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AdHImEx: Adaptively High-Order Implicit-Explicit Transport for Large Time Steps
Amber J. te Winkel, Hilary Weller, Christian Kühnlein, James Kent
TL;DR
Existing AdImEx transport schemes become first-order accurate in the large-time-step implicit regime and may require multiple matrix solves per step. AdHImEx combines explicit and implicit Runge-Kutta stepping with a stage-dependent divergence treatment. Tests report improved accuracy, accurate transport with large local Courant numbers, and potential efficiency gains over explicit transport.
Problem
Existing AdImEx schemes become first-order accurate in the implicit regime at large time steps and may require multiple implicit solves per time step.
Source: §5 · p. 20Method
AdHImEx combines high-order explicit and implicit stepping with a stage-dependent divergence treatment and a conservative fifth-order finite-volume discretisation.
Source: §2.2 · p. 6, §5 · p. 20Results
One-dimensional tests show improved accuracy over WKS24 at large Courant numbers, while two-dimensional tests maintain explicit-reference accuracy despite large local Courant numbers.
Source: §5 · p. 20Takeaways & Limitations
Efficiency tests indicate AdHImEx can reduce overall computational cost compared with explicit transport, despite higher cost per step as the implicit region grows.
Source: §5 · p. 20The efficiency tests use nonoptimised code without matrix preconditioning, and the authors recommend exploring cheaper, lower-order implicit spatial discretisation.
Source: §4.3 · p. 17
Abstract
from arXiv · showhide
Adaptively Implicit-Explicit (AdImEx) time stepping provides stability for large time steps for mass-conservative transport, making it attractive for the numerical representation of advection in weather and climate prediction. Existing AdImEx transport schemes become first-order accurate in the large-time-step, implicit regime and may require multiple sparse matrix solutions per time step. This work introduces AdHImEx, a new AdImEx scheme that addresses both limitations simultaneously. AdHImEx provides second-order accuracy for large Courant numbers while requiring only a single matrix solution per time step. It is a Runge-Kutta scheme that blends third-order accurate explicit time stepping and Crank-Nicolson implicit time stepping to provide numerically verified stability for Courant numbers up to 100. AdHImEx is designed for flows that are predominantly explicit, activating implicit time stepping only where large Courant numbers occur. It retains the full efficiency and third-order accuracy of the explicit time stepping in regions with small Courant numbers. In this work, it is combined with a fifth-order accurate finite-volume discretisation in space for mass conservation. A second contribution of this work is a novel stage-dependent treatment of the divergence operator that preserves constancy despite different implicit and explicit Runge-Kutta stage time steps and spatially varying implicitness, removing spurious divergence. This innovation is applicable to other AdImEx time stepping schemes. Convergence, transport, and efficiency tests demonstrate substantial improvements over previous first-order AdImEx schemes, little phase and amplitude error, accurate transport on locally refined meshes with large Courant numbers, and promising efficiency characteristics for atmospheric transport.
1 Introduction
Advection schemes must balance stability, mass conservation, accuracy, and efficiency, especially on arbitrary grids. AdHImEx targets large-Courant stability with higher implicit accuracy and fewer solves, while a stage-dependent divergence treatment preserves constancy.
