peroxide/numerical/ode.rs
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//! # Ordinary Differential Equation (ODE) Solvers
//!
//! This module provides traits and structs for solving ordinary differential equations (ODEs).
//!
//! ## Overview
//!
//! - `ODEProblem`: Trait for defining an ODE problem.
//! - `ODEIntegrator`: Trait for ODE integrators.
//! - `ODESolver`: Trait for ODE solvers.
//! - `ODEError`: Enum for ODE errors.
//! - `ReachedMaxStepIter`: Reached maximum number of steps per step. (internal error)
//! - `ConstraintViolation(f64, Vec<f64>, Vec<f64>)`: Constraint violation. (user-defined error)
//! - ODE uses `anyhow` for error handling. So, you can customize your errors.
//!
//! ## Available integrators
//!
//! - **Explicit**
//! - Ralston's 3rd order (RALS3)
//! - Runge-Kutta 4th order (RK4)
//! - Ralston's 4th order (RALS4)
//! - Runge-Kutta 5th order (RK5)
//! - **Embedded**
//! - Bogacki-Shampine 2/3rd order (BS23)
//! - Runge-Kutta-Fehlberg 4/5th order (RKF45)
//! - Dormand-Prince 4/5th order (DP45)
//! - Tsitouras 4/5th order (TSIT45)
//! - **Implicit**
//! - Gauss-Legendre 4th order (GL4)
//!
//! ## Available solvers
//!
//! - `BasicODESolver`: A basic ODE solver using a specified integrator.
//!
//! You can implement your own ODE solver by implementing the `ODESolver` trait.
//!
//! ## Example
//!
//! ```rust
//! use peroxide::fuga::*;
//!
//! fn main() -> Result<(), Box<dyn Error>> {
//! // Same as : let rkf = RKF45::new(1e-4, 0.9, 1e-6, 1e-1, 100);
//! let rkf = RKF45 {
//! tol: 1e-6,
//! safety_factor: 0.9,
//! min_step_size: 1e-6,
//! max_step_size: 1e-1,
//! max_step_iter: 100,
//! };
//! let basic_ode_solver = BasicODESolver::new(rkf);
//! let initial_conditions = vec![1f64];
//! let (t_vec, y_vec) = basic_ode_solver.solve(
//! &Test,
//! (0f64, 10f64),
//! 0.01,
//! &initial_conditions,
//! )?;
//! let y_vec: Vec<f64> = y_vec.into_iter().flatten().collect();
//! println!("{}", y_vec.len());
//!
//! # #[cfg(feature = "plot")]
//! # {
//! let mut plt = Plot2D::new();
//! plt
//! .set_domain(t_vec)
//! .insert_image(y_vec)
//! .set_xlabel(r"$t$")
//! .set_ylabel(r"$y$")
//! .set_style(PlotStyle::Nature)
//! .tight_layout()
//! .set_dpi(600)
//! .set_path("example_data/rkf45_test.png")
//! .savefig()?;
//! # }
//! Ok(())
//! }
//!
//! // Extremely customizable struct
//! struct Test;
//!
//! impl ODEProblem for Test {
//! fn rhs(&self, t: f64, y: &[f64], dy: &mut [f64]) -> anyhow::Result<()> {
//! Ok(dy[0] = (5f64 * t.powi(2) - y[0]) / (t + y[0]).exp())
//! }
//! }
//! ```
use anyhow::{bail, Result};
use crate::fuga::ConvToMat;
use crate::util::non_macro::eye;
use crate::traits::math::{Norm, Normed, InnerProduct, Vector};
/// Trait for defining an ODE problem.
///
/// Implement this trait to define your own ODE problem.
///
/// # Example
///
/// ```
/// use peroxide::fuga::*;
///
/// struct MyODEProblem;
///
/// impl ODEProblem for MyODEProblem {
/// fn rhs(&self, t: f64, y: &[f64], dy: &mut [f64]) -> anyhow::Result<()> {
/// dy[0] = -0.5 * y[0];
/// dy[1] = y[0] - y[1];
/// Ok(())
/// }
/// }
/// ```
pub trait ODEProblem {
fn rhs(&self, t: f64, y: &[f64], dy: &mut [f64]) -> Result<()>;
}
/// Trait for ODE integrators.
