feat: implement spline collocation methods

Author: alexfikl

src/pycaputo/fode/spline_collocation.py

@@ -0,0 +1,95 @@
+# SPDX-FileCopyrightText: 2024 Alexandru Fikl <[email protected]>
+# SPDX-License-Identifier: MIT
+
+from __future__ import annotations
+
+from collections.abc import Iterator
+from dataclasses import dataclass
+from functools import cached_property
+from typing import Any, NamedTuple
+
+import numpy as np
+
+from pycaputo.derivatives import CaputoDerivative, Side
+from pycaputo.events import Event, StepCompleted
+from pycaputo.history import History, ProductIntegrationHistory
+from pycaputo.logging import get_logger
+from pycaputo.stepping import (
+    FractionalDifferentialEquationMethod,
+    evolve,
+    make_initial_condition,
+)
+from pycaputo.typing import Array, StateFunctionT
+
+logger = get_logger(__name__)
+
+
+class AdvanceResult(NamedTuple):
+    """Result of :func:`~pycaputo.stepping.advance` for
+    :class:`ProductIntegrationMethod` subclasses."""
+
+    y: Array
+    """Estimated solution at the next time step."""
+    trunc: Array
+    """Estimated truncation error at the next time step."""
+    storage: Array
+    """Array to add to the history storage."""
+
+
+@dataclass(frozen=True)
+class SplineCollocationMethod(FractionalDifferentialEquationMethod[StateFunctionT]):
+    """A spline collocation method for"""
+
+    @cached_property
+    def d(self) -> tuple[CaputoDerivative, ...]:
+        return tuple([
+            CaputoDerivative(alpha=alpha, side=Side.Left)
+            for alpha in self.derivative_order
+        ])
+
+
+@make_initial_condition.register(SplineCollocationMethod)
+def _make_initial_condition_caputo(
+    m: SplineCollocationMethod[StateFunctionT],
+) -> Array:
+    return m.y0[0]
+
+
+@evolve.register(SplineCollocationMethod)
+def _evolve_pi(
+    m: SplineCollocationMethod[StateFunctionT],
+    *,
+    history: History[Any] | None = None,
+    dtinit: float | None = None,
+) -> Iterator[Event]:
+    from pycaputo.controller import estimate_initial_time_step
+
+    if history is None:
+        history = ProductIntegrationHistory.empty_like(m.y0[0])
+
+    # initialize
+    c = m.control
+    n = 0
+    t = c.tstart
+
+    # determine the initial condition
+    yprev = make_initial_condition(m)
+    history.append(t, m.source(t, yprev))
+
+    # determine initial time step
+    if dtinit is None:
+        dt = estimate_initial_time_step(
+            t, yprev, m.source, m.smallest_derivative_order, trunc=m.order + 1
+        )
+    else:
+        dt = dtinit
+
+    yield StepCompleted(
+        t=t,
+        iteration=n,
+        dt=dt,
+        y=yprev,
+        eest=0.0,
+        q=1.0,
+        trunc=np.zeros_like(yprev),
+    )