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use crate::{DenseMatrix, Vector};
use num_traits::{FromPrimitive, One, Zero};
/// A butcher tableau for a Runge-Kutta method.
///
/// The tableau is defined by the matrices `a`, `b`, `c` and `d` and the order of the method.
/// The butchers tableau is often depicted like this example of a 3-stage method:
///
/// ```text
/// c1 | a11 0 0
/// c2 | a21 a22 0
/// c3 | a31 a32 a33
/// -------------------
/// | b1 b2 b3
/// | be1 be2 be3
/// -------------------
/// | d1 d2 d3
/// ```
///
/// where `be` is the embedded method for error control and `d` is the difference between the main and embedded method.
///
/// For continous extension methods, the beta matrix is also included.
///
#[derive(Clone)]
pub struct Tableau<M: DenseMatrix> {
a: M,
b: M::V,
c: M::V,
d: M::V,
order: usize,
beta: Option<M>,
}
impl<M: DenseMatrix> Tableau<M> {
/// TR-BDF2 method
/// from R.E. Bank, W.M. Coughran Jr, W. Fichtner, E.H. Grosse, D.J. Rose and R.K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. Comput.-Aided Design 4 (1985) 436-451.
/// analysed in M.E. Hosea and L.F. Shampine. Analysis and implementation of TR-BDF2. Applied Numerical Mathematics, 20:21–37, 1996.
///
/// continuous extension from :
/// from Jørgensen, J. B., Kristensen, M. R., & Thomsen, P. G. (2018). A family of ESDIRK integration methods. arXiv preprint arXiv:1803.01613.
pub fn tr_bdf2(ctx: M::C) -> Self {
let gamma = M::T::from_f64(2.0 - 2.0_f64.sqrt()).unwrap();
let d = gamma / M::T::from_f64(2.0).unwrap();
let w = M::T::from_f64(2.0_f64.sqrt() / 4.0).unwrap();
let a = M::from_vec(
3,
3,
vec![
M::T::zero(),
d,
w,
M::T::zero(),
d,
w,
M::T::zero(),
M::T::zero(),
d,
],
ctx.clone(),
);
let b = M::V::from_vec(vec![w, w, d], ctx.clone());
let b_hat = M::V::from_vec(
vec![
(M::T::one() - w) / M::T::from_f64(3.0).unwrap(),
(M::T::from_f64(3.0).unwrap() * w + M::T::one()) / M::T::from_f64(3.0).unwrap(),
d / M::T::from_f64(3.0).unwrap(),
],
ctx.clone(),
);
let mut d = M::V::zeros(3, ctx.clone());
for i in 0..3 {
d.set_index(i, b.get_index(i) - b_hat.get_index(i));
}
let beta = M::from_vec(
3,
2,
vec![
M::T::from_f64(2.0).unwrap() * w,
M::T::from_f64(2.0).unwrap() * w,
gamma - M::T::one(),
-w,
-w,
M::T::from_f64(2.0).unwrap() * w,
],
ctx.clone(),
);
let c = M::V::from_vec(vec![M::T::zero(), gamma, M::T::one()], ctx.clone());
let order = 2;
Self::new(a, b, c, d, order, Some(beta))
}
/// A third order ESDIRK method
/// from Jørgensen, J. B., Kristensen, M. R., & Thomsen, P. G. (2018). A family of ESDIRK integration methods. arXiv preprint arXiv:1803.01613.