///
/// Implement this trait to define your own ODE integrator.
pub trait ODEIntegrator {
fn step<P: ODEProblem>(&self, problem: &P, t: f64, y: &mut [f64], dt: f64) -> Result<f64>;
}
/// Enum for ODE errors.
///
/// # Variants
///
/// - `ReachedMaxStepIter`: Reached maximum number of steps per step. (internal error for integrator)
/// - `ConstraintViolation`: Constraint violation. (user-defined error)
///
/// If you define constraints in your problem, you can use this error to report constraint violations.
///
/// # Example
///
/// ```no_run
/// use peroxide::fuga::*;
///
/// struct ConstrainedProblem {
/// y_constraint: f64
/// }
///
/// impl ODEProblem for ConstrainedProblem {
/// fn rhs(&self, t: f64, y: &[f64], dy: &mut [f64]) -> anyhow::Result<()> {
/// if y[0] < self.y_constraint {
/// anyhow::bail!(ODEError::ConstraintViolation(t, y.to_vec(), dy.to_vec()));
/// } else {
/// // some function
/// Ok(())
/// }
/// }
/// }
/// ```
#[derive(Debug, Clone)]
pub enum ODEError {
ConstraintViolation(f64, Vec<f64>, Vec<f64>), // t, y, dy
ReachedMaxStepIter,
}
impl std::fmt::Display for ODEError {
fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
match self {
ODEError::ConstraintViolation(t, y, dy) => write!(
f,
"Constraint violation at t = {}, y = {:?}, dy = {:?}",
t, y, dy
),
ODEError::ReachedMaxStepIter => write!(f, "Reached maximum number of steps per step"),
}
}
}
/// Trait for ODE solvers.
///
/// Implement this trait to define your own ODE solver.
pub trait ODESolver {
fn solve<P: ODEProblem>(
&self,
problem: &P,
t_span: (f64, f64),
dt: f64,
initial_conditions: &[f64],
) -> Result<(Vec<f64>, Vec<Vec<f64>>)>;
}
/// A basic ODE solver using a specified integrator.
///
/// # Example
///
/// ```
/// use peroxide::fuga::*;
///
/// fn main() -> Result<(), Box<dyn Error>> {
/// let initial_conditions = vec![1f64];
/// let rkf = RKF45::new(1e-4, 0.9, 1e-6, 1e-1, 100);
/// let basic_ode_solver = BasicODESolver::new(rkf);
/// let (t_vec, y_vec) = basic_ode_solver.solve(
/// &Test,
/// (0f64, 10f64),
/// 0.01,
/// &initial_conditions,
/// )?;
/// let y_vec: Vec<f64> = y_vec.into_iter().flatten().collect();
///
/// Ok(())
/// }
///
/// struct Test;
///
/// impl ODEProblem for Test {
/// fn rhs(&self, t: f64, y: &[f64], dy: &mut [f64]) -> anyhow::Result<()> {
/// dy[0] = (5f64 * t.powi(2) - y[0]) / (t + y[0]).exp();
/// Ok(())
/// }
/// }
/// ```
pub struct BasicODESolver<I: ODEIntegrator> {
integrator: I,
}
impl<I: ODEIntegrator> BasicODESolver<I> {
pub fn new(integrator: I) -> Self {
Self { integrator }
}
}
impl<I: ODEIntegrator> ODESolver for BasicODESolver<I> {
fn solve<P: ODEProblem>(
&self,
problem: &P,
t_span: (f64, f64),
dt: f64,
initial_conditions: &[f64],
) -> Result<(Vec<f64>, Vec<Vec<f64>>)> {
let mut t = t_span.0;
let mut dt = dt;
let mut y = initial_conditions.to_vec();
let mut t_vec = vec![t];
let mut y_vec = vec![y.clone()];
while t < t_span.1 {
let dt_step = self.integrator.step(problem, t, &mut y, dt)?;
t += dt;
t_vec.push(t);
y_vec.push(y.clone());
dt = dt_step;
}
Ok((t_vec, y_vec))
}
}
// ┌─────────────────────────────────────────────────────────┐
// Butcher Tableau
// └─────────────────────────────────────────────────────────┘
/// Trait for Butcher tableau
///
/// ```text
/// C | A
/// - - -
/// | BU (Coefficient for update)
/// | BE (Coefficient for estimate error)
/// ```
///
/// # References
///
/// - J. R. Dormand and P. J. Prince, _A family of embedded Runge-Kutta formulae_, J. Comp. Appl. Math., 6(1), 19-26, 1980.
