def stability_matrix(self, z):
r"""
Constructs the stability matrix of a two-step Runge-Kutta method.
Right now just for a specific value of z.
We ought to use Sage to do it symbolically.
**Output**:
M -- stability matrix evaluated at z
WARNING: This only works for Type I & Type II methods
right now!!!
"""
s = self.Ahat.shape[1]
if self.type == 'General':
# J Y^n = K Y^{n-1}
K1 = np.column_stack((z * self.Ahat, self.d, 1 - self.d))
K2 = snp.zeros(s + 2)
K2[-1] = 1
K3 = np.concatenate(
(z * self.bhat, np.array((self.theta, 1 - self.theta))))
K = np.vstack((K1, K2, K3))
J = snp.eye(s + 2)
J[:s, :s] = J[:s, :s] - z * self.A
J[-1, :s] = z * self.b
M = snp.solve(J.astype('complex64'), K.astype('complex64'))
#M = snp.solve(J, K) # This version is slower
else:
D = np.hstack([1. - self.d, self.d])
thet = np.hstack([1. - self.theta, self.theta])
A, b = self.A, self.b
if self.type == 'Type II':
ahat = np.zeros([self.s, 1])
ahat[:, 0] = self.Ahat[:, 0]
bh = np.zeros([1, 1])
bh[0, 0] = self.bhat[0]
A = np.hstack([ahat, self.A])
A = np.vstack([np.zeros([1, self.s + 1]), A])
b = np.vstack([bh, self.b])
M1 = np.linalg.solve(np.eye(self.s) - z * self.A, D)
L1 = thet + z * np.dot(self.b.T, M1)
M = np.vstack([L1, [1., 0.]])
return M
def stability_matrix(self,z):
r"""
Constructs the stability matrix of a two-step Runge-Kutta method.
Right now just for a specific value of z.
We ought to use Sage to do it symbolically.
**Output**:
M -- stability matrix evaluated at z
WARNING: This only works for Type I & Type II methods
right now!!!
"""
s = self.Ahat.shape[1]
if self.type == 'General':
# J Y^n = K Y^{n-1}
K1 = np.column_stack((z*self.Ahat,self.d,1-self.d))
K2 = snp.zeros(s+2); K2[-1] = 1
K3 = np.concatenate((z*self.bhat,np.array((self.theta,1-self.theta))))
K = np.vstack((K1,K2,K3))
J = snp.eye(s+2)
J[:s,:s] = J[:s,:s] - z*self.A
J[-1,:s] = z*self.b
M = snp.solve(J.astype('complex64'),K.astype('complex64'))
#M = snp.solve(J, K) # This version is slower
else:
D=np.hstack([1.-self.d,self.d])
thet=np.hstack([1.-self.theta,self.theta])
A,b=self.A,self.b
if self.type=='Type II':
ahat = np.zeros([self.s,1]); ahat[:,0] = self.Ahat[:,0]
bh = np.zeros([1,1]); bh[0,0]=self.bhat[0]
A = np.hstack([ahat,self.A])
A = np.vstack([np.zeros([1,self.s+1]),A])
b = np.vstack([bh,self.b])
M1=np.linalg.solve(np.eye(self.s)-z*self.A,D)
L1=thet+z*np.dot(self.b.T,M1)
M=np.vstack([L1,[1.,0.]])
return M
def downwind_shu_osher_to_butcher(alpha, alphat, beta, betat):
r""" Accepts a Shu-Osher representation of a downwind Runge-Kutta
method and returns the Butcher coefficients
\\begin{align*}
A = & (I-\\alpha_0-\\alphat_0)^{-1} \\beta_0 \\\\
At = & (I-\\alpha_0-\\alphat_0)^{-1} \\betat_0 \\\\
b = & \\beta_1 + (\\alpha_1 + \\alphat_1) * A
\\end{align*}
**References**:
#. [gottlieb2009]_
"""
m = np.size(alpha, 1)
if not np.all([np.size(alpha, 0), np.size(beta, 0), np.size(beta, 1)] == [m + 1, m + 1, m]):
raise Exception("Inconsistent dimensions of Shu-Osher arrays")
X = snp.eye(m) - alpha[0:m, :] - alphat[0:m, :]
A = snp.solve(X, beta[0:m, :])
At = snp.solve(X, betat[0:m, :])
b = beta[m, :] + np.dot(alpha[m, :] + alphat[m, :], A)
bt = betat[m, :] + np.dot(alpha[m, :] + alphat[m, :], At)
return A, At, b, bt
def downwind_shu_osher_to_butcher(alpha, alphat, beta, betat):
r""" Accepts a Shu-Osher representation of a downwind Runge-Kutta
method and returns the Butcher coefficients
\\begin{align*}
A = & (I-\\alpha_0-\\alphat_0)^{-1} \\beta_0 \\\\
At = & (I-\\alpha_0-\\alphat_0)^{-1} \\betat_0 \\\\
b = & \\beta_1 + (\\alpha_1 + \\alphat_1) * A
\\end{align*}
**References**:
#. [gottlieb2009]_
"""
m = np.size(alpha, 1)
if not np.all([np.size(alpha, 0),
np.size(beta, 0),
np.size(beta, 1)] == [m + 1, m + 1, m]):
raise Exception('Inconsistent dimensions of Shu-Osher arrays')
X = snp.eye(m) - alpha[0:m, :] - alphat[0:m, :]
A = snp.solve(X, beta[0:m, :])
At = snp.solve(X, betat[0:m, :])
b = beta[m, :] + np.dot(alpha[m, :] + alphat[m, :], A)
bt = betat[m, :] + np.dot(alpha[m, :] + alphat[m, :], At)
return A, At, b, bt