Methods Module

Contents

modespy.methods.Multistep

This is the abstract base class for all multi-step ODE methods.

modespy.methods.SolverType

Sets uniform constant labels for different solution methods.

Members

This module provides functions that define \(\theta\) coefficients, corresponding values of p-order / k-step, and other parameters associated with different Explicit, Implicit and Implicit + 1 methods.

  • Explicit methods of order \(p = k\):

    • Families of methods:

      • Adams-Bashforth: \(p \in [1, \infty]\)

      • EDF: \(p \in [1, \infty ]\)

      • Nystrom: \(p \in [2, 5]\)

      • EDC1: \(p \in [3, 6]\) containing EDC22, EDC23, EDC24, EDC45

      • EDC2: \(p \in [3, 6]\) containing EDC22, EDC23, EDC34, EDC45

      • EDC3: \(p \in [3, 6]\) containing EDC22, EDC33, EDC24, EDC45

      • EDC4: \(p \in [3, 6]\) containing EDC22, EDC33, EDC34, EDC45

    • Individual / single-order methods:

      • EDC22: p = 3

      • EDC23: p = 4

      • EDC33: p = 4

      • EDC24: p = 5

      • EDC34: p = 5

      • EDC45: p = 6

  • Implicit methods of order \(p = k\):

    • Families of methods:

      • Backward Differentiation Formula (BDF) \(p \in [1, 5]\)

      • Rockswold: \(p \in [2, 4]\)

    • Individual / single-order methods:

      • Kregel: p = 3

  • Implicit methods of order \(p = k+1\):

    • Families of methods:

      • Adams-Moulton: \(p \in [3, \infty]\)

      • dcBDF: \(p \in [2, \infty]\)

      Note

      (Version 0.9.0) dcBDF \(\theta\) values require checking as the original MATALB code has conflicting statements in this area.

      • IDC1: \(p \in [4, 7]\) containing IDC23, IDC24, IDC45, IDC56

      • IDC2: \(p \in [4, 7]\) containing IDC23, IDC34, IDC45, IDC56

    • Individual / single-order methods:

      • Milne2 p = 4

      Note

      (Version 0.9.0) Milne2 p = 4 source code is not implemented / quarantined. Refer to source code for explanation.

      • Milne4: p = 5

      Note

      (Version 0.9.0) Milne4 is implemented but does not appear to be stable or work with the current algorithm.

      • IDC23: p = 4

      • IDC24: p = 5

      • IDC34: p = 5

      • IDC45: p = 6

      • IDC56: p = 7

class modespy.methods.Multistep(solver_type: SolverType, p_theta: Dict[int, Sequence[float]] = None)

Bases: ABC

This is the abstract base class for all multi-step ODE methods.

p_defaults(int, int)

Recommended p range (low, high).

p_limits(int, int) or (int, None)

Absoluste p limits (low, high) or alternatively (low, None) if there is noupper limit.

Note

p_default and p_limit values must separately provided for each instance if p_theta is not supplied during creation.

__init__(solver_type: SolverType, p_theta: Dict[int, Sequence[float]] = None)
Parameters:

p_theta (Dict[int, array-like], optional) – Dict containing each order p as a key and a corresponding sequence of theta values that define the method.

_abc_impl = <_abc._abc_data object>
k(p: int) → int

Returns the number of steps k used by this method to generate a new point based the given order p.

theta(p: int) → ndarray

Returns an array of \(\theta\) values corresponding to the given order p.

class modespy.methods.SolverType(value, names=None, *values, module=None, qualname=None, type=None, start=1, boundary=None)

Bases: Enum

Sets uniform constant labels for different solution methods.

EXPLICIT = 1
IMPLICIT = 2
IMPLICIT_PLUS = 3