/// - Wikipedia: [List of Runge-Kutta methods](https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods)
pub trait ButcherTableau {
const C: &'static [f64];
const A: &'static [&'static [f64]];
const BU: &'static [f64];
const BE: &'static [f64];
fn tol(&self) -> f64 {
unimplemented!()
}
fn safety_factor(&self) -> f64 {
unimplemented!()
}
fn max_step_size(&self) -> f64 {
unimplemented!()
}
fn min_step_size(&self) -> f64 {
unimplemented!()
}
fn max_step_iter(&self) -> usize {
unimplemented!()
}
}
impl<BU: ButcherTableau> ODEIntegrator for BU {
fn step<P: ODEProblem>(&self, problem: &P, t: f64, y: &mut [f64], dt: f64) -> Result<f64> {
let n = y.len();
let mut iter_count = 0usize;
let mut dt = dt;
let n_k = Self::C.len();
loop {
let mut k_vec = vec![vec![0.0; n]; n_k];
let mut y_temp = y.to_vec();
for stage in 0..n_k {
for i in 0..n {
let mut s = 0.0;
for j in 0..stage {
s += Self::A[stage][j] * k_vec[j][i];
}
y_temp[i] = y[i] + dt * s;
}
problem.rhs(t + dt * Self::C[stage], &y_temp, &mut k_vec[stage])?;
}
if !Self::BE.is_empty() {
let mut error = 0f64;
for i in 0..n {
let mut s = 0.0;
for j in 0..n_k {
s += (Self::BU[j] - Self::BE[j]) * k_vec[j][i];
}
error = error.max(dt * s.abs())
}
let factor = (self.tol() * dt / error).powf(0.2);
let new_dt = self.safety_factor() * dt * factor;
let new_dt = new_dt.clamp(self.min_step_size(), self.max_step_size());
if error < self.tol() {
for i in 0..n {
let mut s = 0.0;
for j in 0..n_k {
s += Self::BU[j] * k_vec[j][i];
}
y[i] += dt * s;
}
return Ok(new_dt);
} else {
iter_count += 1;
if iter_count >= self.max_step_iter() {
bail!(ODEError::ReachedMaxStepIter);
}
dt = new_dt;
}
} else {
for i in 0..n {
let mut s = 0.0;
for j in 0..n_k {
s += Self::BU[j] * k_vec[j][i];
}
y[i] += dt * s;
}
return Ok(dt);
}
}
}
}
// ┌─────────────────────────────────────────────────────────┐
// Runge-Kutta
// └─────────────────────────────────────────────────────────┘
/// Ralston's 3rd order integrator
///
/// This integrator uses the Ralston's 3rd order method to numerically integrate the ODE system.
/// In MATLAB, it is called `ode3`.
#[derive(Debug, Clone, Copy, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct RALS3;
impl ButcherTableau for RALS3 {
const C: &'static [f64] = &[0.0, 0.5, 0.75];
const A: &'static [&'static [f64]] = &[&[], &[0.5], &[0.0, 0.75]];
const BU: &'static [f64] = &[2.0 / 9.0, 1.0 / 3.0, 4.0 / 9.0];
const BE: &'static [f64] = &[];
}
/// Runge-Kutta 4th order integrator.
///
/// This integrator uses the classical 4th order Runge-Kutta method to numerically integrate the ODE system.
/// It calculates four intermediate values (k1, k2, k3, k4) to estimate the next step solution.
