Notebooks · lib/ode/nonstiff_events

lib/ode on non-stiff problems, events, dense output and batches

The checks with an exact answer:

the Kepler orbit (eccentricity 0.5, ten periods): the energy is conserved and the orbit closes, so after ten periods the state is the initial one (SciPy's RK45 at the same tolerances takes the same 2247 steps and drifts the same 1.4 × 10⁻⁹ in energy);
Lotka-Volterra: V = d x - c ln x + b y - a ln y is conserved;
dense output: y' = -y + sin t has a closed form, checked between the steps (t_eval) for each method;
events: a ball dropped from 10 m lands at sqrt(2 h / g), a terminal event located on the dense output; the prey of Lotka-Volterra crosses x = 5 upward once per cycle, a non-terminal event with a direction;
a batch: 40 one-compartment pharmacokinetic models against Bateman's closed form;
AUTO: Dormand-Prince on Van der Pol with eps = 10⁻⁶ detects stiffness and hands over to Radau IIA, and the answer is Radau's own.

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from ode.solve import ode_batch, ode_cfg, ode_solve, no_events, no_jac
from ode.core import AUTO, BDF, DONE, DOPRI5, JAC_FD, JAC_GIVEN, RADAU, TERMINAL_EVENT
from ode.problems import ball, ball_hits, kepler, kepler_energy, lotka, pk, vdpol, vdpol_jac

MAX_STEPS: Int = 1000000
NO_PARAMS = zeros(0)
NONE = zeros(0)

#: the Kepler orbit, eccentricity 0.5, a = 1: period 2 pi
ECC = 0.5
PERIODS = 10.0
k0 = [1.0 - ECC, 0.0, 0.0, sqrt((1.0 + ECC) / (1.0 - ECC))]
k_end = 2.0 * pi * PERIODS
ck = ode_cfg(DOPRI5, k_end, 1.0e-10, 1.0e-12 * ones(4), 0.0, inf, MAX_STEPS, JAC_FD)
rk = ode_solve(kepler, no_jac, no_events, 0.0, k_end, k0, NO_PARAMS, ck, [k_end], NONE, NONE)
yk = rk.y[0, :]
drift = abs(kepler_energy(yk) - kepler_energy(k0)) / abs(kepler_energy(k0))
closure = amax(abs(yk - k0))
print(f"Kepler, 10 periods: {rk.nsteps} steps, {rk.nfev} evaluations, energy drift {drift:.1e}, return error {closure:.1e}")

#: Lotka-Volterra and its invariant
LV_P = [1.5, 1.0, 3.0, 1.0]
def lv_invariant(y: Arr, p: Arr) -> Num {
    return p[3] * y[0] - p[2] * log(y[0]) + p[1] * y[1] - p[0] * log(y[1])
}
def prey_at_five(t: Num, y: Arr, p: Arr) -> Arr {
    return [y[0] - 5.0]
}
lv0 = [10.0, 5.0]
LV_END = 15.0
cl = ode_cfg(DOPRI5, LV_END, 1.0e-10, 1.0e-12 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
rl = ode_solve(lotka, no_jac, prey_at_five, 0.0, LV_END, lv0, LV_P, cl, NONE, [1.0], [0.0])
nl = len(rl.t)
lv_drift = abs(lv_invariant(rl.y[nl - 1, :], LV_P) - lv_invariant(lv0, LV_P))
print(f"Lotka-Volterra: {rl.nsteps} steps, invariant drift {lv_drift:.1e}, prey rose through 5 at {rl.t_events}")
# the event's state has the prey at 5, and its time is between two output steps
ev_miss = amax(abs(rl.y_events[:, 0] - 5.0))

#: dense output: y' = -y + sin t, y(0) = 1: y = 1.5 e^-t + (sin t - cos t) / 2
def forced(t: Num, y: Arr, p: Arr) -> Arr {
    return [-y[0] + sin(t)]
}
def forced_row(y: Arr, t: Num, p: Arr, i) -> Num {
    return forced(t, y, p)[i]
}
def forced_jac(t: Num, y: Arr, p: Arr) -> Arr {
    J = zeros(1, 1)
    J[0, :] = grad(forced_row, y, t, p, 0)
    return J
}
FORCED_END = 10.0
# output every 0.1 time units
FORCED_POINTS: Int = 101
tq = linspace(0.0, FORCED_END, FORCED_POINTS)
exact = 1.5 * exp(-tq) + 0.5 * (sin(tq) - cos(tq))
dense_methods = [DOPRI5, RADAU, BDF]
dense_err = zeros(len(dense_methods))
mi = 0
for mm in dense_methods {
    m: Int = int(mm)
    cf = ode_cfg(m, FORCED_END, 1.0e-9, 1.0e-12 * ones(1), 0.0, inf, MAX_STEPS, JAC_GIVEN)
    rf = ode_solve(forced, forced_jac, no_events, 0.0, FORCED_END, [1.0], NO_PARAMS, cf, tq, NONE, NONE)
    dense_err[mi] = amax(abs(rf.y[:, 0] - exact))
    mi += 1
}
print(f"dense output on 101 points, max error: dopri5 {dense_err[0]:.1e}, radau {dense_err[1]:.1e}, bdf {dense_err[2]:.1e}")

