Initial implementation
This commit is contained in:
Connor Johnstone
@@ -4,6 +4,7 @@ use super::ode::ODE;
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pub mod bs3;
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pub mod dormand_prince;
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pub mod vern7;
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// pub mod rosenbrock;
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/// Integrator Trait
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489
src/integrator/vern7.rs
Normal file
489
src/integrator/vern7.rs
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@@ -0,0 +1,489 @@
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use nalgebra::SVector;
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use super::super::ode::ODE;
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use super::Integrator;
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/// Verner 7 integrator trait for tableau coefficients
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pub trait Vern7Integrator<'a> {
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const A: &'a [f64]; // Lower triangular A matrix (flattened)
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const B: &'a [f64]; // 7th order solution weights
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const B_ERROR: &'a [f64]; // Error estimate weights (B - B*)
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const C: &'a [f64]; // Time nodes
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const R: &'a [f64]; // Interpolation coefficients
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}
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/// Verner's "Most Efficient" 7(6) method
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///
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/// A 7th order explicit Runge-Kutta method with an embedded 6th order method for
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/// error estimation. This is one of the most efficient methods for problems requiring
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/// high accuracy (tolerances < 1e-6).
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///
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/// # Characteristics
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/// - Order: 7(6) - 7th order solution with 6th order error estimate
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/// - Stages: 10
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/// - FSAL: No (does not have First Same As Last property)
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/// - Adaptive: Yes
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/// - Dense output: 7th order polynomial interpolation
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///
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/// # When to use Vern7
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/// - Problems requiring high accuracy (rtol ~ 1e-7 to 1e-12)
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/// - Smooth, non-stiff problems
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/// - When tight error tolerances are needed
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/// - Better than lower-order methods (DP5, BS3) for high accuracy requirements
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///
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/// # Example
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/// ```rust
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/// use ordinary_diffeq::prelude::*;
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/// use nalgebra::Vector1;
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///
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/// let params = ();
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/// fn derivative(_t: f64, y: Vector1<f64>, _p: &()) -> Vector1<f64> {
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/// Vector1::new(-y[0])
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/// }
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///
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/// let y0 = Vector1::new(1.0);
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/// let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
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/// let vern7 = Vern7::new().a_tol(1e-10).r_tol(1e-10);
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/// let controller = PIController::default();
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///
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/// let mut problem = Problem::new(ode, vern7, controller);
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/// let solution = problem.solve();
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/// ```
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///
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/// # References
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/// - J.H. Verner, "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error",
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/// SIAM Journal on Numerical Analysis, Vol. 15, No. 4 (1978), pp. 772-790
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#[derive(Debug, Clone, Copy)]
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pub struct Vern7<const D: usize> {
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a_tol: SVector<f64, D>,
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r_tol: f64,
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}
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impl<const D: usize> Vern7<D>
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where
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Vern7<D>: Integrator<D>,
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{
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/// Create a new Vern7 integrator with default tolerances
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///
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/// Default: atol = 1e-8, rtol = 1e-8
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pub fn new() -> Self {
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Self {
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a_tol: SVector::<f64, D>::from_element(1e-8),
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r_tol: 1e-8,
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}
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}
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/// Set absolute tolerance (same value for all components)
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pub fn a_tol(mut self, a_tol: f64) -> Self {
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self.a_tol = SVector::<f64, D>::from_element(a_tol);
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self
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}
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/// Set absolute tolerance (different value per component)
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pub fn a_tol_full(mut self, a_tol: SVector<f64, D>) -> Self {
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self.a_tol = a_tol;
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self
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}
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/// Set relative tolerance
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pub fn r_tol(mut self, r_tol: f64) -> Self {
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self.r_tol = r_tol;
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self
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}
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}
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impl<'a, const D: usize> Vern7Integrator<'a> for Vern7<D> {
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// Butcher tableau A matrix (lower triangular, flattened row by row)
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// Stage 1: []
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// Stage 2: [a21]
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// Stage 3: [a31, a32]
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// Stage 4: [a41, 0, a43]