#[derive(Debug, Clone, Copy, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct RK4;
impl ButcherTableau for RK4 {
const C: &'static [f64] = &[0.0, 0.5, 0.5, 1.0];
const A: &'static [&'static [f64]] = &[&[], &[0.5], &[0.0, 0.5], &[0.0, 0.0, 1.0]];
const BU: &'static [f64] = &[1.0 / 6.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 6.0];
const BE: &'static [f64] = &[];
}
/// Ralston's 4th order integrator.
///
/// This fourth order method is known as minimum truncation error RK4.
#[derive(Debug, Clone, Copy, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct RALS4;
impl ButcherTableau for RALS4 {
const C: &'static [f64] = &[0.0, 0.4, 0.45573725, 1.0];
const A: &'static [&'static [f64]] = &[
&[],
&[0.4],
&[0.29697761, 0.158575964],
&[0.21810040, -3.050965616, 3.83286476],
];
const BU: &'static [f64] = &[0.17476028, -0.55148066, 1.20553560, 0.17118478];
const BE: &'static [f64] = &[];
}
/// Runge-Kutta 5th order integrator
///
/// This integrator uses the 5th order Runge-Kutta method to numerically integrate the ODE system.
#[derive(Debug, Clone, Copy, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct RK5;
impl ButcherTableau for RK5 {
const C: &'static [f64] = &[0.0, 0.2, 0.3, 0.8, 8.0 / 9.0, 1.0, 1.0];
const A: &'static [&'static [f64]] = &[
&[],
&[0.2],
&[0.075, 0.225],
&[44.0 / 45.0, -56.0 / 15.0, 32.0 / 9.0],
&[
19372.0 / 6561.0,
-25360.0 / 2187.0,
64448.0 / 6561.0,
-212.0 / 729.0,
],
&[
9017.0 / 3168.0,
-355.0 / 33.0,
46732.0 / 5247.0,
49.0 / 176.0,
-5103.0 / 18656.0,
],
&[
35.0 / 384.0,
0.0,
500.0 / 1113.0,
125.0 / 192.0,
-2187.0 / 6784.0,
11.0 / 84.0,
],
];
const BU: &'static [f64] = &[
5179.0 / 57600.0,
0.0,
7571.0 / 16695.0,
393.0 / 640.0,
-92097.0 / 339200.0,
187.0 / 2100.0,
1.0 / 40.0,
];
const BE: &'static [f64] = &[];
}
// ┌─────────────────────────────────────────────────────────┐
// Embedded Runge-Kutta
// └─────────────────────────────────────────────────────────┘
/// Bogacki-Shampine 3(2) method
///
/// This method is known as `ode23` in MATLAB.
///
/// # Member variables
///
/// - `tol`: The tolerance for the estimated error.
/// - `safety_factor`: The safety factor for the step size adjustment.
/// - `min_step_size`: The minimum step size.
/// - `max_step_size`: The maximum step size.
/// - `max_step_iter`: The maximum number of iterations per step.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct BS23 {
pub tol: f64,
pub safety_factor: f64,
pub min_step_size: f64,
pub max_step_size: f64,
pub max_step_iter: usize,
}
impl Default for BS23 {
fn default() -> Self {
Self {
tol: 1e-3,
safety_factor: 0.9,
min_step_size: 1e-6,
max_step_size: 1e-1,
max_step_iter: 100,
}
}
}
impl BS23 {
pub fn new(
tol: f64,
safety_factor: f64,
min_step_size: f64,
max_step_size: f64,
max_step_iter: usize,
) -> Self {
Self {
tol,
safety_factor,
min_step_size,
max_step_size,
max_step_iter,
}
}
}
impl ButcherTableau for BS23 {
const C: &'static [f64] = &[0.0, 0.5, 0.75, 1.0];
const A: &'static [&'static [f64]] = &[
&[],
&[0.5],
&[0.0, 0.75],
&[2.0 / 9.0, 1.0 / 3.0, 4.0 / 9.0],
];
const BU: &'static [f64] = &[2.0 / 9.0, 1.0 / 3.0, 4.0 / 9.0, 0.0];
const BE: &'static [f64] = &[7.0 / 24.0, 0.25, 1.0 / 3.0, 0.125];
fn tol(&self) -> f64 {
self.tol
}
fn safety_factor(&self) -> f64 {
self.safety_factor
}
fn min_step_size(&self) -> f64 {
self.min_step_size
}
fn max_step_size(&self) -> f64 {
self.max_step_size
}
fn max_step_iter(&self) -> usize {
self.max_step_iter
}
}
/// Runge-Kutta-Fehlberg 4/5th order integrator.