pub fn esdirk34(ctx: M::C) -> Self {
let gamma = M::T::from_f64(0.435_866_521_508_459).unwrap();
let a = M::from_vec(
4,
4,
vec![
M::T::zero(),
gamma,
M::T::from_f64(0.140_737_774_724_706_2).unwrap(),
M::T::from_f64(0.102_399_400_619_911).unwrap(),
M::T::zero(),
gamma,
M::T::from_f64(-0.108_365_551_381_320_8).unwrap(),
M::T::from_f64(-0.376_878_452_255_556_1).unwrap(),
M::T::zero(),
M::T::zero(),
gamma,
M::T::from_f64(0.838_612_530_127_186_1).unwrap(),
M::T::zero(),
M::T::zero(),
M::T::zero(),
gamma,
],
ctx.clone(),
);
let b = M::V::from_vec(
vec![
a.get_index(3, 0),
a.get_index(3, 1),
a.get_index(3, 2),
a.get_index(3, 3),
],
ctx.clone(),
);
let c = M::V::from_vec(
vec![
M::T::zero(),
M::T::from_f64(0.871_733_043_016_918).unwrap(),
M::T::from_f64(0.468_238_744_851_844_4).unwrap(),
M::T::one(),
],
ctx.clone(),
);
let d = M::V::from_vec(
vec![
M::T::from_f64(-0.054_625_497_240_413_94).unwrap(),
M::T::from_f64(-0.494_208_893_625_994_96).unwrap(),
M::T::from_f64(0.221_934_499_735_064_66).unwrap(),
M::T::from_f64(0.326_899_891_131_344_27).unwrap(),
],
ctx.clone(),
);
Self::new(a, b, c, d, 3, None)
}
pub fn tsit45(ctx: M::C) -> Self {
let c = M::V::from_vec(
vec![
M::T::zero(),
M::T::from_f64(0.161).unwrap(),
M::T::from_f64(0.327).unwrap(),
M::T::from_f64(0.9).unwrap(),
M::T::from_f64(0.9800255409045097).unwrap(),
M::T::one(),
M::T::one(),
],
ctx.clone(),
);
let b = M::V::from_vec(
vec![
M::T::from_f64(0.09646076681806523).unwrap(),
M::T::from_f64(0.01).unwrap(),
M::T::from_f64(0.4798896504144996).unwrap(),
M::T::from_f64(1.379008574103742).unwrap(),
M::T::from_f64(-3.290069515436081).unwrap(),
M::T::from_f64(2.324710524099774).unwrap(),
M::T::zero(),
],
ctx.clone(),
);
let d = M::V::from_vec(
vec![
M::T::from_f64(-0.001_780_011_052_225_777).unwrap(),
M::T::from_f64(-0.0008164344596567469).unwrap(),
M::T::from_f64(0.007880878010261995).unwrap(),
M::T::from_f64(-0.1447110071732629).unwrap(),
M::T::from_f64(0.5823571654525552).unwrap(),
M::T::from_f64(-0.45808210592918697).unwrap(),
M::T::from_f64(0.015151515151515152).unwrap(),
],
ctx.clone(),
);
// a matrix
// [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
// [ c[1], 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ],
// [ c[2] - c[1], 0.335480655492357, 0.0, 0.0, 0.0, 0.0, 0.0 ],
// [ c[3] - c[1] - c[2], -6.359448489975075, 4.362295432869581, 0.0, 0.0, 0.0, 0.0 ],
// [ c[4] - c[1] - c[2] - c[3], -11.74888356406283, 7.495539342889836, -0.09249506636175525, 0.0, 0.0, 0.0 ],
// [ c[5] - c[1] - c[2] - c[3] - c[4], -12.92096931784711, 8.159367898576159, -0.071584973281401, -0.02826905039406838, 0.0, 0.0 ],
// [ b[0], b[1], b[2], b[3], b[4], b[5], 0.0 ]
let mut a = M::zeros(7, 7, ctx.clone());
a.set_index(2, 1, M::T::from_f64(0.335_480_655_492_357).unwrap());
a.set_index(3, 1, M::T::from_f64(-6.359448489975075).unwrap());
a.set_index(4, 1, M::T::from_f64(-11.74888356406283).unwrap());
a.set_index(5, 1, M::T::from_f64(-12.92096931784711).unwrap());
a.set_index(3, 2, M::T::from_f64(4.362295432869581).unwrap());
a.set_index(4, 2, M::T::from_f64(7.495539342889836).unwrap());
a.set_index(5, 2, M::T::from_f64(8.159367898576159).unwrap());
a.set_index(4, 3, M::T::from_f64(-0.09249506636175525).unwrap());
a.set_index(5, 3, M::T::from_f64(-0.071_584_973_281_401).unwrap());
a.set_index(5, 4, M::T::from_f64(-0.02826905039406838).unwrap());