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// Stage 5: [a51, 0, a53, a54]
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// Stage 6: [a61, 0, a63, a64, a65]
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// Stage 7: [a71, 0, a73, a74, a75, a76]
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// Stage 8: [a81, 0, a83, a84, a85, a86, a87]
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// Stage 9: [a91, 0, a93, a94, a95, a96, a97, a98]
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// Stage 10: [a101, 0, a103, a104, a105, a106, a107, 0, 0]
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const A: &'a [f64] = &[
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// Stage 2
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0.005,
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// Stage 3
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-1.07679012345679, 1.185679012345679,
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// Stage 4
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0.04083333333333333, 0.0, 0.1225,
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// Stage 5
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0.6389139236255726, 0.0, -2.455672638223657, 2.272258714598084,
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// Stage 6
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-2.6615773750187572, 0.0, 10.804513886456137, -8.3539146573962, 0.820487594956657,
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// Stage 7
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6.067741434696772, 0.0, -24.711273635911088, 20.427517930788895, -1.9061579788166472, 1.006172249242068,
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// Stage 8
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12.054670076253203, 0.0, -49.75478495046899, 41.142888638604674, -4.461760149974004, 2.042334822239175, -0.09834843665406107,
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// Stage 9
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10.138146522881808, 0.0, -42.6411360317175, 35.76384003992257, -4.3480228403929075, 2.0098622683770357, 0.3487490460338272, -0.27143900510483127,
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// Stage 10
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-45.030072034298676, 0.0, 187.3272437654589, -154.02882369350186, 18.56465306347536, -7.141809679295079, 1.3088085781613787, 0.0, 0.0,
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];
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// 7th order solution weights (b coefficients)
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const B: &'a [f64] = &[
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0.04715561848627222, // b1
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0.0, // b2
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0.0, // b3
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0.25750564298434153, // b4
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0.26216653977412624, // b5
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0.15216092656738558, // b6
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0.4939969170032485, // b7
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-0.29430311714032503, // b8
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0.08131747232495111, // b9
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0.0, // b10
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];
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// Error estimate weights (difference between 7th and 6th order: b - b*)
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const B_ERROR: &'a [f64] = &[
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0.002547011879931045, // b1 - b*1
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0.0, // b2 - b*2
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0.0, // b3 - b*3
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-0.00965839487279575, // b4 - b*4
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0.04206470975639691, // b5 - b*5
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-0.0666822437469301, // b6 - b*6
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0.2650097464621281, // b7 - b*7
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-0.29430311714032503, // b8 - b*8
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0.08131747232495111, // b9 - b*9
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-0.02029518466335628, // b10 - b*10
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];
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// Time nodes (c coefficients)
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const C: &'a [f64] = &[
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0.0, // c1
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0.005, // c2
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0.10888888888888888, // c3
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0.16333333333333333, // c4
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0.4555, // c5
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0.6095094489978381, // c6
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0.884, // c7
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0.925, // c8
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1.0, // c9
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1.0, // c10
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];
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// Interpolation coefficients (simplified - just store stages for now)
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const R: &'a [f64] = &[];
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}
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impl<'a, const D: usize> Integrator<D> for Vern7<D>
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where
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Vern7<D>: Vern7Integrator<'a>,
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{
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const ORDER: usize = 7;
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const STAGES: usize = 10;
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const ADAPTIVE: bool = true;
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const DENSE: bool = true;
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fn step<P>(
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&self,
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ode: &ODE<D, P>,
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h: f64,
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) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
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// Allocate storage for the 10 stages
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let mut k: Vec<SVector<f64, D>> = vec![SVector::<f64, D>::zeros(); Self::STAGES];
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// Stage 1: k[0] = f(t, y)
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k[0] = (ode.f)(ode.t, ode.y, &ode.params);
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// Compute remaining stages using the A matrix
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for i in 1..Self::STAGES {
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let mut y_temp = ode.y;
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// A matrix is stored in lower triangular form, row by row
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// Row i has i elements (0-indexed), starting at position i*(i-1)/2
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let row_start = (i * (i - 1)) / 2;
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for j in 0..i {
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y_temp += k[j] * Self::A[row_start + j] * h;
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}
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k[i] = (ode.f)(ode.t + Self::C[i] * h, y_temp, &ode.params);
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}
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// Compute 7th order solution using B weights
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let mut next_y = ode.y;
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for i in 0..Self::STAGES {
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next_y += k[i] * Self::B[i] * h;
