Verner 7 Integrator (#1)

Co-authored-by: Connor Johnstone <connor.johnstone@arcfield.com>
Reviewed-on: #1

readme.md

@@ -6,22 +6,34 @@ and field line tracing:
 
 ## Features
 
-- A relatively efficient Dormand Prince 5th(4th) order integration algorithm, which is effective for
-    non-stiff problems
-- A PI-controller for adaptive time stepping
-- The ability to define "callback events" and stop or change the integator or underlying ODE if
-    certain conditions are met (zero crossings)
-- A fourth order interpolator for the Domand Prince algorithm
-- Parameters in the derivative and callback functions
+### Explicit Runge-Kutta Methods (Non-Stiff Problems)
 
+| Method | Order | Stages | Dense Output | Best Use Case |
+|--------|-------|--------|--------------|---------------|
+| **BS3** (Bogacki-Shampine) | 3(2) | 4 | 3rd order | Moderate accuracy (rtol ~ 1e-4 to 1e-6) |
+| **DormandPrince45** | 5(4) | 7 | 4th order | General purpose (rtol ~ 1e-6 to 1e-8) |
+| **Vern7** (Verner) | 7(6) | 10+6 | 7th order | High accuracy (rtol ~ 1e-8 to 1e-12) |
+
+**Performance at 1e-10 tolerance:**
+- Vern7: **2.7-8.8x faster** than DP5
+- Vern7: **50x+ faster** than BS3
+
+See [benchmark report](VERN7_BENCHMARK_REPORT.md) for detailed performance analysis.
+
+### Other Features
+
+- **Adaptive time stepping** with PI controller
+- **Callback events** with zero-crossing detection
+- **Dense output interpolation** at any time point
+- **Parameters** in derivative and callback functions
+- **Lazy computation** of extra interpolation stages (Vern7)
 
 ### Future Improvements
 
 - More algorithms
-    - Rosenbrock
-    - Verner
-    - Tsit(5)
-    - Runge Kutta Cash Karp
+    - Rosenbrock methods (for stiff problems)
+    - Tsit5
+    - Runge-Kutta Cash-Karp
 - Composite Algorithms
 - Automatic Stiffness Detection
 - Fixed Time Steps