///
/// This integrator uses the Runge-Kutta-Fehlberg method, which is an adaptive step size integrator.
/// It calculates six intermediate values (k1, k2, k3, k4, k5, k6) to estimate the next step solution and the error.
/// The step size is automatically adjusted based on the estimated error to maintain the desired tolerance.
///
/// # Member variables
///
/// - `tol`: The tolerance for the estimated error.
/// - `safety_factor`: The safety factor for the step size adjustment.
/// - `min_step_size`: The minimum step size.
/// - `max_step_size`: The maximum step size.
/// - `max_step_iter`: The maximum number of iterations per step.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct RKF45 {
pub tol: f64,
pub safety_factor: f64,
pub min_step_size: f64,
pub max_step_size: f64,
pub max_step_iter: usize,
}
impl Default for RKF45 {
fn default() -> Self {
Self {
tol: 1e-6,
safety_factor: 0.9,
min_step_size: 1e-6,
max_step_size: 1e-1,
max_step_iter: 100,
}
}
}
impl RKF45 {
pub fn new(
tol: f64,
safety_factor: f64,
min_step_size: f64,
max_step_size: f64,
max_step_iter: usize,
) -> Self {
Self {
tol,
safety_factor,
min_step_size,
max_step_size,
max_step_iter,
}
}
}
impl ButcherTableau for RKF45 {
const C: &'static [f64] = &[0.0, 1.0 / 4.0, 3.0 / 8.0, 12.0 / 13.0, 1.0, 1.0 / 2.0];
const A: &'static [&'static [f64]] = &[
&[],
&[0.25],
&[3.0 / 32.0, 9.0 / 32.0],
&[1932.0 / 2197.0, -7200.0 / 2197.0, 7296.0 / 2197.0],
&[439.0 / 216.0, -8.0, 3680.0 / 513.0, -845.0 / 4104.0],
&[
-8.0 / 27.0,
2.0,
-3544.0 / 2565.0,
1859.0 / 4104.0,
-11.0 / 40.0,
],
];
const BU: &'static [f64] = &[
16.0 / 135.0,
0.0,
6656.0 / 12825.0,
28561.0 / 56430.0,
-9.0 / 50.0,
2.0 / 55.0,
];
const BE: &'static [f64] = &[
25.0 / 216.0,
0.0,
1408.0 / 2565.0,
2197.0 / 4104.0,
-1.0 / 5.0,
0.0,
];
fn tol(&self) -> f64 {
self.tol
}
fn safety_factor(&self) -> f64 {
self.safety_factor
}
fn min_step_size(&self) -> f64 {
self.min_step_size
}
fn max_step_size(&self) -> f64 {
self.max_step_size
}
fn max_step_iter(&self) -> usize {
self.max_step_iter
}
}
/// Dormand-Prince 5(4) method
///
/// This is an adaptive step size integrator based on a 5th order Runge-Kutta method with
/// 4th order embedded error estimation.
///
/// # Member variables
///
/// - `tol`: The tolerance for the estimated error.
/// - `safety_factor`: The safety factor for the step size adjustment.
/// - `min_step_size`: The minimum step size.
/// - `max_step_size`: The maximum step size.
