f(1, 2)*g(1, 2)/h_x + f(h_x + 1, 2)*g(h_x + 1, 2)/h_x)/h_y + (f(1, h_y + 2)*g(1, h_y + 2)/h_x + f(h_x + 1, h_y + 2)*g(h_x + 1, h_y + 2)/h_x)/h_y

devito.finite_differences.finite_difference.
generic_derivative
(expr, dim, fd_order, deriv_order, symbolic=False, matvec=direct[1], x0=None)[source]¶ Arbitraryorder derivative of a given expression.
 Parameters
expr (exprlike) – 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 secondorder derivative.
stagger (Side, optional) – Shift of the finitedifference approximation.
x0 (dict, optional) – Origin of the finitedifference scheme as a map dim: origin_dim.
 Returns
derivorder
derivative ofexpr
. Return type
exprlike