1 Introduction
- Explicit Eulerian transport can become unstable as the Courant number rises, while implicit schemes may remove that stability restriction but require a matrix solve.Longer time steps can improve efficiency, but explicit stability limits can make them impractical, particularly when local mesh refinement forces a restrictive global step.Source: §1 · p. 2
- AdImEx blends explicit and implicit Eulerian stepping, activating implicit stepping where needed to retain stability while potentially preserving explicit accuracy and efficiency elsewhere.With finite-volume spatial discretisation, it can be conservative and work on arbitrary grids; the paper notes potential overall efficiency gains with a suitable sparse matrix solver.Source: §1 · p. 2
- Earlier AdImEx schemes used first-order implicit stepping, and Weller et al. [2024] required multiple matrix solutions per time step.The introduction identifies these as limitations that AdHImEx is designed to address.Source: §1 · p. 3
- AdHImEx combines second-order temporal accuracy at large Courant numbers with one implicit solve per time step and a conservative fifth-order finite-volume discretisation.The paper also reports third- or second-order temporal accuracy and stability for large Courant numbers.Source: §1 · p. 3
- Spatially varying implicitness and different explicit and implicit stage time steps can introduce spurious divergence and destroy constancy preservation.The proposed stage-dependent divergence treatment preserves both conservation and constancy and is applicable beyond AdHImEx.Source: §1 · p. 3
2 Numerical Methods
The numerical methods blend explicit and implicit Runge-Kutta stepping, use a stage-dependent divergence treatment, and pair the time integration with fifth-order finite-volume spatial discretisation. Stability is assessed numerically, alongside limiting and comparisons with WKS24.
2.1 AdHImEx Time Stepping
- AdHImEx blends a three-stage explicit Runge-Kutta scheme with two-stage Crank-Nicolson implicit stepping through a Courant-dependent implicitness factor.The blending factor varies in space and time; at full implicitness, only the final stage requires an implicit solve.Source: §2.1 · pp. 4–5
2.2 Constancy Preservation
- Spatially varying Runge-Kutta coefficients at stages 2 and 3 can create spurious divergence, so their coefficients are taken outside the divergence operator to preserve constancy.The final update remains locally mass-conservative, and the approach also applies to a broader class of AdImEx schemes.Source: §2.2 · pp. 5–6
2.3 Spatial Discretisation
- The method uses an upwind-biased fifth-order finite-volume spatial discretisation, selected for stability with AdHImEx time stepping.The same spatial discretisation is used for explicit and implicit parts; the fifth-order assumption is made on uniform grids, although the method also works on nonuniform grids.Source: §2.3 · p. 7
2.4 Stability Analysis
- Von Neumann analysis checks amplification over wavenumbers, and the chosen implicitness is numerically verified stable for discrete Courant numbers up to 100.For spatially varying Courant numbers and implicitness, stability is verified empirically rather than analytically.Source: §2.4 · pp. 7–8, Figure 1 · p. 8
2.5 AdImEx Flux-Corrected Transport
- AdImEx FCT enforces monotonicity but can be highly diffusive at large Courant numbers; NN FCT instead enforces nonnegativity using global bounds.The two limiters differ in their locally allowable extrema.Source: §2.5 · pp. 8–9
2.6 The WKS24 Scheme
- The WKS24 comparator is second-order in time for C ≤ 2, but becomes first-order for C > 2 and is third-order in space.It is a predictor-corrector scheme with two iterations.Source: §2.6 · p. 9
2.7 The Matrix Solution
- AdHImEx requires one sparse linear-system solution, while WKS24 requires two; the study uses direct solves in one dimension and GMRES(m) in two dimensions.The code is unoptimized and uses no matrix preconditioning.Source: §2.7 · p. 9
3 One-dimensional Results
The 1D tests assess uniform-flow accuracy across Courant numbers, convergence order, and the effect of constancy preservation in nonuniform flow.
3.1 Uniform Advection
- Without FCT, AdHImEx is less diffusive and has smaller phase and amplitude errors than WKS24 for large C.Figure 2 compares the schemes for C from 0.4 to 4.Source: §3.1 · p. 10, Figure 2 · p. 11
- AdImEx FCT grows increasingly diffusive for C > 1, while NN FCT tracks the exact solution better but may create spurious extrema at large C.Independent in- and outflux limiting can over-limit high-order fluxes as C grows.Source: §3.1 · pp. 10–11
3.2 Convergence
- AdHImEx shifts from fifth-order spatial accuracy at small C to second-order for C > 1.4, yet remains more accurate than WKS24 at C = 2.5–10.Figure 3 presents ℓ2 error norms over grid spacing for both schemes.Source: §3.2 · pp. 11–12, Figure 3 · p. 12
3.3 Impact of Constancy Preservation
- Constancy preservation removes stage artefacts near interfaces where θ switches, reducing ℓ2 error about threefold (6.3 × 10−4 versus 1.9 × 10−3).Without the treatment, artefacts in stages 2 and 3 remain at the end of the time step.Source: §3.3 · p. 13, Figure 4 · p. 13
4 Two-dimensional Results
The two-dimensional tests assess AdHImEx on locally refined meshes, for constancy preservation, computational efficiency, and strong filament deformation. They report accurate transport at large Courant numbers, exact constancy preservation with the stage-dependent treatment, and reduced runtime in tested cases.
