High-order shock-capturing schemes of the fractional-step (Strang-splitting) type are a standard tool for simulating detonation and other high-speed reacting flows governed by the reactive Euler equations with stiff Arrhenius source terms. Yee, Kotov, Wang and Shu (2013) showed that on underresolved grids these methods exhibit systematic spurious behavior — wrong propagation speed of discontinuities, spurious oscillations, and nonphysical states — with a counter-intuitive dependence on the time step: within the classical CFL limit, the set of time steps yielding the correct shock speed consists of disjoint intervals, error is non-monotone in the CFL number, and smaller time steps can make results worse. Of the open issues the authors listed, two have since been settled by follow-up work: subcell resolution was generalized from single-reaction to multispecies, multireaction flows, and positivity/bound-preserving limiters were characterized for multispecies detonation models. The surviving open core, formulated here as the problem, is (i) a predictive stability theory of the fully discretized split system (the convection scheme, the reaction-step ODE solver, and any safeguard logic) that explains and predicts the disjoint CFL stability intervals and non-monotone error behavior of the standard schemes, and (ii) the construction of stiff-source evaluations compatible with high-order convection operators that demonstrably reduce spurious numerics on underresolved grids.

Background

In a reacting gas the flow is described by the reactive Euler equations: the compressible Euler equations of gas dynamics, written for the conserved variables (density, momentum, energy) together with one or more reaction-progress variables λ\lambda that interpolate between unburnt (λ=1\lambda=1) and burnt (λ=0\lambda=0) states. In one space dimension the system has the form

∂U∂t+∂F(U)∂x=S(U),\frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x} = S(U),

where the source S(U)S(U) contains chemical heat-release rates of Arrhenius type, e.g. −Kλe−Ea/T-K\lambda e^{-E_a/T} with rate constant KK, activation energy EaE_a, and temperature TT. When the reaction time scale is orders of magnitude shorter than the flow time scale the problem is said to be stiff; in a detonation — a combustion wave sustained by the shock it drives — the reaction zone can then be far thinner than any affordable grid cell.

A standard solution strategy is the fractional-step (operator-splitting) approach: each time step is split into a convection substep, solved by a shock-capturing scheme, and a reaction substep, in which the stiff ordinary differential equation Ut=S(U)U_t = S(U) is integrated at each grid point, typically arranged by Strang splitting (convection–reaction–convection with half steps) to retain second-order accuracy. WENO (weighted essentially non-oscillatory) schemes are high-order shock-capturing methods — fifth order is denoted WENO5 — that adaptively combine reconstruction stencils to avoid oscillations near discontinuities. Variants studied by the source authors include WENO5/SR, in which the reaction step is modified by Harten's subcell resolution (SR): the idea of using the smooth reconstructions in the two neighboring cells to relocate the discontinuity inside a cell before applying the source operator; and nonlinear filter schemes (WENO5fi), which apply an explicit dissipation-only correction near shocks, combined with the Ducros-type splitting of the source. The CFL (Courant–Friedrichs–Lewy) number is the dimensionless time step, bounded above by the classical stability limit of the convection scheme.

Since LeVeque and Yee (1990) it has been known that on underresolved grids such methods can converge to a weak solution with the wrong propagation speed of the discontinuity: numerical smearing of the shock introduces nonequilibrium (partially burnt) states that switch the stiff source on prematurely, so the numerical detonation travels at a speed set by the discretization rather than the physics. Griffiths, Stuart and Yee (1992) analyzed model equations with low-order schemes as discrete dynamical systems, uncovering spurious steady states and spurious waves, and Lafon and Yee (1996) found that pointwise evaluation of the source term is the least stable discretization choice. Wang, Shu, Yee and Sjögreen (2012) proposed the high-order subcell-resolution method for advection equations with stiff sources, initially for two-species single-reaction models. Yee, Kotov, Wang and Shu (2013) then performed a systematic CFL-sweep study — on the 1D Chapman–Jouguet (C-J) detonation with Arrhenius kinetics (baseline rate constant K0=16,418K_0 = 16{,}418 with ignition temperature Tign=25T_{\mathrm{ign}} = 25, domain [0,30][0, 30]; stiffness levels K0K_0, 100K0100K_0 and 1000K01000K_0) and on a 2D Heaviside-form detonation in a channel (domain [0,0.025]×[0,0.005][0, 0.025] \times [0, 0.005], γ=1.4\gamma = 1.4) — comparing WENO5, WENO5/SR, WENO5fi+split and WENO5fi/SR+split with pointwise source evaluation. They found that the improved methods delay the onset of wrong-speed behavior on coarse grids but all fail somewhere, and that the time-step dependence is counter-intuitive: the CFL values that avoid spurious solutions form disjoint intervals inside the classical CFL limit, smaller time steps can be worse, and the linearized stability region of the nonhomogeneous system is not a single interval, so in their words the traditional concept of the CFL guideline needs to be revised. Their stated future agenda centered on non-pointwise source evaluation for higher-than-second-order methods, on a dynamical-systems-style analysis of the discretized reactive system, and on diagnostic questions (Roe-average-induced oscillations, coupling of flow sensors with subcell modification, and combining adaptive stiff ODE solvers with subcell resolution).

