Added energy conservation test

src/integrator/vern7.rs

@@ -764,4 +764,59 @@ mod tests {
         // 7th order interpolation should be very accurate
         assert_relative_eq!(y_interp[0], exact, epsilon = 1e-8);
     }
+
+    #[test]
+    fn test_vern7_long_term_energy_conservation() {
+        // Test energy conservation over 1000 periods of harmonic oscillator
+        // This verifies that Vern7 maintains accuracy over long integrations
+        type Params = ();
+
+        fn derivative(_t: f64, y: Vector2<f64>, _p: &Params) -> Vector2<f64> {
+            // Harmonic oscillator: y'' + y = 0
+            // As system: y1' = y2, y2' = -y1
+            Vector2::new(y[1], -y[0])
+        }
+
+        let y0 = Vector2::new(1.0, 0.0);  // Start at maximum displacement, zero velocity
+
+        // Period of harmonic oscillator is 2π
+        let period = 2.0 * std::f64::consts::PI;
+        let num_periods = 1000.0;
+        let t_end = num_periods * period;
+
+        let ode = ODE::new(&derivative, 0.0, t_end, y0, ());
+        let vern7 = Vern7::new().a_tol(1e-10).r_tol(1e-10);
+        let controller = PIController::default();
+
+        let mut problem = Problem::new(ode, vern7, controller);
+        let solution = problem.solve();
+
+        // Check solution at the end
+        let y_final = solution.states.last().unwrap();
+
+        // Energy of harmonic oscillator: E = 0.5 * (y1^2 + y2^2)
+        let energy_initial = 0.5 * (y0[0] * y0[0] + y0[1] * y0[1]);
+        let energy_final = 0.5 * (y_final[0] * y_final[0] + y_final[1] * y_final[1]);
+
+        // After 1000 periods, energy drift should be minimal
+        let energy_drift = (energy_final - energy_initial).abs() / energy_initial;
+
+        println!("Initial energy: {}", energy_initial);
+        println!("Final energy: {}", energy_final);
+        println!("Energy drift after {} periods: {:.2e}", num_periods, energy_drift);
+        println!("Number of steps: {}", solution.times.len());
+
+        // Energy should be conserved to high precision (< 1e-7 relative error over 1000 periods)
+        // This is excellent for a non-symplectic method!
+        assert!(
+            energy_drift < 1e-7,
+            "Energy drift too large: {:.2e}",
+            energy_drift
+        );
+
+        // Also check that we return near the initial position after 1000 periods
+        // (should be back at (1, 0))
+        assert_relative_eq!(y_final[0], 1.0, epsilon = 1e-6);
+        assert_relative_eq!(y_final[1], 0.0, epsilon = 1e-6);
+    }
 }