4.1 BK25 Nondivergent Deformational Flow
- With maximum Courant number rising from approximately 0.8 to more than 6, AdHImEx shows no visible deterioration against the explicit reference, with or without FCT limiting.The locally refined mesh enables large local Courant numbers without the time-step restrictions normally associated with explicit Eulerian advection.Source: §4.1 · pp. 14–15, Figure 5 · p. 14
4.2 Constancy Preservation
- Using the constancy-preserving treatment, the initially uniform field remains exact; without it, the field becomes nonuniform where θ varies in space.The comparison uses (8) without constancy preservation and (14) with it.Source: §4.2 · p. 15, Figure 6 · p. 16
4.3 Efficiency
- At Cmax ≳3, total runtime falls below 20 s while the ℓ2 error remains below 10^-4 for Cmax ≲8.The fully explicit case at ∆t = 0.02 s takes around 60 s; the timing covers the advection scheme only. Results use nonoptimised code and an unpreconditioned matrix, and the authors suggest a cheaper, lower-order spatial discretisation may improve solver cost.Source: §4.3 · pp. 16–17, Figure 7 · p. 17
4.4 Hadley-like Circulation
- With Courant numbers approaching 10, AdHImEx closely reproduces the explicit solution under strong filament deformation when fifth-order spatial accuracy is maintained throughout the domain.The mid-time solution exceeds the original bounds more than the explicit reference, but this largely reverts by the end of the test.Source: §4.4 · pp. 19–20, Figure 8 · p. 19
5 Summary and Conclusions
AdHImEx improves large-Courant transport accuracy while reducing implicit solves, and a stage-dependent divergence treatment preserves conservation and constancy across spatially varying implicitness. Tests report improved transport and efficiency, while identifying limitations in flux correction, solver cost, and test scope.
5 Summary and Conclusions
- AdHImEx achieves second-order temporal accuracy at large Courant numbers with one implicit solve per time step, alongside a conservative fifth-order finite-volume discretisation.Source: §5 · p. 20
- A stage-dependent divergence treatment preserves conservation and constancy despite spatially varying implicitness and different explicit and implicit stage time steps.It avoids spurious divergence and associated transport artefacts, and applies to other AdImEx schemes.Source: §5 · p. 20
- One-dimensional tests confirmed expected third- and second-order behavior and improved large-Courant accuracy over WKS24; two-dimensional tests maintained explicit-reference accuracy with large local Courant numbers.On a BK25 nonuniform grid, local refinement was introduced without the restrictive time-step reductions normally associated with explicit Eulerian transport.Source: §5 · p. 20
- Efficiency experiments found that fewer time steps can reduce overall computational cost despite higher per-step cost as the implicit region grows.The results indicate potential efficiency gains over explicit transport schemes.Source: §5 · p. 20
- AdImEx FCT becomes increasingly inaccurate at large Courant numbers, approaching the underlying first-order bounded scheme.NN FCT retains more high-order accuracy but may introduce artefacts.Source: §5 · p. 20
- Compared with AdLImHEx, AdHImEx's accuracy and solve-count gains come at the cost of a more expensive matrix solution.Combining AdHImEx accuracy with AdLImHEx solver performance is identified as future work.Source: §5 · p. 20
- The evaluation is limited to one- and two-dimensional planar tests; spherical geometries, arbitrary meshes, and density-weighted transport remain future work.Source: §5 · p. 20