def sym_wave(dim, sym_phi):
"""Return symbolic expressions for the wave equation system given a desired
solution. (Note: In order to support manufactured solutions, we modify the wave
equation to add a source term (f). If the solution is exact, this term should
be 0.)
"""
sym_c = pmbl.var("c")
sym_coords = prim.make_sym_vector("x", dim)
sym_t = pmbl.var("t")
# f = phi_tt - c^2 * div(grad(phi))
sym_f = sym.diff(sym_t)(sym.diff(sym_t)(sym_phi)) - sym_c**2\
* sym.div(sym.grad(dim, sym_phi))
# u = phi_t
sym_u = sym.diff(sym_t)(sym_phi)
# v = c*grad(phi)
sym_v = [sym_c * sym.diff(sym_coords[i])(sym_phi) for i in range(dim)]
# rhs(u part) = c*div(v) + f
# rhs(v part) = c*grad(u)
sym_rhs = flat_obj_array(sym_c * sym.div(sym_v) + sym_f,
make_obj_array([sym_c]) * sym.grad(dim, sym_u))
return sym_u, sym_v, sym_f, sym_rhs
def test_symbolic_div():
"""
Compute the symbolic divergence of a vector expression and compare it to an
expected result.
"""
# (Equivalent to make_obj_array([pmbl.var("x")[i] for i in range(3)]))
sym_coords = pmbl.make_sym_vector("x", 3)
sym_x = sym_coords[0]
sym_y = sym_coords[1]
sym_f = make_obj_array([sym_x, sym_x * sym_y, sym_y])
sym_div_f = sym.div(sym_f)
expected_sym_div_f = 1 + sym_x + 0
assert sym_div_f == expected_sym_div_f
def sym_diffusion(dim, sym_alpha, sym_u):
"""Return a symbolic expression for the diffusion operator applied to a function.
"""
return sym.div([sym_alpha * grad_i for grad_i in sym.grad(dim, sym_u)])
def sym_diffusion(dim, alpha, sym_u):
"""Return a symbolic expression for the diffusion operator applied to a function.
"""
return alpha * sym.div(sym.grad(dim, sym_u))