Chemical Explosive Mode Analysis (CEMA)

Updated 28 November 2025

Chemical Explosive Mode Analysis (CEMA) is a diagnostic technique that utilizes the leading eigenvalue of the chemical Jacobian to detect ignition and combustion transitions.
It integrates dynamical-systems theory with spectral analysis to identify autoignition events and heat-release spikes through precise eigenvalue computation.
CEMA supports scalable, high-fidelity simulations by leveraging analytical Jacobian assembly and quantile-sampling, enabling adaptive workflow controls in combustion modeling.

Chemical Explosive Mode Analysis (CEMA) is a diagnostic technique in reacting-flow and combustion modeling that leverages local dynamical-systems theory to identify autoignition, flame propagation, and critical chemical transients by analyzing the spectrum of the chemical Jacobian. CEMA centers on the leading eigenvalue with maximal real part of the chemical source Jacobian and enables identification of ignition propensity, heat-release spikes, and shifts between pre- and post-ignition behavior. Through its mathematical framework, it has become integral to high-fidelity simulations, adaptive workflows, and modern exascale computational combustion.

1. Mathematical Formulation and Jacobian Construction

CEMA is applied to the system of equations governing species transport and energy balance in reacting flows. In direct numerical simulation (DNS) or zero-dimensional (homogeneous) models, the evolution of the conserved variables $y(t,x)$ —including temperature and species mass fractions—can be written as: $\frac{D y}{Dt} = \omega(y) + s(y)$ where $\omega(y)$ denotes the chemical source terms and $s(y)$ the mixing or transport contributions (Gadalla, 2022, Bennett et al., 2015, Rabbani et al., 2023).

The chemical Jacobian $J_\omega(y) = \frac{\partial\omega(y)}{\partial y}$ is crucial, as it encodes the linearized response of the chemical source terms to perturbations in state. For practical computations, an analytical assembly is preferred: for each elementary reaction, contributions to $J_\omega$ are obtained as sparse stencils involving derivatives of forward and backward rates with respect to concentrations and temperature. This approach, implemented in combination with the PyJac code generator, yields substantial performance gains versus traditional finite-difference Jacobians in massively detailed mechanisms (Gadalla, 2022).

2. Spectral Analysis, Explosive Mode, and Physical Interpretation

At each spatial grid point or ODE state, CEMA computes the spectrum $\{\lambda_i\}$ of $J_\omega$ . Ordering the eigenvalues by decreasing real part, $\mathrm{Re}(\lambda_1)\geq \mathrm{Re}(\lambda_2)\geq\ldots$ , the leading eigenvalue $\lambda_e\equiv \lambda_1$ determines the most dynamically unstable direction accessible to infinitesimal perturbations via chemistry alone (Bennett et al., 2015, Rabbani et al., 2023). If $\mathrm{Re}(\lambda_e)>0$ , the chemical system is locally explosive (pre-ignition), whereas $\mathrm{Re}(\lambda_e)<0$ denotes post-ignition or deflagrative conditions.

The eigenvectors $v$ and $b$ (right and left, respectively) associated with $\lambda_e$ define the direction of fastest local chemical amplification. In premixed flames, the locus where $\lambda_e=0$ essentially traces the instantaneous flame front (Gadalla, 2022). Modal diagnostic tools from computational singular perturbation (CSP) theory, such as the Amplitude Participation Index (API), Time-Scale Participation Index (TPI), and species/temperature pointers, further decompose the contributions of individual reactions and state variables to the explosive mode (Rabbani et al., 2023).

3. Algorithmic Realizations and Approximations

Direct evaluation of $\lambda_e$ at every simulation point has computational costs scaling as $\mathcal{O}(N \cdot M^3)$ per timestep (with $N$ grid points or cells and $M$ chemical variables), which is often prohibitive for large systems (Bennett et al., 2015). To circumvent this, a quantile-sampling approach achieves scalable approximation:

Draw $k$ random samples per timestep, compute $J_\omega$ and leading eigenvalue at these locations.
Sort sampled $\lambda_e$ values and estimate desired percentiles (e.g., median, upper tails).
Rigorous error/confidence intervals for percentile estimation are given by Hoeffding’s bound, with $k$ determined by prescribed tolerance $\epsilon$ and probability $\delta$ but independent of $N$ .
The method extends to composite indicators, such as the P-indicator, based on spreads of upper-tail percentiles.