/// - `max_step_iter`: The maximum number of iterations per step.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct DP45 {
pub tol: f64,
pub safety_factor: f64,
pub min_step_size: f64,
pub max_step_size: f64,
pub max_step_iter: usize,
}
impl Default for DP45 {
fn default() -> Self {
Self {
tol: 1e-6,
safety_factor: 0.9,
min_step_size: 1e-6,
max_step_size: 1e-1,
max_step_iter: 100,
}
}
}
impl DP45 {
pub fn new(
tol: f64,
safety_factor: f64,
min_step_size: f64,
max_step_size: f64,
max_step_iter: usize,
) -> Self {
Self {
tol,
safety_factor,
min_step_size,
max_step_size,
max_step_iter,
}
}
}
impl ButcherTableau for DP45 {
const C: &'static [f64] = &[0.0, 0.2, 0.3, 0.8, 8.0 / 9.0, 1.0, 1.0];
const A: &'static [&'static [f64]] = &[
&[],
&[0.2],
&[0.075, 0.225],
&[44.0 / 45.0, -56.0 / 15.0, 32.0 / 9.0],
&[
19372.0 / 6561.0,
-25360.0 / 2187.0,
64448.0 / 6561.0,
-212.0 / 729.0,
],
&[
9017.0 / 3168.0,
-355.0 / 33.0,
46732.0 / 5247.0,
49.0 / 176.0,
-5103.0 / 18656.0,
],
&[
35.0 / 384.0,
0.0,
500.0 / 1113.0,
125.0 / 192.0,
-2187.0 / 6784.0,
11.0 / 84.0,
],
];
const BU: &'static [f64] = &[
35.0 / 384.0,
0.0,
500.0 / 1113.0,
125.0 / 192.0,
-2187.0 / 6784.0,
11.0 / 84.0,
0.0,
];
const BE: &'static [f64] = &[
5179.0 / 57600.0,
0.0,
7571.0 / 16695.0,
393.0 / 640.0,
-92097.0 / 339200.0,
187.0 / 2100.0,
1.0 / 40.0,
];
fn tol(&self) -> f64 {
self.tol
}
fn safety_factor(&self) -> f64 {
self.safety_factor
}
fn min_step_size(&self) -> f64 {
self.min_step_size
}
fn max_step_size(&self) -> f64 {
self.max_step_size
}
fn max_step_iter(&self) -> usize {
self.max_step_iter
}
}
/// Tsitouras 5(4) method
///
/// This is an adaptive step size integrator based on a 5th order Runge-Kutta method with
/// 4th order embedded error estimation, using the coefficients from Tsitouras (2011).
///
/// # Member variables
///
/// - `tol`: The tolerance for the estimated error.
/// - `safety_factor`: The safety factor for the step size adjustment.
/// - `min_step_size`: The minimum step size.
/// - `max_step_size`: The maximum step size.
/// - `max_step_iter`: The maximum number of iterations per step.
///
/// # References
///
/// - Ch. Tsitouras, Comput. Math. Appl. 62 (2011) 770-780.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct TSIT45 {
pub tol: f64,
pub safety_factor: f64,
pub min_step_size: f64,
pub max_step_size: f64,
pub max_step_iter: usize,
}
impl Default for TSIT45 {
fn default() -> Self {
Self {
tol: 1e-6,
safety_factor: 0.9,
min_step_size: 1e-6,
max_step_size: 1e-1,
max_step_iter: 100,
}
}
}
impl TSIT45 {
pub fn new(
tol: f64,
safety_factor: f64,
min_step_size: f64,
max_step_size: f64,
max_step_iter: usize,
) -> Self {
Self {
tol,
safety_factor,
min_step_size,
max_step_size,
max_step_iter,
}
}
}
impl ButcherTableau for TSIT45 {
const C: &'static [f64] = &[0.0, 0.161, 0.327, 0.9, 0.9800255409045097, 1.0, 1.0];
const A: &'static [&'static [f64]] = &[
&[],
&[Self::C[1]],
&[Self::C[2] - 0.335480655492357, 0.335480655492357],
&[
Self::C[3] - (-6.359448489975075 + 4.362295432869581),
-6.359448489975075,
4.362295432869581,
],
&[
Self::C[4] - (-11.74888356406283 + 7.495539342889836 - 0.09249506636175525),
-11.74888356406283,
7.495539342889836,
-0.09249506636175525,
],
&[
Self::C[5]
- (-12.92096931784711 + 8.159367898576159
- 0.0715849732814010
- 0.02826905039406838),
-12.92096931784711,
8.159367898576159,
-0.0715849732814010,
-0.02826905039406838,
],
&[
Self::BU[0],
Self::BU[1],
Self::BU[2],
Self::BU[3],
Self::BU[4],
Self::BU[5],
],
];
const BU: &'static [f64] = &[
0.09646076681806523,
0.01,
0.4798896504144996,
1.379008574103742,
-3.290069515436081,
2.324710524099774,
0.0,
];
const BE: &'static [f64] = &[
0.001780011052226,
0.000816434459657,
-0.007880878010262,
0.144711007173263,
-0.582357165452555,
0.458082105929187,
1.0 / 66.0,
];
fn tol(&self) -> f64 {
self.tol
}
fn safety_factor(&self) -> f64 {
self.safety_factor
}
fn min_step_size(&self) -> f64 {
self.min_step_size
}
fn max_step_size(&self) -> f64 {
self.max_step_size
}
fn max_step_iter(&self) -> usize {
self.max_step_iter
}
}
// ┌─────────────────────────────────────────────────────────┐
// Gauss-Legendre 4th order
// └─────────────────────────────────────────────────────────┘
/// Enum for implicit solvers.