for i in 1..7 {
let mut a_sum = M::T::zero();
for j in 1..i {
a_sum += a.get_index(i, j);
}
a.set_index(i, 0, c.get_index(i) - a_sum);
}
for j in 0..6 {
a.set_index(6, j, b.get_index(j));
}
// b0 = -1.05308849772902*t**4 + 2.91325546182191*t**3 - 2.76370619727483*t**2 + 1.0*t
// b1 = 0.1017*t**4 - 0.2234*t**3 + 0.1317*t**2
// b2 = 2.49062728565125*t**4 - 5.9410338721315*t**3 + 3.93029623689475*t**2
// b3 = -16.5481028892449*t**4 + 30.3381886302823*t**3 - 12.4110771669337*t**2
// b4 = 47.3795219628193*t**4 - 88.1789048947664*t**3 + 37.509313416511*t**2
// b5 = -34.8706578614966*t**4 + 65.0918946747937*t**3 - 27.8965262891973*t**2
// b6 = 2.5*t**4 - 4.0*t**3 + 1.5*t**2
//r11 = convert(T, 1.0)
//r12 = convert(T, -2.763706197274826)
//r22 = convert(T, 0.13169999999999998)
//r32 = convert(T, 3.9302962368947516)
//r42 = convert(T, -12.411077166933676)
//r52 = convert(T, 37.50931341651104)
//r62 = convert(T, -27.896526289197286)
//r72 = convert(T, 1.5)
//r13 = convert(T, 2.9132554618219126)
//r23 = convert(T, -0.2234)
//r33 = convert(T, -5.941033872131505)
//r43 = convert(T, 30.33818863028232)
//r53 = convert(T, -88.1789048947664)
//r63 = convert(T, 65.09189467479366)
//r73 = convert(T, -4)
//r14 = convert(T, -1.0530884977290216)
//r24 = convert(T, 0.1017)
//r34 = convert(T, 2.490627285651253)
//r44 = convert(T, -16.548102889244902)
//r54 = convert(T, 47.37952196281928)
//r64 = convert(T, -34.87065786149661)
//r74 = convert(T, 2.5)
let beta = M::from_vec(
7,
4,
vec![
M::T::one(),
M::T::zero(),
M::T::zero(),
M::T::zero(),
M::T::zero(),
M::T::zero(),
M::T::zero(),
M::T::from_f64(-2.76370619727483).unwrap(),
M::T::from_f64(0.1317).unwrap(),
M::T::from_f64(3.93029623689475).unwrap(),
M::T::from_f64(-12.4110771669337).unwrap(),
M::T::from_f64(37.509313416511).unwrap(),
M::T::from_f64(-27.8965262891973).unwrap(),
M::T::from_f64(1.5).unwrap(),
M::T::from_f64(2.91325546182191).unwrap(),
M::T::from_f64(-0.2234).unwrap(),
M::T::from_f64(-5.9410338721315).unwrap(),
M::T::from_f64(30.3381886302823).unwrap(),
M::T::from_f64(-88.1789048947664).unwrap(),
M::T::from_f64(65.0918946747937).unwrap(),
M::T::from_f64(-4.0).unwrap(),
M::T::from_f64(-1.05308849772902).unwrap(),
M::T::from_f64(0.1017).unwrap(),
M::T::from_f64(2.49062728565125).unwrap(),
M::T::from_f64(-16.5481028892449).unwrap(),
M::T::from_f64(47.3795219628193).unwrap(),
M::T::from_f64(-34.8706578614966).unwrap(),
M::T::from_f64(2.5).unwrap(),
],
ctx.clone(),
);
let order = 4;
Self::new(a, b, c, d, order, Some(beta))
}
pub fn new(a: M, b: M::V, c: M::V, d: M::V, order: usize, beta: Option<M>) -> Self {
let s = c.len();
assert_eq!(a.ncols(), s, "Invalid number of rows in a, expected {s}");
assert_eq!(a.nrows(), s, "Invalid number of columns in a, expected {s}",);
assert_eq!(b.len(), s, "Invalid number of elements in b, expected {s}",);
assert_eq!(c.len(), s, "Invalid number of elements in c, expected {s}",);
if let Some(beta) = &beta {
assert_eq!(
beta.nrows(),
s,
"Invalid number of rows in beta, expected {s}",
);
}
Self {
a,
b,
c,
d,
order,
beta,
}
}
pub fn order(&self) -> usize {
self.order
}
pub fn s(&self) -> usize {
self.c.len()
}
pub fn a(&self) -> &M {
&self.a
}
pub fn b(&self) -> &M::V {
&self.b
}
pub fn c(&self) -> &M::V {
&self.c
}
pub fn d(&self) -> &M::V {
&self.d
}
pub fn beta(&self) -> Option<&M> {
self.beta.as_ref()
}
}