linear_multistep_method
¶
Examples:
>>> import nodepy.linear_multistep_method as lm
>>> ab3=lm.Adams_Bashforth(3)
>>> ab3.order()
3
>>> bdf2=lm.backward_difference_formula(2)
>>> bdf2.order()
2
>>> bdf2.is_zero_stable()
True
>>> bdf7=lm.backward_difference_formula(7)
>>> bdf7.is_zero_stable()
False
>>> bdf3=lm.backward_difference_formula(3)
>>> bdf3.A_alpha_stability()
86
>>> ssp32=lm.elm_ssp2(3)
>>> ssp32.order()
2
>>> ssp32.ssp_coefficient()
1/2
>>> ssp32.plot_stability_region()
<Figure...
- class nodepy.linear_multistep_method.LinearMultistepMethod(alpha, beta, name='Linear multistep method', shortname='LMM', description='')[source]¶
Implementation of linear multistep methods in the form:
$$ \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots + \alpha_0 y_n = h ( \beta_k f_{n+k} + \cdots + \beta_0 f_n ) $$
Methods are automatically normalized so that \(\alpha_k=1\).
Notes: Representation follows Hairer & Wanner p. 368, NOT Butcher.
- characteristic_polynomials()[source]¶
Returns the characteristic polynomials (also known as generating polynomials) of a linear multistep method. They are:
$$ \rho(z) = \sum_{j=0}^k \alpha_k z^k $$
$$ \sigma(z) = \sum_{j=0}^k \beta_k z^k $$
Examples:
>>> from nodepy import lm >>> ab5 = lm.Adams_Bashforth(5) >>> rho,sigma = ab5.characteristic_polynomials() >>> print(rho) 5 4 1 x - 1 x >>> print(sigma) 4 3 2 2.64 x - 3.853 x + 3.633 x - 1.769 x + 0.3486
Reference: [HNorW93] p. 370, eq. 2.4
- order(tol=1e-10)[source]¶
Return the order of the local truncation error of a linear multistep method.
Examples:
>>> from nodepy import lm >>> am3=lm.Adams_Moulton(3) >>> am3.order() 4
- property p¶
- latex()[source]¶
Print a LaTeX representation of a linear multistep formula.
Example:
>>> from nodepy import lm >>> print(lm.Adams_Bashforth(2).latex()) \begin{align} y_{n + 2} - y_{n + 1} = \frac{3}{2}h f(y_{n + 1}) - \frac{1}{2}h f(y_{n})\end{align}
- ssp_coefficient()[source]¶
Return the SSP coefficient of the method.
The SSP coefficient is given by
$$ \min_{0 \le j < k} - \alpha_k/ \beta_k $$
if \(\alpha_j<0\) and \(\beta_j>0\) for all \(j\), and is equal to zero otherwise.
Examples:
>>> from nodepy import lm >>> ssp32=lm.elm_ssp2(3) >>> ssp32.ssp_coefficient() 1/2 >>> bdf2=lm.backward_difference_formula(2) >>> bdf2.ssp_coefficient() 0
- plot_stability_region(N=100, bounds=None, color='r', filled=True, alpha=1.0, to_file=False, longtitle=False)[source]¶
The region of absolute stability of a linear multistep method is the set
$$ \{ z \in C \mid \rho(\zeta) - z \sigma(\zeta) \text{ satisfies the root condition} \} $$
where \(\rho(\zeta)\) and \(\sigma(\zeta)\) are the characteristic functions of the method.
Also plots the boundary locus, which is given by the set of points z:
$$ \{z \mid z = \rho(\exp(\imath \theta)) / \sigma(\exp(\imath\theta)), 0 \le \theta \le 2\pi \} $$
Here \(\rho\) and \(\sigma\) are the characteristic polynomials of the method.
Reference: [LeV07] section 7.6.1
- Input: (all optional)
N – Number of gridpoints to use in each direction
bounds – limits of plotting region
color – color to use for this plot
filled – if true, stability region is filled in (solid); otherwise it is outlined
- plot_boundary_locus(N=1000, figsize=None)[source]¶
Plot the boundary locus, which is given by the set of points
$$ \{z \mid z = \rho(\exp(\imath\theta)) / \sigma(\exp(\imath\theta)), 0\le \theta \le 2\pi \} $$
where \(\rho\) and \(\sigma\) are the characteristic polynomials of the method.
Reference: [LeV07] section 7.6.1
- class nodepy.linear_multistep_method.AdditiveLinearMultistepMethod(alpha, beta, gamma, name='Additive linear multistep method')[source]¶
Method for solving equations of the form
$$ y’(t) = f(y) + g(y) $$
The method takes the form
$$ \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots + \alpha_0 y_n = h ( \beta_k f_{n+k} + \cdots + \beta_0 f_n + \gamma_k f_{n+k} + \cdots + \gamma_0 f_n ) $$
Methods are automatically normalized so that \(\alpha_k=1\).
The usual reference for these is Ascher, Ruuth, and Whetton. But we follow a different notation (as just described).
- order(tol=1e-10)[source]¶
Return the order of the local truncation error of an additive linear multistep method. The output is the minimum of the order of the component methods.