Follow-up work settled parts of this agenda. Subcell resolution was extended to multispecies and multireaction reactive Euler flows in one and two dimensions by the same group (Wang, Shu, Yee, Kotov, Sjögreen, 2015), and bound- and positivity-preserving limiters were developed and characterized for multispecies and multireaction detonation models (Du, Wang, Qian, Yang 2019; Du and Yang 2023), building on positivity-preserving flux limiters of Hu, Adams and Shu. A family of empirical remedies for wrong-speed propagation also exists: the equilibrium state method (Zhang, Liu, Chen, Wang 2014), the threshold values method with a physical-argument analysis of the spurious wave pattern (Yu, Li, Zhang, Wang 2017), a new stiff-detonation capturing method (Deng, Xie, Xiao, Teng 2018), split random time-stepping (Wang, Pan, Hu, Adams 2019), and deterministic projection methods (Chertock, Chu, Kurganov 2024). None of these, however, supplies a stability theory predicting when the standard fractional-step schemes produce correct or spurious solutions, and the non-pointwise source-evaluation direction has not been developed for high-order reactive-flow schemes.

Problem Statement

For high-order fractional-step (Strang-split) shock-capturing discretizations of the reactive Euler equations with stiff Arrhenius-type source terms — in the benchmark setting of the source study: the 1D Chapman–Jouguet detonation with Arrhenius kinetics and the 2D Heaviside-form detonation in a channel, at stiffness levels K0K_0, 100K0100K_0, 1000K01000K_0 (multiples of the source's baseline reaction-rate constant; for the 1D Arrhenius case K0=16,418K_0 = 16{,}418 with ignition temperature Tign=25T_{\mathrm{ign}} = 25), on grids too coarse to resolve the reaction zone — establish a principled understanding and design of the convection–source coupling, specifically:

  1. Predictive stability theory. Develop a mathematical analysis of the fully discretized split system — the convection scheme, the reaction-step ODE solver, and any safeguard logic such as density/pressure cut-offs or positivity limiters — that explains the counter-intuitive time-step dependence documented by Yee, Kotov, Wang and Shu (2013) and predicts, for standard schemes such as WENO5, WENO5/SR, WENO5fi+split and WENO5fi/SR+split, the stability/accuracy intervals as functions of stiffness and grid size on the benchmark problems. The phenomena to be explained are: the set of CFL values yielding the correct discontinuity speed forming disjoint intervals within the classical CFL limit; non-monotone dependence of the error on the CFL number; smaller time steps sometimes producing more spurious behavior; and scheme-dependent reversals of relative performance between 1D and 2D. A complete theory must also settle the source's narrower diagnostic questions within the same framework: whether the spurious oscillations near discontinuities stem from the use of Roe-average states, and why the nonlinear-filter flow sensor and the subcell modification interact badly in some 2D cases.

  2. Convection-compatible source evaluation. Construct and analyze a non-pointwise evaluation of the stiff source term compatible with higher-than-second-order convection difference operators — the direction the source authors singled out as most promising — that demonstrably reduces wrong-speed propagation and spurious oscillations on underresolved grids relative to pointwise evaluation, together with an understanding of its coupling with subcell resolution and with adaptive stiff ODE integration in the reaction step.

The common scientific objective is to replace the currently empirical time-step selection and ad hoc safeguards of fractional-step reactive-flow simulation by a predictive theory and a matching discretization methodology; the two deliverables are linked, since the same discretized convection–source interaction is the object of both the analysis and the design.

References

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