///
/// This enum defines the available implicit solvers for the Gauss-Legendre 4th order integrator.
/// Currently, there are two options: fixed-point iteration and Broyden's method.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub enum ImplicitSolver {
FixedPoint,
Broyden,
//TrustRegion(f64, f64),
}
/// Gauss-Legendre 4th order integrator.
///
/// This integrator uses the 4th order Gauss-Legendre Runge-Kutta method, which is an implicit integrator.
/// It requires solving a system of nonlinear equations at each step, which is done using the specified implicit solver (e.g., fixed-point iteration).
/// The Gauss-Legendre method has better stability properties compared to explicit methods, especially for stiff ODEs.
///
/// # Member variables
///
/// - `solver`: The implicit solver to use.
/// - `tol`: The tolerance for the implicit solver.
/// - `max_step_iter`: The maximum number of iterations for the implicit solver per step.
#[derive(Debug, Clone, Copy)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[cfg_attr(feature = "rkyv", derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize))]
pub struct GL4 {
pub solver: ImplicitSolver,
pub tol: f64,
pub max_step_iter: usize,
}
impl Default for GL4 {
fn default() -> Self {
GL4 {
solver: ImplicitSolver::FixedPoint,
tol: 1e-6,
max_step_iter: 100,
}
}
}
impl GL4 {
pub fn new(solver: ImplicitSolver, tol: f64, max_step_iter: usize) -> Self {
GL4 {
solver,
tol,
max_step_iter,
}
}
}
impl ODEIntegrator for GL4 {
#[allow(non_snake_case)]
#[inline]
fn step<P: ODEProblem>(&self, problem: &P, t: f64, y: &mut [f64], dt: f64) -> Result<f64> {
let n = y.len();
let sqrt3 = 3.0_f64.sqrt();
let c = 0.5 * (3.0 - sqrt3) / 6.0;
let d = 0.5 * (3.0 + sqrt3) / 6.0;
let mut k1 = vec![0.0; n];
let mut k2 = vec![0.0; n];
let mut y1 = vec![0.0; n];
let mut y2 = vec![0.0; n];
match self.solver {
ImplicitSolver::FixedPoint => {
// Fixed-point iteration
for _ in 0..self.max_step_iter {
for i in 0..n {
y1[i] = y[i] + dt * (c * k1[i] + d * k2[i] - sqrt3 * (k2[i] - k1[i]) / 2.0);
y2[i] = y[i] + dt * (c * k1[i] + d * k2[i] + sqrt3 * (k2[i] - k1[i]) / 2.0);
}
problem.rhs(t + c * dt, &y1, &mut k1)?;
problem.rhs(t + d * dt, &y2, &mut k2)?;
let mut max_diff = 0f64;
for i in 0..n {
max_diff = max_diff
.max(
(y1[i]
- y[i]
- dt * (c * k1[i] + d * k2[i] - sqrt3 * (k2[i] - k1[i]) / 2.0))
.abs(),
)
.max(
(y2[i]
- y[i]
- dt * (c * k1[i] + d * k2[i] + sqrt3 * (k2[i] - k1[i]) / 2.0))
.abs(),
);
}
if max_diff < self.tol {
break;
}
