A priori prediction of the spurious weak-detonation regime in fractional-step shock-capturing computation of stiff detonations

When a detonation wave in a reactive-flow system with a stiff source term is computed by a fractional-step (operator-splitting) shock-capturing scheme on an under-resolved grid, the physical strong detonation can degenerate into a spurious pattern: a fast weak detonation followed by a slower shock, propagating at the wrong speed. Zhang and Wang (J. Comput. Phys. 291:52-59, 2015) explained the mechanism in two representative one-dimensional models: the numerically generated intermediate state between the weak detonation and the shock controls the speed gap $DS=S_{wd}-S_s$ (weak-detonation speed minus shock speed), and the spurious bifurcation occurs exactly when $DS>0$. They further showed the counter-intuitive fact that $DS$, and hence the propagation error, oscillates non-monotonically as the time step or the mesh is refined. The note itself, however, left the predictive content open: the dependence of the intermediate state on the discretization parameters was not described by any analytical model, evaluating the criterion required a post-hoc exact Riemann solve of the specific problem, and only two particular 1D cases were treated. The open problem is to establish a general a priori predictive criterion (ideally an analytical bound) that determines, from the discretization (advection scheme and limiter, time step/CFL, mesh), the source-term model, and the initial data, whether the computation enters the spurious regime, and that analytically accounts for the observed oscillatory refinement behavior; how far the intermediate-state mechanism extends to other schemes, more complex combustion models, and multidimensional settings remains an open direction of the problem. Subsequent work has introduced corrective schemes (the threshold values method, split random time-stepping, deterministic projection, subcell resolution, and boundary-variation-diminishing (BVD) hybrid schemes, which blend two reconstructions to minimize the variation at cell interfaces) rather than a general predictive criterion, so the analytical core remains open.

Background

A detonation is a combustion wave in which a shock and a coupled exothermic reaction propagate together. In the reactive Euler equations, a single reaction-progress variable tracks unburnt-to-burnt conversion through a stiff source term (for example an Arrhenius rate or an ignition-temperature switch with a large rate constant $Da$, the Damköhler number measuring reaction speed relative to fluid motion). The reaction zone behind the front is then extremely thin, and practical computations are frequently under-resolved: the cell size and time step exceed the reaction-zone width and time.

The standard discretization is the fractional-step or operator-splitting approach (in its Strang form: half a reaction substep, a full advection substep, half a reaction substep): the homogeneous hyperbolic system is advanced by a shock-capturing scheme — a conservative finite-volume/difference method whose built-in numerical dissipation spreads a discontinuity over a few cells, with the limiter (flux/slope limiter, e.g. TVD total-variation-diminishing limiting or AUSM+TVD — the advection upstream splitting method for the inviscid fluxes paired with a TVD limiter) controlling that spreading — and the stiff reaction ODE is integrated cell by cell. Since Colella, Majda and Roytburd (1986) and LeVeque and Yee (1990), a well-known failure mode of this combination is known: the advection substep smears the front, the reaction substep then acts prematurely on the smeared, non-physical intermediate states, and the strong detonation degenerates into a bifurcating pattern consisting of a weak detonation (a combustion wave on the inadmissible weak branch of the detonation solution family — the Chapman-Jouguet detonation is the minimum-speed self-sustaining detonation, at which the burnt gas is sonic relative to the front, and both the admissible strong branch and the weak branch travel at or above that speed, the weak branch being excluded for these single-step irreversible reaction models) followed by a shock, so the front propagates at the wrong speed. Extensive remedies exist — front tracking and level-set methods, Helzel-LeVeque-Warnecke's modified fractional step, Bao and Jin's random projection method, Tosatto and Vigevano's MinMax reconstruction, Wang-Shu-Yee-Sjögreen's subcell resolution, the equilibrium state method of Zhang, Liu, Chen and Wang — each with documented limitations in stiffness range, dimensionality, or kinetics complexity.