The following pseudocode, employing MPI parallelism, captures the outlined procedure:

for each analysis time step t:
  for each processor p in 0..M-1:
    sample s local grid indices
    for i in 1..s:
      compute J_omega at location i
      find lambda_e via eigendecomposition
    store s values to local array S_p
  gather S_p across all processors to global sample S
  sort S, compute percentiles for alpha, beta, gamma
  form P-indicator: (p_alpha - p_gamma)/(p_beta - p_gamma)
  if P-indicator < threshold: trigger adaptivity

(Bennett et al., 2015)

4. Applications and Workflow Integration

CEMA’s principal utility lies in robust, physics-based detection of combustion events—such as heat-release spikes and ignition transitions—for enabling adaptive scientific workflows in high-performance computing environments. For example, percentile-based CEMA triggers can:

Detect the onset of critical events (bursts of heat release, ignition fronts) in turbulent combustion,
Dynamically adjust simulation controls—mesh refinement, output frequency, or I/O rates—based on the evolving chemical state (Bennett et al., 2015),
Isolate chemical or thermal runaway regions, as established in comparative analyses of detailed NH $_3$ /air mechanisms (Rabbani et al., 2023).

In OpenFOAM, analytical CEMA diagnostics are integrated as runtime fields, enabling direct extraction of spatial fields of the explosive mode and seamless visualization and triggering during standard CFD runs (Gadalla, 2022).

5. Diagnostic and Mechanistic Insights

CEMA not only provides triggering and localization but also delivers mechanistic understanding by quantifying which reactions and species dominate pre-ignition and ignition dynamics:

In autoignition studies of ammonia oxidation, the early explosive mode is universally tied to the initiation reaction NH $_3$ +O $_2\rightleftharpoons$ NH $_2$ +HO $_2$ . Differences in mechanism behavior—such as the presence or absence of extended “chemical runaway” and the role of two-nitrogen chemistry—manifest directly in CEMA’s modal decomposition (Rabbani et al., 2023).
Late-stage ignition is dominated by OH-producing and OH-consuming reactions, with the explosive mode strongly projecting onto temperature and radical pools, enabling correlation of CEMA diagnostics with experimentally relevant observables such as ignition delay and heat release.

These detailed analyses facilitate the rational design of reduced or optimized kinetic models, as well as direct performance comparisons across mechanisms.

6. Computational Performance, Validation, and Limitations

Analytical Jacobian assembly (via PyJac) reduces per-cell Jacobian construction to $\mathcal{O}(N_r)$ , where $N_r$ is the number of reactions, and minimizes overhead relative to finite-difference techniques (Gadalla, 2022). Full CEMA eigenanalysis adds only 5–10% per-timestep cost in typical premixed-flame benchmarks, while percentile-based quantile sampling reduces wall-time overhead to as little as 1% when applied adaptively (Bennett et al., 2015).

Quantitative validation demonstrates that CEMA-based fields consistently agree with standard heat-release measures and that percentile-sampling errors are nearly always below $10^{-3}$ in rank, with predicted triggers matching expert windows in over 99% of cases.

Limitations of current approaches include:

The focus on global percentile trends rather than rare localized anomalies, which may go undetected if under-sampled.
Neglect of explicit time-variation in the CEMA eigenbasis (“frozen” approximation), which may induce errors in highly transient, stiff-chemistry situations.
The analytical Jacobian currently addresses chemical processes but not full diffusive coupling and, in multiphase or heterogeneous systems, requires further code extension (Gadalla, 2022).

7. Generalizability Across Indicators and Future Extensions

The quantile-sampling and percentile-spread paradigm pioneered for CEMA extends naturally to any scalar indicator derived from sorted field statistics (e.g., vorticity, gradients, alternate eigenvalues). Its error bounds, independent of problem size, guarantee scalability to exascale simulation environments (Bennett et al., 2015).

Potential future directions include:

Incorporation of projected CEMA diagnostics, such as Explosion Index (EI) and Participation Index (PI),
Deployment of CEMA-based analysis and triggers for GPU-accelerated workflows,
Extension to treat multi-phase or surface-driven reaction systems,
Development of advanced statistical techniques to improve rare-event or anomaly detection without sacrificing scalability.

Through its marriage of dynamical-systems theory, spectral analysis, and scalable numerical implementation, CEMA provides an essential, rigorously grounded toolset for modern high-fidelity combustion diagnostics and simulation-based scientific discovery (Bennett et al., 2015, Gadalla, 2022, Rabbani et al., 2023).