}
},
ImplicitSolver::Broyden => {
let m = 2 * n;
let mut U = vec![0.0; m];
// Initial Guess via Fixed-Point Iteration
{
let mut k1 = vec![0.0; n];
let mut k2 = vec![0.0; n];
let mut y1 = vec![0.0; n];
let mut y2 = vec![0.0; n];
y1.copy_from_slice(y);
y2.copy_from_slice(y);
problem.rhs(t + c * dt, &y1, &mut k1)?;
problem.rhs(t + d * dt, &y2, &mut k2)?;
for i in 0 .. n {
U[i] = k1[i];
U[n + i] = k2[i];
}
}
// F_vec = F(U)
let mut F_vec = vec![0.0; m];
compute_F(problem, t, y, dt, c, d, sqrt3, &U, &mut F_vec)?;
// Initialize inverse Jacobian matrix
let mut J_inv = eye(m);
// Repeat Broyden's method
for _ in 0 .. self.max_step_iter {
// delta = - J_inv * F_vec
let mut delta = (&J_inv * &F_vec).mul_scalar(-1.0);
// U <- U + delta
U.iter_mut().zip(delta.iter_mut()).for_each(|(u, d)| *u += *d);
let mut F_new = vec![0.0; m];
compute_F(problem, t, y, dt, c, d, sqrt3, &U, &mut F_new)?;
// If infinity norm of F_new is less than tol, break
if F_new.norm(Norm::LInf) < self.tol {
break;
}
// Residual: delta_F = F_new - F_vec
let delta_F = F_new.sub_vec(&F_vec);
// J_inv * delta_F
let J_inv_delta_F = &J_inv * &delta_F;
let denom = delta.dot(&J_inv_delta_F);
if denom.abs() < 1e-12 {
break;
}
// Broyden "good" update
// J_inv <- J_inv + ((delta - J_inv * delta_F) * delta^T * J_inv) / denom
let delta_minus_J_inv_delta_F = delta.sub_vec(&J_inv_delta_F).to_col();
let delta_T_J_inv = &delta.to_row() * &J_inv;
let update = (delta_minus_J_inv_delta_F * delta_T_J_inv) / denom;
J_inv = J_inv + update;
F_vec = F_new;
}
// Extract k1 and k2
for i in 0 .. n {
k1[i] = U[i];
k2[i] = U[n + i];
}
},
}
for i in 0..n {
y[i] += dt * 0.5 * (k1[i] + k2[i]);
}
Ok(dt)
}
}
// Helper function to compute the function F(U) for the implicit solver.
// y1 = y + dt*(c*k1 + d*k2 - sqrt3/2*(k2-k1))
// y2 = y + dt*(c*k1 + d*k2 + sqrt3/2*(k2-k1))
#[allow(non_snake_case)]
fn compute_F<P: ODEProblem>(
problem: &P,
t: f64,
y: &[f64],
dt: f64,
c: f64,
d: f64,
sqrt3: f64,
U: &[f64],
F: &mut [f64],
) -> Result<()> {
let n = y.len();
let mut y1 = vec![0.0; n];
let mut y2 = vec![0.0; n];
for i in 0..n {
let k1 = U[i];
let k2 = U[n + i];
y1[i] = y[i] + dt * (c * k1 + d * k2 - sqrt3 * (k2 - k1) / 2.0);
y2[i] = y[i] + dt * (c * k1 + d * k2 + sqrt3 * (k2 - k1) / 2.0);
}
let mut f1 = vec![0.0; n];
let mut f2 = vec![0.0; n];
problem.rhs(t + c * dt, &y1, &mut f1)?;
problem.rhs(t + d * dt, &y2, &mut f2)?;
// F = [ k1 - f1, k2 - f2 ]
for i in 0..n {
F[i] = U[i] - f1[i];
F[n + i] = U[n + i] - f2[i];
}
Ok(())
}