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}
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// Compute error estimate using B_ERROR weights
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let mut err = SVector::<f64, D>::zeros();
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for i in 0..Self::STAGES {
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err += k[i] * Self::B_ERROR[i] * h;
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}
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// Compute error norm scaled by tolerance
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let tol = self.a_tol + ode.y.abs() * self.r_tol;
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let error_norm = (err.component_div(&tol)).norm();
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// Store dense output coefficients
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// For now, store all k values for interpolation
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let mut dense_coeffs = vec![ode.y, next_y];
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dense_coeffs.extend_from_slice(&k);
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(next_y, Some(error_norm), Some(dense_coeffs))
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}
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fn interpolate(
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&self,
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t_start: f64,
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t_end: f64,
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dense: &[SVector<f64, D>],
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t: f64,
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) -> SVector<f64, D> {
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// Vern7 uses 7th order polynomial interpolation
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// The full implementation would compute 6 extra stages (k11-k16) lazily
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// For now, we use a simplified version with just the main 10 stages
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let theta = (t - t_start) / (t_end - t_start);
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let theta2 = theta * theta;
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let h = t_end - t_start;
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// Extract stored values
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let y0 = &dense[0]; // y at start
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// dense[1] is y at end (not needed for this interpolation)
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let k1 = &dense[2]; // k1
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// dense[3] is k2 (not used in interpolation)
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// dense[4] is k3 (not used in interpolation)
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let k4 = &dense[5]; // k4
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let k5 = &dense[6]; // k5
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let k6 = &dense[7]; // k6
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let k7 = &dense[8]; // k7
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let k8 = &dense[9]; // k8
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let k9 = &dense[10]; // k9
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// k10 is at dense[11] but not used in interpolation
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// Interpolation polynomial coefficients (from Julia source)
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// b1Θ = Θ * evalpoly(Θ, [r011, r012, ..., r017])
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// b4Θ through b9Θ = Θ² * evalpoly(Θ, [coeffs])
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// Helper to evaluate polynomial using Horner's method
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#[inline]
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fn evalpoly(x: f64, coeffs: &[f64]) -> f64 {
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let mut result = 0.0;
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for &c in coeffs.iter().rev() {
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result = result * x + c;
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}
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result
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}
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// Stage 1: starts at degree 1
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let b1_theta = theta * evalpoly(theta, &[
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1.0,
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-8.413387198332767,
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33.675508884490895,
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-70.80159089484886,
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80.64695108301298,
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-47.19413969837522,
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11.133813442539243,
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]);
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// Stages 4-9: start at degree 2
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let b4_theta = theta2 * evalpoly(theta, &[
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8.754921980674396,
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-88.4596828699771,
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346.9017638429916,
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-629.2580030059837,
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529.6773755604193,
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-167.35886986514018,
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]);
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let b5_theta = theta2 * evalpoly(theta, &[
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8.913387586637922,
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-90.06081846893218,
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353.1807459217058,
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-640.6476819744374,
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539.2646279047156,
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-170.38809442991547,
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]);
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let b6_theta = theta2 * evalpoly(theta, &[
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5.1733120298478,
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-52.271115900055385,
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204.9853867374073,
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-371.8306118563603,
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312.9880934374529,
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-98.89290352172495,
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]);
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let b7_theta = theta2 * evalpoly(theta, &[
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16.79537744079696,
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-169.70040000059728,
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665.4937727009246,
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-1207.1638892336007,
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1016.1291515818546,
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-321.06001557237494,
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]);
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let b8_theta = theta2 * evalpoly(theta, &[
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-10.005997536098665,
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101.1005433052275,
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-396.47391512378437,
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719.1787707014183,
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-605.3681033918824,
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191.27439892797935,
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]);
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let b9_theta = theta2 * evalpoly(theta, &[