- plot_imex_stability_region(both_real=False, N=100, color='r', filled=True, alpha=1.0, fignum=None, bounds=[-10, 1, -5, 5])[source]¶
- Input: (all optional)
N – Number of gridpoints to use in each direction
bounds – limits of plotting region
color – color to use for this plot
filled – if true, stability region is filled in (solid); otherwise it is outlined
- nodepy.linear_multistep_method.Adams_Bashforth(k)[source]¶
Construct the k-step, Adams-Bashforth method. The methods are explicit and have order k. They have the form:
$$ y_{n+1} = y_n + h \sum_{j=0}^{k-1} \beta_j f(y_{n-k+j+1}) $$
They are generated using equations (1.5) and (1.7) from [HNorW93] III.1, along with the binomial expansion.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> ab3=lm.Adams_Bashforth(3) >>> print(ab3) 3-step Adams-Bashforth Explicit 3-step method of order 3 alpha = [ 0 0 -1 1 ] beta = [ 5/12 -4/3 23/12 0 ] >>> ab3.order() 3
Reference: [HNorW93]
- nodepy.linear_multistep_method.Nystrom(k)[source]¶
Construct the \(k\)-step explicit Nystrom linear multistep method. The methods are explicit and have order \(k\).
They have the form:
$$y_{n+1} = y_{n-1} + h \sum_{j=0}^{k-1} \beta_j f(y_{n-k+j+1})$$
They are generated using equations (1.13) and (1.7) from [HNorW93] III.1, along with the binomial expansion and the relation in exercise 4 on p. 367.
Note that the term “Nystrom method” is also commonly used to refer to a class of methods for second-order ODEs; those are NOT the methods generated by this function.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> nys3=lm.Nystrom(6) >>> nys3.order() 6
Reference: [HNorW93]
- nodepy.linear_multistep_method.Adams_Moulton(k)[source]¶
Construct the \(k\)-step, Adams-Moulton method. The methods are implicit and have order \(k+1\). They have the form:
$$ y_{n+1} = y_n + h \sum_{j=0}^{k} \beta_j f(y_{n-k+j+1}) $$
They are generated using equation (1.9) and the equation in Exercise 3 from Hairer & Wanner III.1, along with the binomial expansion.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> am3=lm.Adams_Moulton(3) >>> am3.order() 4
Reference: [HNorW93]
- nodepy.linear_multistep_method.Milne_Simpson(k)[source]¶
Construct the \(k\)-step, Milne-Simpson method. The methods are implicit and (for \(k \ge 3\)) have order \(k+1\). They have the form:
$$ y_{n+1} = y_{n-1} + h \sum_{j=0}^{k} \beta_j f(y_{n-k+j+1}) $$
They are generated using equation (1.15), the equation in Exercise 3, and the relation in exercise 4, all from Hairer & Wanner III.1, along with the binomial expansion.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> ms3=lm.Milne_Simpson(3) >>> ms3.order() 4
Reference: [HNorW93]
- nodepy.linear_multistep_method.backward_difference_formula(k)[source]¶
Construct the \(k\)-step backward differentiation method. The methods are implicit and have order \(k\).
They have the form:
$$ \sum_{j=0}^{k} \alpha_j y_{n+k-j+1} = h \beta_j f(y_{n+1}) $$
They are generated using equation (1.22’) from Hairer & Wanner III.1, along with the binomial expansion.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> bdf4=lm.backward_difference_formula(4) >>> bdf4.A_alpha_stability() 73
Reference: [HNorW93] pp. 364-365
- nodepy.linear_multistep_method.elm_ssp2(k)[source]¶
Returns the optimal SSP \(k\)-step linear multistep method of order 2.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> lm10=lm.elm_ssp2(10) >>> lm10.ssp_coefficient() 8/9
- nodepy.linear_multistep_method.sand_cc(s)[source]¶
Construct Sand’s circle-contractive method of order \(p=2(s+1)\) that uses \(2^s + 1\) steps.
Examples:
>>> import nodepy.linear_multistep_method as lm >>> cc4 = lm.sand_cc(4) >>> cc4.order() 10 >>> cc4.ssp_coefficient() 1/8
Reference: [San86]
- nodepy.linear_multistep_method.arw2(gam, c)[source]¶
Returns the second order IMEX additive multistep method based on the parametrization in Section 3.2 of Ascher, Ruuth, & Whetton. The parameters are gam and c. Known methods are obtained with the following values:
(1/2,0): CNAB (1/2,1/8): MCNAB (0,1): CNLF (1,0): SBDF
Examples:
>>> from nodepy import lm >>> import sympy >>> CNLF = lm.arw2(0,sympy.Rational(1)) >>> CNLF.order() 2 >>> CNLF.method1.ssp_coefficient() 1 >>> CNLF.method2.ssp_coefficient() 0 >>> print(CNLF.stiff_damping_factor()) 0.999...
- nodepy.linear_multistep_method.arw3(gam, theta, c)[source]¶
Returns the third order IMEX additive multistep method based on the parametrization in Section 3.3 of Ascher, Ruuth, & Whetton. The parameters are gam, theta, and c. Known methods are obtained with the following values:
(1,0,0): SBDF3
Note that there is one sign error in the ARW paper; it is corrected here.
- nodepy.linear_multistep_method.loadLMM(which='All')[source]¶
Load a set of standard linear multistep methods for testing.
Examples:
>>> from nodepy import lm >>> ebdf5 = lm.loadLMM('eBDF5') >>> print(ebdf5) eBDF 5 Explicit 5-step method of order 5 alpha = [ -12/137 75/137 -200/137 300/137 -300/137 1 ] beta = [ 60/137 -300/137 600/137 -600/137 300/137 0 ] >>> ebdf5.is_zero_stable() True