def test_laplacian():
'''
SEM on the plane: laplacian(), f=x^3+2*y^2 on [-1,1]^2
'''
ngq = 4
gq_pts, gq_wts = gausslobatto(ngq-1)
x = gq_pts[:,np.newaxis]
y = gq_pts[np.newaxis,:]
scalar = np.zeros((ngq,ngq), 'f8')
ret = np.zeros((ngq,ngq), 'f8')
sep = SpectralElementPlane(ngq)
scalar[:] = x*x*x + 2*y*y
sep.laplacian(scalar, ret)
aa_equal(ret, 6*x+0*y+4, 13)
def test_vorticity():
'''
SEM on the plane: vorticity(), F=(2*y^2,x^3) on [-1,1]^2
'''
ngq = 4
gq_pts, gq_wts = gausslobatto(ngq-1)
x = gq_pts[:,np.newaxis]
y = gq_pts[np.newaxis,:]
vector = np.zeros((ngq,ngq,2), 'f8')
ret = np.zeros((ngq,ngq), 'f8')
sep = SpectralElementPlane(ngq)
vector[:,:,0] = 2*y*y
vector[:,:,1] = x*x*x
sep.vorticity(vector, ret)
aa_equal(ret, 3*x*x-4*y, 15)
def test_divergence():
'''
SEM on the plane: divergence(), F=(x^3,2*y^2) on [-1,1]^2
'''
ngq = 4
gq_pts, gq_wts = gausslobatto(ngq-1)
x = gq_pts[:,np.newaxis]
y = gq_pts[np.newaxis,:]
vector = np.zeros((ngq,ngq,2), 'f8')
ret = np.zeros((ngq,ngq), 'f8')
sep = SpectralElementPlane(ngq)
vector[:,:,0] = x*x*x
vector[:,:,1] = 2*y*y
sep.divergence(vector, ret)
aa_equal(ret, 3*x*x+4*y, 15)
def test_gradient():
'''
SEM on the plane: gradient(), f=x^2+2*y^2 on [-1,1]^2
'''
ngq = 4
gq_pts, gq_wts = gausslobatto(ngq-1)
x = gq_pts[:,np.newaxis]
y = gq_pts[np.newaxis,:]
scalar = np.zeros((ngq,ngq), 'f8')
ret = np.zeros((ngq,ngq,2), 'f8')
sep = SpectralElementPlane(ngq)
scalar[:] = x*x + 2*y*y
sep.gradient(scalar, ret)
aa_equal(ret[:,:,0], 2*x+0*y, 15)
aa_equal(ret[:,:,1], 0*x+4*y, 15)
def init_dvv(self):
ngq = self.ngq
p_order = ngq-1
gq_pts, gq_wts = gausslobatto(p_order)
dvv = self.dvv
for i in xrange(ngq):
for j in xrange(ngq):
if i != j:
dvv[i,j] = 1/(gq_pts[i] - gq_pts[j]) * \
( legendre(p_order,gq_pts[i]) / legendre(p_order,gq_pts[j]) )
else:
if i == 0:
dvv[i,j] = - p_order*(p_order+1)/4
elif i == p_order:
dvv[i,j] = p_order*(p_order+1)/4
elif 0 < i < p_order:
dvv[i,j] = 0
sep.laplacian(scalar, ret)
aa_equal(ret, 6*x+0*y+4, 13)
def test_zero_identity():
'''
SEM on the plane: zero identity, vorticity(gradient())=0
'''
ngq = 4
sep = SpectralElementPlane(ngq)
scalar = np.random.rand(ngq,ngq)
ret1 = np.zeros((ngq,ngq,2), 'f8')
ret2 = np.zeros((ngq,ngq), 'f8')
sep.gradient(scalar, ret1)
sep.vorticity(ret1, ret2)
aa_equal(ret2, np.zeros((ngq,ngq), 'f8'), 14)
if __name__ == '__main__':
ngq = 4
gq_pts, gq_wts = gausslobatto(ngq-1)
sep = SpectralElementPlane(ngq)
print 'derivative matrix (dvvT)\n', sep.dvv.T