#: the bouncing ball: dropped from 10 m, it lands at sqrt(2 h / g)
G = 9.81
DROP = 10.0
cb = ode_cfg(DOPRI5, 10.0, 1.0e-6, 1.0e-9 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
rb = ode_solve(ball, no_jac, ball_hits, 0.0, 10.0, [DROP, 0.0], [G], cb, NONE, [-1.0], [1.0])
landing = sqrt(2.0 * DROP / G)
print(f"ball: lands at {rb.t_events[0]:.15f} (exact {landing:.15f}), at {rb.y_events[0, 1]:.6f} m/s; status {rb.status}")

#: a batch: 40 oral doses of 100 mg, absorption and elimination rates varied
NB: Int = 40
DOSE = 100.0
Y0 = zeros(NB, 2)
# the parameters: absorption rate ka, elimination rate ke, volume V
PK_PARAMS: Int = 3
P = zeros(NB, PK_PARAMS)
for b to NB {
    Y0[b, 0] = DOSE
    P[b, :] = [0.5 + 0.05 * b, 0.1 + 0.004 * b, 20.0]
}
PK_END = 24.0
# output every half hour
PK_POINTS: Int = 49
tpk = linspace(0.0, PK_END, PK_POINTS)
cp = ode_cfg(RADAU, PK_END, 1.0e-9, 1.0e-12 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
Ypk, spk, fpk = ode_batch(pk, no_jac, no_events, 0.0, PK_END, Y0, P, cp, tpk, NONE, NONE)
#| Bateman's concentration at every output time, for one row's parameters.
def bateman_curve(t: Arr, dose: Num, p: Arr) -> Arr {
    return dose * p[0] / (p[2] * (p[0] - p[1])) * (exp(-p[1] * t) - exp(-p[0] * t))
}
pk_dev = zeros(NB)
for b to NB {
    conc = reshape(Ypk[b, :], len(tpk), 2)[:, 1]
    pk_dev[b] = amax(abs(conc - bateman_curve(tpk, DOSE, P[b, :])))
}
pk_err = amax(pk_dev)
print(f"batch of {NB} PK models: max |C - Bateman| = {pk_err:.1e} mg/L, {sum(spk[:, 4])} steps in all")

#: AUTO on Van der Pol, eps = 1e-6: Dormand-Prince until the stiffness test fires
VDP_EPS = [1.0e-6]
ca = ode_cfg(AUTO, 2.0, 1.0e-6, 1.0e-6 * ones(2), 0.0, inf, MAX_STEPS, JAC_GIVEN)
ra = ode_solve(vdpol, vdpol_jac, no_events, 0.0, 2.0, [2.0, 0.0], VDP_EPS, ca, [2.0], NONE, NONE)
cr = ode_cfg(RADAU, 2.0, 1.0e-10, 1.0e-10 * ones(2), 0.0, inf, MAX_STEPS, JAC_GIVEN)
rr = ode_solve(vdpol, vdpol_jac, no_events, 0.0, 2.0, [2.0, 0.0], VDP_EPS, cr, [2.0], NONE, NONE)
auto_diff = amax(abs(ra.y[0, :] - rr.y[0, :]))
print(f"auto: stiff at t = {ra.stiff_at:.4f}, {ra.nsteps} steps, differs from Radau at rtol 1e-10 by {auto_diff:.1e}")

test "the Kepler orbit keeps its energy and closes after ten periods" {
    assert_lt(drift, 2.0e-9)
    assert_lt(closure, 5.0e-7)
}
test "Lotka-Volterra keeps its invariant, and the prey crosses 5 upward once per cycle" {
    assert_lt(lv_drift, 1.0e-8)
    assert_near(len(rl.t_events), 3.0, 0.0)
    assert_lt(ev_miss, 1.0e-9)
}
test "the dense output of every method is accurate between steps" {
    assert_lt(amax(dense_err), 1.0e-7)
}
test "the ball's landing is a terminal event at sqrt(2 h / g)" {
    assert_near(rb.t_events[0], landing, 1.0e-12)
    assert_near(rb.status, TERMINAL_EVENT, 0.0)
    assert_near(rb.t[len(rb.t) - 1], landing, 0.0)
}
test "a batch of PK models matches the Bateman solution" {
    assert_lt(pk_err, 1.0e-6)
}
test "AUTO detects stiffness and its answer is Radau's" {
    assert_gt(ra.stiff_at, 0.0)
    assert_lt(auto_diff, 1.0e-4)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

