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A new computational methodology, termed ‘PeleLM-FDF’ is developed and utilised for high fidelity large eddy simulation (LES) of complex turbulent combustion systems. This methodology is constructed via a hybrid scheme combining the Eulerian PeleLM base flow solver with the Lagrangian Monte Carlo simulator of the filtered density func- tion (FDF) for the subgrid scale reactive scalars. The resulting methodology is capable of simulating some of the most intricate physics of complex turbulence-combustion interactions. This is demonstrated by LES of a non-premixed CO/H2 temporally evolv- ing jet flame. The chemistry is modelled via a skeletal kinetics model, and the results are appraised via a posteriori comparisons against direct numerical simulation (DNS) data of the same flame. Excellent agreements are observed for the time evolution of various statistics of the thermo-chemical quantities, including the manifolds of the multi-scalar mixing. The new methodology is capable of capturing the complex phe- nomena of flame-extinction and re-ignition at a 1/512 of the computational cost of the DNS. The high fidelity and the computational affordability of the new PeleLM-FDF solver warrants its consideration for LES of practical turbulent combustion systems. 

We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.