High-order implicit-explicit additive Runge-Kutta schemes for numerical combustion with adaptive mesh refinement

The objective of this study is to develop and apply efficient solution techniques for numerical modeling of combustion with stiff chemical kinetics in practical combustors. The new technique combines a fourth-order implicit-explicit (IMEX) additive Runge-Kutta scheme (ARK) with adaptive mesh refinement (AMR). The IMEX component treats the stiff reactions implicitly but integrates convection and diffusion explicitly in time, and thus permits the solution to advance with larger time-step sizes than that of explicit time-marching methods alone. The AMR further adds computational efficiency by effectively placing high spatial resolution meshes in regions with strong gradients, such as flame fronts. The novelty of this study is in the integration of a fourth-order IMEX ARK method with AMR for a high-order finite-volume scheme and the application to solving complex reacting flows governed by the compressible Navier-Stokes equations with very stiff chemistry in a practical combustor geometry. The effectiveness and performance of the adaptive ARK4 is assessed for complex reacting flows by examining properties, such as the presence of shock waves, the time-scale changes in response to AMR levels, and the size and stiffness of reaction mechanisms for various fuels such as H2$$ {\mathrm{H}}_2 $$, CH4$$ {\mathrm{CH}}_4 $$, and C3H8$$ {\mathrm{C}}_3{\mathrm{H}}_8 $$. The new adaptive ARK4 method is verified and validated using a convection-diffusion-reaction problem and shock-driven combustion, respectively. The validated algorithm is then applied to solve the stiff C3H8$$ {\mathrm{C}}_3{\mathrm{H}}_8 $$-air combustion in a bluff-body combustor. A significant speedup of three orders of magnitude is achieved in comparison to the standard ERK4 method at the given solution accuracy.