Implicit–explicit time stepping

The split

After spatial discretisation an initial-value problem becomes, in each group,

\[ M \frac{dX}{dt} + L X = F(X, t), \]

where \(X\) holds the spectral coefficients and the taus, \(M\) and \(L\) are the sparse matrices of the linear terms (the boundary and gauge rows have no \(M\) part), and \(F\) collects the nonlinear and forcing terms, evaluated pseudospectrally. Stiff linear terms (viscosity, diffusion, pressure) are treated implicitly and \(F\) explicitly.

In the native layer the split follows one rule: a term that is linear in the unknowns, with constant, parameter or given-field coefficients, goes to \(L\); a nonlinear term or a source goes to \(F\); each term moves across the equals sign with its sign. explicit(...) and implicit(...) in the equation text override the rule for one term (an implicit nonlinear term is refused). explain() lists where every term went.

Schemes

Two families are available, with Dedalus’s names so that results can be compared:

  • IMEX Runge–Kutta RK111, RK222, RK443 (Ascher, Ruuth and Spiteri 1997). A stage reads

    \[ (M + k H_{ii} L)\, X_i = M X_0 + k \sum_{j<i} A_{ij} F(X_j, t_j) - k \sum_{j<i} H_{ij} L X_j , \]
    with \((A, H)\) the explicit and implicit tableaux. In these tableaux all \(H_{ii}\) are equal, so one factorisation of \(M + k H_{ii} L\) per group serves every stage of the run.

  • Semi-implicit BDF SBDF1 to SBDF4 (Wang and Ruuth 2008): multistep, one solve per step; the first steps are taken with a declared start-up procedure, printed by explain().

The time step is fixed and chosen by you. A CFL estimate (sim.cfl("u", safety=0.5), the safety factor yours) is an indicator of stability, not a control of the time error; asking a run to stop at a time that is not a whole number of steps is refused rather than silently changing the last step.

Factorisations

Each group’s matrix \(M + k H L\) is banded with a border (the tau columns and the boundary rows). It is factorised once by a banded LU with the border eliminated through a small Schur complement (capacitance) system; all groups are solved together in one batched call, on the CPU or on the GPU.

Watching the taus

The tau variables absorb the residual of the truncated equations. Their size along a run is a cheap diagnostic of resolution: a growing tau means the top modes are not resolved. Conatus can monitor them (TauWatch) and warn above a declared threshold.

Exact restart

The state of a run (coefficients, time, step count and, for multistep schemes, the history) is written in full, so that a restarted run continues bit for bit.

References

  • U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997) 151–167.

  • D. Wang and S. J. Ruuth, Variable step-size implicit-explicit linear multistep methods for time-dependent partial differential equations, J. Comput. Math. 26 (2008) 838–855.

  • K. J. Burns et al., Phys. Rev. Research 2 (2020) 023068.