from ode.solve import ode_batch, ode_cfg, ode_solve, no_events, no_jac
from ode.core import AUTO, BDF, DONE, DOPRI5, JAC_FD, JAC_GIVEN, RADAU, TERMINAL_EVENT
from ode.problems import ball, ball_hits, kepler, kepler_energy, lotka, pk, vdpol, vdpol_jac

MAX_STEPS: Int = 1000000
NO_PARAMS = zeros(0)
NONE = zeros(0)

#: the Kepler orbit, eccentricity 0.5, a = 1: period 2 pi
ECC = 0.5
PERIODS = 10.0
k0 = [1.0 - ECC, 0.0, 0.0, sqrt((1.0 + ECC) / (1.0 - ECC))]
k_end = 2.0 * pi * PERIODS
ck = ode_cfg(DOPRI5, k_end, 1.0e-10, 1.0e-12 * ones(4), 0.0, inf, MAX_STEPS, JAC_FD)
rk = ode_solve(kepler, no_jac, no_events, 0.0, k_end, k0, NO_PARAMS, ck, [k_end], NONE, NONE)
yk = rk.y[0, :]
drift = abs(kepler_energy(yk) - kepler_energy(k0)) / abs(kepler_energy(k0))
closure = amax(abs(yk - k0))
print(f"Kepler, 10 periods: {rk.nsteps} steps, {rk.nfev} evaluations, energy drift {drift:.1e}, return error {closure:.1e}")

#: Lotka-Volterra and its invariant
LV_P = [1.5, 1.0, 3.0, 1.0]
def lv_invariant(y: Arr, p: Arr) -> Num {
    return p[3] * y[0] - p[2] * log(y[0]) + p[1] * y[1] - p[0] * log(y[1])
}
def prey_at_five(t: Num, y: Arr, p: Arr) -> Arr {
    return [y[0] - 5.0]
}
lv0 = [10.0, 5.0]
LV_END = 15.0
cl = ode_cfg(DOPRI5, LV_END, 1.0e-10, 1.0e-12 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
rl = ode_solve(lotka, no_jac, prey_at_five, 0.0, LV_END, lv0, LV_P, cl, NONE, [1.0], [0.0])
nl = len(rl.t)
lv_drift = abs(lv_invariant(rl.y[nl - 1, :], LV_P) - lv_invariant(lv0, LV_P))
print(f"Lotka-Volterra: {rl.nsteps} steps, invariant drift {lv_drift:.1e}, prey rose through 5 at {rl.t_events}")
# the event's state has the prey at 5, and its time is between two output steps
ev_miss = amax(abs(rl.y_events[:, 0] - 5.0))

#: dense output: y' = -y + sin t, y(0) = 1: y = 1.5 e^-t + (sin t - cos t) / 2
def forced(t: Num, y: Arr, p: Arr) -> Arr {
    return [-y[0] + sin(t)]
}
def forced_row(y: Arr, t: Num, p: Arr, i) -> Num {
    return forced(t, y, p)[i]
}
def forced_jac(t: Num, y: Arr, p: Arr) -> Arr {
    J = zeros(1, 1)
    J[0, :] = grad(forced_row, y, t, p, 0)
    return J
}
FORCED_END = 10.0
# output every 0.1 time units
FORCED_POINTS: Int = 101
tq = linspace(0.0, FORCED_END, FORCED_POINTS)
exact = 1.5 * exp(-tq) + 0.5 * (sin(tq) - cos(tq))
dense_methods = [DOPRI5, RADAU, BDF]
dense_err = zeros(len(dense_methods))
mi = 0
for mm in dense_methods {
    m: Int = int(mm)
    cf = ode_cfg(m, FORCED_END, 1.0e-9, 1.0e-12 * ones(1), 0.0, inf, MAX_STEPS, JAC_GIVEN)
    rf = ode_solve(forced, forced_jac, no_events, 0.0, FORCED_END, [1.0], NO_PARAMS, cf, tq, NONE, NONE)
    dense_err[mi] = amax(abs(rf.y[:, 0] - exact))
    mi += 1
}
print(f"dense output on 101 points, max error: dopri5 {dense_err[0]:.1e}, radau {dense_err[1]:.1e}, bdf {dense_err[2]:.1e}")