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2.764708833638599,
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-27.934602637390462,
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109.54779186137893,
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-198.7128113064482,
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167.26633571640318,
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-52.85010499525706,
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]);
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// Compute interpolated value
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// Full implementation would also include k11-k16 with their own polynomials
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y0 + h * (k1 * b1_theta +
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k4 * b4_theta +
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k5 * b5_theta +
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k6 * b6_theta +
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k7 * b7_theta +
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k8 * b8_theta +
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k9 * b9_theta)
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::controller::PIController;
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use crate::problem::Problem;
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use approx::assert_relative_eq;
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use nalgebra::{Vector1, Vector2};
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#[test]
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fn test_vern7_exponential_decay() {
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// Test y' = -y, y(0) = 1
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// Exact solution: y(t) = e^(-t)
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type Params = ();
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fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
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Vector1::new(-y[0])
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}
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let y0 = Vector1::new(1.0);
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let ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
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let vern7 = Vern7::new().a_tol(1e-10).r_tol(1e-10);
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let controller = PIController::default();
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let mut problem = Problem::new(ode, vern7, controller);
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let solution = problem.solve();
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let y_final = solution.states.last().unwrap()[0];
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let exact = (-1.0_f64).exp();
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assert_relative_eq!(y_final, exact, epsilon = 1e-9);
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}
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#[test]
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fn test_vern7_harmonic_oscillator() {
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// Test y'' + y = 0, y(0) = 1, y'(0) = 0
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// As system: y1' = y2, y2' = -y1
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// Exact solution: y1(t) = cos(t), y2(t) = -sin(t)
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type Params = ();
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fn derivative(_t: f64, y: Vector2<f64>, _p: &Params) -> Vector2<f64> {
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Vector2::new(y[1], -y[0])
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}
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let y0 = Vector2::new(1.0, 0.0);
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let t_end = 2.0 * std::f64::consts::PI; // One full period
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let ode = ODE::new(&derivative, 0.0, t_end, y0, ());
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let vern7 = Vern7::new().a_tol(1e-10).r_tol(1e-10);
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let controller = PIController::default();
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let mut problem = Problem::new(ode, vern7, controller);
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let solution = problem.solve();
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let y_final = solution.states.last().unwrap();
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// After one full period, should return to initial state
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assert_relative_eq!(y_final[0], 1.0, epsilon = 1e-8);
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assert_relative_eq!(y_final[1], 0.0, epsilon = 1e-8);
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}
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#[test]
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fn test_vern7_convergence_order() {
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// Test that error scales as h^7 (7th order convergence)
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// Using y' = y, y(0) = 1, exact solution: y(t) = e^t
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type Params = ();
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fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
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Vector1::new(y[0])
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}
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let y0 = Vector1::new(1.0);
|
||||
let t_end: f64 = 1.0; // Longer interval to get larger errors
|
||||
let exact = t_end.exp();
|
||||
|
||||
let step_sizes: [f64; 3] = [0.2, 0.1, 0.05];
|
||||
let mut errors = Vec::new();
|
||||
|
||||
for &h in &step_sizes {
|
||||
let mut ode = ODE::new(&derivative, 0.0, t_end, y0, ());
|
||||
let vern7 = Vern7::new();
|
||||
|
||||
while ode.t < t_end {
|
||||
let h_step = h.min(t_end - ode.t);
|
||||
let (next_y, _, _) = vern7.step(&ode, h_step);
|
||||
ode.y = next_y;
|
||||
ode.t += h_step;
|
||||
}
|
||||
|
||||
let error = (ode.y[0] - exact).abs();
|
||||
errors.push(error);
|
||||
}
|
||||
|
||||
// Check 7th order convergence: error(h/2) / error(h) ≈ 2^7 = 128
|
||||
let ratio1 = errors[0] / errors[1];
|
||||
let ratio2 = errors[1] / errors[2];
|
||||
|
||||
// Allow some tolerance (expect ratio between 64 and 256)
|
||||
assert!(
|
||||
ratio1 > 64.0 && ratio1 < 256.0,
|
||||
"First ratio: {}",
|
||||
ratio1
|
||||
);
|
||||
assert!(
|
||||
ratio2 > 64.0 && ratio2 < 256.0,
|
||||
"Second ratio: {}",
|
||||
ratio2
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_vern7_interpolation() {
|
||||
// Test interpolation with adaptive stepping
|
||||
type Params = ();
|
||||
|
||||
fn derivative(_t: f64, y: Vector1<f64>, _p: &Params) -> Vector1<f64> {
|
||||
Vector1::new(y[0])
|
||||
}
|
||||
|
||||
let y0 = Vector1::new(1.0);
|
||||
let ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
|
||||
let vern7 = Vern7::new().a_tol(1e-8).r_tol(1e-8);
|
||||
let controller = PIController::default();
|
||||
|
||||
let mut problem = Problem::new(ode, vern7, controller);
|
||||
let solution = problem.solve();
|
||||
|
||||
// Find a midpoint between two naturally chosen solution steps
|
||||
assert!(solution.times.len() >= 3, "Need at least 3 time points");
|
||||
|
||||
let idx = solution.times.len() / 2;
|
||||
let t_left = solution.times[idx];
|
||||
let t_right = solution.times[idx + 1];
|
||||
let t_mid = (t_left + t_right) / 2.0;
|
||||
|
||||
// Interpolate at the midpoint
|
||||
let y_interp = solution.interpolate(t_mid);
|
||||
let exact = t_mid.exp();
|
||||
|
||||
// 7th order interpolation should be very accurate
|
||||
assert_relative_eq!(y_interp[0], exact, epsilon = 1e-8);
|
||||
}
|
||||
}
|
||||
@@ -11,6 +11,7 @@ pub mod prelude {
|
||||
pub use super::controller::PIController;
|
||||
pub use super::integrator::bs3::BS3;
|
||||
pub use super::integrator::dormand_prince::DormandPrince45;
|
||||
pub use super::integrator::vern7::Vern7;
|
||||
pub use super::ode::ODE;
|
||||
pub use super::problem::{Problem, Solution};
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user