def diffusion_stencil():
"""Create stencil and substitutions for the diffusion equation"""
p = Function('p')
dx2 = as_finite_diff(p(x, y, t).diff(x, x), [x - h, x, x + h])
dy2 = as_finite_diff(p(x, y, t).diff(y, y), [y - h, y, y + h])
dt = as_finite_diff(p(x, y, t).diff(t), [t, t + s])
eqn = Eq(dt, a * (dx2 + dy2))
stencil = solve(eqn, p(x, y, t + s))[0]
return stencil, (p(x, y, t), p(x + h, y, t), p(x - h, y, t),
p(x, y + h, t), p(x, y - h, t), s, h)
def get_space_derivatives(self, derivative_order, accuracy_order):
global t, x, y, z
#TODO: Generalize
if accuracy_order != 2:
raise ValueError('Only implemented for accuracy order 2')
if derivative_order > 2:
raise ValueError('Derivates only available until second order')
h = self.h
print self.fields
if derivative_order==1:
return (as_finite_diff(self.fields[0].function(x,y,z,t).diff(x), [x-h, x+h]), as_finite_diff(self.fields[0].function(x,y,z,t).diff(y), [y-h, y+h]), as_finite_diff(self.fields[0].function(x,y,z,t).diff(z), [z-h, z+h]))
else:
return (as_finite_diff(self.fields[0].function(x,y,z,t).diff(x,x), [x-h,x, x+h]), as_finite_diff(self.fields[0].function(x,y,z,t).diff(y,y), [y-h,y, y+h]), as_finite_diff(self.fields[0].function(x,y,z,t).diff(z,z), [z-h,z, z+h]))
def get_time_derivative(self, derivative_order, accuracy_order):
global x, y, z
t=self.t
#TODO: Generalize
if accuracy_order != 2:
raise ValueError('Only implemented for accuracy order 2')
if derivative_order > 2:
raise ValueError('Derivates only available until second order')
s = self.s
print self.fields
if derivative_order==1:
return as_finite_diff(self.fields[0].function(x,y,z,t).diff(t), [t-s, t+s])
else:
return as_finite_diff(self.fields[0].function(x,y,z,t).diff(t,t), [t-s,t, t+s])
def dt2(self):
"""Symbol for the second derivative wrt the t dimension"""
_t = self.indices[0]
width_t = int(self.time_order / 2)
indt = [(_t + i * s) for i in range(-width_t, width_t + 1)]
return as_finite_diff(self.diff(_t, _t), indt)
def test_stencil_derivative(shape, SymbolType, dimension):
"""Test symbolic behaviour when expanding stencil derivatives"""
u = SymbolType(name='u', shape=shape)
u.data[:] = 66.6
dx = u.diff(x)
dxx = u.diff(x, x)
# Check for sympy Derivative objects
assert (isinstance(dx, Derivative) and isinstance(dxx, Derivative))
s_dx = as_finite_diff(dx, [x - x.spacing, x])
s_dxx = as_finite_diff(dxx, [x - x.spacing, x, x + x.spacing])
# Check stencil length of first and second derivatives
assert (len(s_dx.args) == 2 and len(s_dxx.args) == 3)
u_dx = s_dx.args[0].args[1]
u_dxx = s_dx.args[0].args[1]
# Ensure that devito meta-data survived symbolic transformation
assert (u_dx.shape[-2:] == shape and u_dxx.shape[-2:] == shape)
assert (np.allclose(u_dx.data, 66.6))
assert (np.allclose(u_dxx.data, 66.6))
def dt(self):
"""Symbol for the first derivative wrt the time dimension"""
if self.time_order == 1:
# This hack is needed for the first-order diffusion test
indices = [t, t + s]
else:
width = int(self.time_order / 2)
indices = [(t + i * s) for i in range(-width, width + 1)]
return as_finite_diff(self.diff(t), indices)
def test_second_derivatives_space(derivative, dimension, order):
"""Test second derivative expressions against native sympy"""
u = TimeData(name='u', shape=(20, 20, 20), time_order=2, space_order=order)
expr = getattr(u, derivative)
# Establish native sympy derivative expression
width = int(order / 2)
indices = [(dimension + i * h) for i in range(-width, width + 1)]
s_expr = as_finite_diff(u.diff(dimension, dimension), indices)
assert (simplify(expr - s_expr) == 0) # Symbolic equality
assert (expr == s_expr) # Exact equailty
def dz4(self):
"""Symbol for the fourth derivative wrt the z dimension"""
width_h = max(int(self.space_order / 2), 2)
indz = [(z + i * z.spacing) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(z, z, z, z), indz)
def dy4(self):
"""Symbol for the fourth derivative wrt the y dimension"""
width_h = max(int(self.space_order / 2), 2)
indy = [(y + i * y.spacing) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(y, y, y, y), indy)
def dx4(self):
"""Symbol for the fourth derivative wrt the x dimension"""
width_h = max(int(self.space_order / 2), 2)
indx = [(x + i * x.spacing) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(x, x, x, x), indx)
def dz(self):
"""Symbol for the first derivative wrt the z dimension"""
width_h = int(self.space_order/2)
indz = [(z + i * h) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(z), indz)
def dy2(self):
"""Symbol for the second derivative wrt the y dimension"""
width_h = int(self.space_order/2)
indy = [(y + i * h) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(y, y), indy)
def dx2(self):
"""Symbol for the second derivative wrt the x dimension"""
width_h = int(self.space_order/2)
indx = [(x + i * h) for i in range(-width_h, width_h + 1)]
return as_finite_diff(self.diff(x, x), indx)
