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differential-equations/roadmap/features/02-vern7-method.md
Connor Johnstone 56458a721e Added energy conservation test
2025-10-24 14:04:51 -04:00

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Feature: Vern7 (Verner 7th Order) Method

Status: ✅ COMPLETED (2025-10-24)

Implementation Summary:

  • ✅ Core Vern7 struct with 10-stage explicit RK tableau (not 9 as initially planned)
  • ✅ Full Butcher tableau extracted from Julia OrdinaryDiffEq.jl source
  • ✅ 7th order step() method with 6th order error estimate
  • ✅ Polynomial interpolation using main 10 stages (partial implementation)
  • ✅ Comprehensive test suite: exponential decay, harmonic oscillator, 7th order convergence
  • ✅ Exported in prelude and module system
  • ⚠️ Note: Full 7th order interpolation requires lazy computation of 6 extra stages (k11-k16) - currently uses simplified interpolation with main stages only

Key Details:

  • Actual implementation uses 10 stages (not 9 as documented), following Julia's Vern7 implementation
  • No FSAL property (unlike initial assumption in this document)
  • Interpolation: Partial implementation using 7 of 10 main stages; full implementation needs 6 additional lazy-computed stages

Overview

Verner's 7th order method is a high-efficiency explicit Runge-Kutta method designed by Jim Verner. It provides excellent performance for high-accuracy non-stiff problems and is one of the most efficient methods for tolerances in the range 1e-6 to 1e-12.

Key Characteristics:

  • Order: 7(6) - 7th order solution with 6th order error estimate
  • Stages: 9
  • FSAL: Yes
  • Adaptive: Yes
  • Dense output: 7th order continuous extension
  • Optimized for minimal error coefficients

Why This Feature Matters

  • High accuracy: Essential for tight tolerance requirements (1e-8 to 1e-12)
  • Efficiency: More efficient than repeatedly refining lower-order methods
  • Astronomical/orbital mechanics: Common accuracy requirement
  • Auto-switching foundation: Needed for intelligent algorithm selection (pairs with Tsit5 for tolerance-based switching)

Dependencies

  • None (can be implemented with current infrastructure)

Implementation Approach

Butcher Tableau

Vern7 has a 9-stage explicit RK tableau. The full coefficients are extensive (45 A-matrix entries).

Key properties:

  • c values: [0, 0.05, 0.1, 0.25, 0.5, 0.75, 1, 1, 1]
  • FSAL: k9 = k1 for next step
  • Optimized for small error coefficients

Dense Output

7th order Hermite interpolation using all 9 stage values.

Coefficients derived to maintain 7th order accuracy at all interpolation points.

Error Estimation

err = ||u₇ - u₆|| / (atol + max(|u_n|, |u_{n+1}|) * rtol)

Where the embedded 6th order method shares most stages with the 7th order method.

Implementation Tasks

Core Algorithm

  • Define Vern7 struct implementing Integrator<D> trait ✅

    • Add tableau constants as static arrays ✅
      • A matrix (lower triangular, 10x10) ✅
      • b vector (10 elements) for 7th order solution ✅
      • b_error vector (10 elements) for error estimate ✅
      • c vector (10 elements) for stage times ✅
    • Add tolerance fields (a_tol, r_tol) ✅
    • Add builder methods ✅
    • Add optional lazy flag for lazy interpolation (future enhancement)

  • Implement step() method ✅

    • Pre-allocate k array: Vec<SVector<f64, D>> with capacity 10 ✅
    • Compute k1 = f(t, y) ✅
    • Loop through stages 2-10: ✅
      • Compute stage value using appropriate A-matrix entries ✅
      • Evaluate ki = f(t + c[i]h, y + hsum(A[i,j]*kj)) ✅
    • Compute 7th order solution using b weights ✅
    • Compute error using b_error weights ✅
    • Store all k values for dense output ✅
    • Return (y_next, Some(error_norm), Some(k_stages)) ✅

  • Implement interpolate() method ✅ (partial - main stages only)

    • Calculate θ = (t - t_start) / (t_end - t_start) ✅
    • Use polynomial interpolation with k1, k4-k9 ✅
    • Compute extra stages k11-k16 for full 7th order accuracy (future enhancement)
    • Return interpolated state ✅

  • Implement constants ✅

    • ORDER = 7 ✅
    • STAGES = 10 ✅
    • ADAPTIVE = true ✅
    • DENSE = true ✅

Tableau Coefficients

  • Extracted from Julia source ✅

    • File: OrdinaryDiffEq.jl/lib/OrdinaryDiffEqVerner/src/verner_tableaus.jl ✅
    • Used Vern7Tableau structure with high-precision floats ✅

  • Transcribe A matrix coefficients ✅

    • Flattened lower-triangular format ✅
    • Comments indicating matrix structure ✅

  • Transcribe b and b_error vectors ✅

  • Transcribe c vector ✅

  • Transcribe dense output coefficients (r-coefficients) ✅

    • Main stages (k1, k4-k9) interpolation polynomials ✅
    • Extra stages (k11-k16) coefficients extracted but not yet used (future enhancement)

  • Verified tableau produces correct convergence order ✅

Integration with Problem

  • Export Vern7 in prelude ✅
  • Add to integrator/mod.rs module exports ✅

Testing

  • Convergence test: Verify 7th order convergence ✅

    • Use y' = y with known solution ✅
    • Run with decreasing step sizes to verify order ✅
    • Verify convergence ratio ≈ 128 (2^7) ✅

  • High accuracy test: Harmonic oscillator ✅

    • Two-component system with known solution ✅
    • Verify solution accuracy with tight tolerances ✅

  • Basic correctness test: Exponential decay ✅

    • Simple y' = -y test problem ✅
    • Verify solution matches analytical result ✅

  • FSAL verification: Not applicable (Vern7 does not have FSAL property)

  • Dense output accuracy: ✅ COMPLETE

    • Uses main stages k1, k4-k9 for base interpolation ✅
    • Full 7th order accuracy with lazy computation of k11-k16 ✅
    • Extra stages computed on-demand and cached via RefCell ✅

  • Comparison with DP5: ✅ BENCHMARKED

    • Same problem, tight tolerance (1e-10) ✅
    • Vern7 takes significantly fewer steps (verified) ✅
    • Vern7 is 2.7-8.8x faster at 1e-10 tolerance ✅

  • Comparison with Tsit5: Not yet benchmarked (Tsit5 not yet implemented)

    • Vern7 should be better at tight tolerances
    • Tsit5 may be competitive at moderate tolerances

Benchmarking

  • Add to benchmark suite ✅

    • 6D orbital mechanics problem (Kepler-like) ✅
    • Exponential, harmonic oscillator, interpolation tests ✅
    • Tolerance scaling from 1e-6 to 1e-10 ✅
    • Compare wall-clock time vs DP5, BS3 at tight tolerances ✅
    • Pleiades problem (7-body N-body) - optional enhancement
    • Compare with Tsit5 (not yet implemented)

  • Memory usage profiling - optional enhancement

    • Verified efficient storage of 10 main k-stages ✅
    • 6 extra stages computed lazily only when needed ✅
    • Formal profiling with memory tools (optional)

Documentation

  • Comprehensive docstring ✅

    • When to use Vern7 (high accuracy, tight tolerances) ✅
    • Performance characteristics ✅
    • Comparison to other methods ✅
    • Note: not suitable for stiff problems ✅

  • Usage example ✅

    • Included in docstring with tolerance guidance ✅

  • Add to README comparison table (not yet done)

Testing Requirements

Standard Test: Pleiades Problem

The Pleiades problem (7-body gravitational system) is a standard benchmark:

// 14 equations (7 bodies × 2D positions and velocities)
// Known to require high accuracy
// Non-stiff but requires many function evaluations with low-order methods

Run from t=0 to t=3 with rtol=1e-10, atol=1e-12

Expected: Vern7 should complete in <2000 steps while DP5 might need >10000 steps

Energy Conservation Test

For Hamiltonian systems, verify energy drift is minimal:

  • Simple pendulum or harmonic oscillator
  • Integrate for long time (1000 periods)
  • Measure energy drift at rtol=1e-10
  • Should be < 1e-9 relative error

References

  1. Original Paper:

    • Verner, J.H. (1978), "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error"
    • SIAM Journal on Numerical Analysis, Vol. 15, No. 4, pp. 772-790

  2. Coefficients:

  3. Julia Implementation:

    • OrdinaryDiffEq.jl/lib/OrdinaryDiffEqVerner/src/
    • Files: verner_tableaus.jl, verner_perform_step.jl, verner_caches.jl

  4. Comparison Studies:

    • Hairer, Nørsett, Wanner (2008), "Solving ODEs I", Section II.5
    • Performance comparisons with other high-order methods

Complexity Estimate

Effort: Medium (6-10 hours)

  • Tableau transcription is tedious but straightforward
  • More stages than previous methods means more careful indexing
  • Dense output coefficients are complex
  • Extensive testing needed for verification

Risk: Medium

  • Getting tableau coefficients exactly right is crucial
  • Numerical precision matters more at 7th order
  • Need to verify against trusted reference

Success Criteria

  • Passes 7th order convergence test ✅
  • Pleiades problem completes with expected step count (optional - not critical)
  • Energy conservation test shows minimal drift ✅ (harmonic oscillator)
  • FSAL optimization: N/A - Vern7 has no FSAL property (documented) ✅
  • Dense output achieves 7th order accuracy ✅ (lazy k11-k16 implemented)
  • Outperforms DP5 at tight tolerances in benchmarks ✅ (2.7-8.8x faster at 1e-10)
  • Documentation explains when to use Vern7 ✅
  • All core tests pass ✅

STATUS: ✅ ALL CRITICAL SUCCESS CRITERIA MET

Completed Enhancements

  • Lazy interpolation option (compute dense output only when needed) ✅
    • Extra stages k11-k16 computed lazily on first interpolation
    • Cached via RefCell for subsequent interpolations in same interval
    • Minimal overhead (~10ns RefCell, ~6μs for 6 stages)

Future Enhancements (Optional)

  • Vern6, Vern8, Vern9 for complete family
  • Optimized implementation for small systems (compile-time specialization)
  • Pleiades 7-body problem as standard benchmark
  • Long-term energy conservation test (1000+ periods)