Zhang and Wang's 2015 note contributed a mechanism for two representative 1D models: a simplified $2\times2$ system (one velocity/one temperature variable with ignition at temperature $T_{ig}$ and heat release $Q$, computed with a second-order UNO (uniformly non-oscillatory slope reconstruction) scheme) and the reactive Euler equations ($\gamma=1.4$, heat release $q_0=20$, $Da=10^4$, $T_{ig}=2$, computed with AUSM+TVD, with standard strong-detonation Riemann initial data (two constant states separated by a single discontinuity): left burnt gas $\rho=2$, $u=4$, $p=20$; right unburnt gas $\rho=1$, $u=0$, $p=1$). After each split step the smeared front contains a numerically generated intermediate state; applying the Rankine-Hugoniot jump conditions (conservation relations across a discontinuity) to this state yields the weak-detonation speed $S_{wd}$ and the following shock speed $S_s$. Their criterion: the spurious bifurcating pattern occurs exactly when the speeds mismatch, $DS=S_{wd}-S_s>0$; for the $2\times2$ system $DS$ is an explicit function of the intermediate temperature $T_x$, namely $S_s=(T_b+T_x)/2$ and $S_{wd}=T_x^2/(2(T_x-Q))$ with $T_b$ the burnt-state temperature. The counter-intuitive behaviors follow: sweeping $\Delta t$ at fixed mesh, $DS$ is nearly constant for $\Delta t<10^{-5}$ but oscillates for larger $\Delta t$; sweeping the grid number $N$, $DS$ decreases only on average with oscillations (crossing zero near $N\approx8950$ in the $2\times2$ case, where the spurious pattern disappears). Hence smaller time steps or finer meshes can make the spurious behavior worse, and the error is non-monotone under refinement. Crucially, the note establishes the mechanism only as a diagnosis: the intermediate state is read off the computed (or exact-Riemann) solution, and the note states that the causal relationship between the discretization parameters ($\Delta t$, $N$) and the intermediate-state parameters has not been substantiated by an analytical model and that no general criterion exists for other fractional-step schemes, source models, or reactive systems with discontinuities; extension to other limiters, more complex combustion models, and multidimensional settings remains open.

Problem Statement

For a reactive-flow system with a stiff source term advanced by a fractional-step shock-capturing scheme, establish a general predictive criterion — ideally an analytical bound — that determines, a priori from the discretization (advection scheme and limiter, time step or CFL number — the Courant–Friedrichs–Lewy number, the ratio of the time step to the wave-transit time across a cell — and the mesh), the source-term model, and the initial data, whether the computation enters the spurious regime in which the strong detonation bifurcates into a weak detonation followed by a shock; equivalently, predict the sign of the speed gap $DS=S_{wd}-S_s$ associated with the numerically generated intermediate state, without solving the exact Riemann problem of the specific problem post hoc. The resulting criterion and its analytical model of the intermediate state must be consistent with, and account for, the counter-intuitive behavior documented by Zhang and Wang (2015) in the two one-dimensional seed models — the simplified $2\times2$ system ($Da=200$, $Q=2$, $T_{ig}=0.05$) and the reactive Euler equations ($\gamma=1.4$, $q_0=20$, $Da=10^4$, $T_{ig}=2$): $DS$ nearly constant for small time steps, oscillatory for larger time steps, and only on average decreasing, with oscillations, under grid refinement. The note established the intermediate-state mechanism only for these two one-dimensional seed models; the required deliverable is the general criterion stated above, expressed in terms of the advection scheme and limiter, the time step or CFL number, the mesh, the source-term model, and the initial data, so that the two seed schemes are special cases, and demonstrated on these two seed models. Establishing the criterion for further schemes and limiters, more complex combustion kinetics, and multidimensional reactive flows with discontinuities — directions whose difficulty is documented in the subsequent literature — remains an open direction of the problem beyond that deliverable. The overall objective is to turn the note's post-hoc mechanistic diagnosis into a predictive theory of the spurious regime of under-resolved stiff reacting-flow computation.

References

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