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ODE Solvers

OASIS provides three families of implicit ODE solvers for time integration. The choice of solver affects accuracy, stability, and computational cost.

Solver Selection

Set timeIntMethod in dataProblem.dat (line 18) or YAML solver.method:

Value Solver Order Adaptive Best for
1 BDF1 2 (internal) Automatic —
2 BDFN 1–6 (configurable) Optional Stiff mooring problems
3 ESDIRK46 4 (embedded 5th) Optional General use (default)

ESDIRK46

Explicit Singly Diagonal Implicit Runge-Kutta method with 6 stages. This is the recommended default solver.

Based on: "Fourth-order Runge–Kutta schemes for fluid mechanics applications" (Carpenter, 2005).

Characteristics:

  • 4th-order main scheme with embedded 5th-order error estimator
  • Handles stiff and non-stiff problems
  • Built-in adaptive time stepping with error control
  • No initialisation required (self-starting)

Configuration:

3               // timeIntMethod (ESDIRK46)
2               // timeIntOrder (unused for ESDIRK)
1               // timeIntAdaptivity (adaptive)
0               // timeIntJacNumStepsMax
1e-5            // timeIntAbsTol
1e-3            // timeIntRelTol
20              // maxIterStep

Adaptivity options (from ESDIRK_adaptive_time_step_data.dat, optional):

Parameter Default Description
adaptivity_type 2 Error estimation method (2 = Noventa 2018)
adaptivity_smooth_flag 2 Smoothing (2 = atan smoothing)
rho_min 2/3 Minimum step multiplier to accept

BDFN

Backward Differentiation Formula with configurable order \(N\) (1 to 6).

Characteristics:

  • Implicit, unconditionally A-stable (order 1-2), A(α)-stable (order 3-6)
  • Higher order = more accurate but potentially less robust
  • Stores \(N+1\) previous states
  • Self-initialises by running \(N\) steps of BDF1

Order coefficients:

Order Formula Stability
1 \(y_{n+1} = y_n + \Delta t \, f_{n+1}\) A-stable
2 \(y_{n+1} = \frac{4}{3}y_n - \frac{1}{3}y_{n-1} + \frac{2}{3}\Delta t \, f_{n+1}\) A-stable
3-6 Higher-order multi-step A(α)-stable

Configuration:

2               // timeIntMethod (BDFN)
3               // timeIntOrder (order 3)
0               // timeIntAdaptivity (fixed step)
0               // timeIntJacNumStepsMax
1e-5            // timeIntAbsTol
1e-3            // timeIntRelTol
20              // maxIterStep

Fixed Time Step Constraints

When timeIntAdaptivity = 0:

  • writeTimeStep must be a multiple of maxTimeStep
  • simulationTime must be a multiple of maxTimeStep

BDF1

First-order BDF (backward Euler). Internally implemented via the BDF2 class.

  • Unconditionally stable but only first-order accurate
  • Use timeIntMethod = 2 with timeIntOrder = 1 for standalone BDF1

Common Parameters

All solvers share these convergence parameters:

Parameter Description Typical value
timeIntAbsTol Absolute error tolerance \(10^{-5}\)
timeIntRelTol Relative error tolerance \(10^{-3}\)
maxIterStep Max Newton iterations per step 20
timeIntJacNumStepsMax Steps between Jacobian recomputation (0 = every step) 0

Solver Guidelines

General dynamics (free bodies + waves):

  • Use ESDIRK46 with adaptivity ON — most robust, excellent error control

Stiff mooring problems:

  • Start with ESDIRK46. If convergence issues, try BDFN order 3–4

Wave excitation analysis (locked bodies):

  • ESDIRK46 with fixed \(\Delta t = 0.1\) s

Convergence Troubleshooting

If the solver fails to converge:

  1. Reduce maxTimeStep by 1–2 orders of magnitude
  2. Increase maxIterStep (e.g., 20 → 50)
  3. Tighten tolerances (timeIntAbsTol \(10^{-5} \to 10^{-6}\))
  4. Change solver order (BDFN: try order 2 or 3)
  5. Switch solver family (ESDIRK46 → BDFN or vice versa)

Debug Output

When debug flags are enabled, the solver writes:

File Columns Description
time_adaptivity_debug.txt \(t\), \(\Delta t\), EWT, LTE, \(\rho\), rejected Adaptive time stepping diagnostics
Example Solver
solvers/bdf1 BDF order 1
solvers/bdfn_order2 BDF order 2
solvers/bdfn_order4 BDF order 4
solvers/esdirk46 ESDIRK 4th order