#: the bouncing ball: dropped from 10 m, it lands at sqrt(2 h / g)
G = 9.81
DROP = 10.0
cb = ode_cfg(DOPRI5, 10.0, 1.0e-6, 1.0e-9 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
rb = ode_solve(ball, no_jac, ball_hits, 0.0, 10.0, [DROP, 0.0], [G], cb, NONE, [-1.0], [1.0])
landing = sqrt(2.0 * DROP / G)
print(f"ball: lands at {rb.t_events[0]:.15f} (exact {landing:.15f}), at {rb.y_events[0, 1]:.6f} m/s; status {rb.status}")

#: a batch: 40 oral doses of 100 mg, absorption and elimination rates varied
NB: Int = 40
DOSE = 100.0
Y0 = zeros(NB, 2)
# the parameters: absorption rate ka, elimination rate ke, volume V
PK_PARAMS: Int = 3
P = zeros(NB, PK_PARAMS)
for b to NB {
    Y0[b, 0] = DOSE
    P[b, :] = [0.5 + 0.05 * b, 0.1 + 0.004 * b, 20.0]
}
PK_END = 24.0
# output every half hour
PK_POINTS: Int = 49
tpk = linspace(0.0, PK_END, PK_POINTS)
cp = ode_cfg(RADAU, PK_END, 1.0e-9, 1.0e-12 * ones(2), 0.0, inf, MAX_STEPS, JAC_FD)
Ypk, spk, fpk = ode_batch(pk, no_jac, no_events, 0.0, PK_END, Y0, P, cp, tpk, NONE, NONE)
#| Bateman's concentration at every output time, for one row's parameters.
def bateman_curve(t: Arr, dose: Num, p: Arr) -> Arr {
    return dose * p[0] / (p[2] * (p[0] - p[1])) * (exp(-p[1] * t) - exp(-p[0] * t))
}
pk_dev = zeros(NB)
for b to NB {
    conc = reshape(Ypk[b, :], len(tpk), 2)[:, 1]
    pk_dev[b] = amax(abs(conc - bateman_curve(tpk, DOSE, P[b, :])))
}
pk_err = amax(pk_dev)
print(f"batch of {NB} PK models: max |C - Bateman| = {pk_err:.1e} mg/L, {sum(spk[:, 4])} steps in all")

#: AUTO on Van der Pol, eps = 1e-6: Dormand-Prince until the stiffness test fires
VDP_EPS = [1.0e-6]
ca = ode_cfg(AUTO, 2.0, 1.0e-6, 1.0e-6 * ones(2), 0.0, inf, MAX_STEPS, JAC_GIVEN)
ra = ode_solve(vdpol, vdpol_jac, no_events, 0.0, 2.0, [2.0, 0.0], VDP_EPS, ca, [2.0], NONE, NONE)
cr = ode_cfg(RADAU, 2.0, 1.0e-10, 1.0e-10 * ones(2), 0.0, inf, MAX_STEPS, JAC_GIVEN)
rr = ode_solve(vdpol, vdpol_jac, no_events, 0.0, 2.0, [2.0, 0.0], VDP_EPS, cr, [2.0], NONE, NONE)
auto_diff = amax(abs(ra.y[0, :] - rr.y[0, :]))
print(f"auto: stiff at t = {ra.stiff_at:.4f}, {ra.nsteps} steps, differs from Radau at rtol 1e-10 by {auto_diff:.1e}")

test "the Kepler orbit keeps its energy and closes after ten periods" {
    assert_lt(drift, 2.0e-9)
    assert_lt(closure, 5.0e-7)
}
test "Lotka-Volterra keeps its invariant, and the prey crosses 5 upward once per cycle" {
    assert_lt(lv_drift, 1.0e-8)
    assert_near(len(rl.t_events), 3.0, 0.0)
    assert_lt(ev_miss, 1.0e-9)
}
test "the dense output of every method is accurate between steps" {
    assert_lt(amax(dense_err), 1.0e-7)
}
test "the ball's landing is a terminal event at sqrt(2 h / g)" {
    assert_near(rb.t_events[0], landing, 1.0e-12)
    assert_near(rb.status, TERMINAL_EVENT, 0.0)
    assert_near(rb.t[len(rb.t) - 1], landing, 0.0)
}
test "a batch of PK models matches the Bateman solution" {
    assert_lt(pk_err, 1.0e-6)
}
test "AUTO detects stiffness and its answer is Radau's" {
    assert_gt(ra.stiff_at, 0.0)
    assert_lt(auto_diff, 1.0e-4)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors