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Efficient and accurate simulation of turbulent combusting flows in complex geometry remains a challenging and computationally expensive proposition. A significant source of computational expense is in the integration of the temporal domain, where small time steps are required for the accurate resolution of chemical reactions and long solution times are needed for many practical applications. To address the small step sizes, a fourthorder implicitexplicit additive RungeKutta (ARK4) method is developed to integrate the stiff chemical reactions implicitly while advancing the convective and diffusive physics explicitly in time. Applications involving complex geometry, stiff reaction mechanisms, and highorder spatial discretizations are challenged by stability issues in the numerical solution of the nonlinear problem that arises from the implicit treatment of the stiff term. Techniques for maintaining a physical thermodynamic state during the numerical solution of the nonlinear problem, such as placing constraints on the nonlinear solver and the use of a nonlinear optimizer to find valid thermodynamic states, are proposed and tested. Verification and validation are performed on the new adaptive ARK4 method for lean premixed flames burning hydrogen, showing preservation of 4thorder error convergence and recovery of literature results. ARK4 is then applied to solve lean, premixed propaneair combustion in a bluffbody combustor geometry. In the twodimensional case, ARK4 provides a 70Ã— speedup over the standard fourstage RungeKutta method and, for the threedimensional case, threeordersofmagnitudelarger time step sizes are achieved. To further increase the computational scaling of the algorithms, parallelintime (PinT) techniques are explored. PinT has the dual benefit of providing parallelization to long temporal domains as well as taking advantage of hardware trends towards more concurrency in modern highperformance computing platforms. Adaptive mesh refinement (AMR) with subcycling is added to multigrid reductionintime (MGRIT) to create a temporallyparallelized algorithm with efficient solutionadaptive grids. The new PinT algorithm is applied to purely diffusive problems, Couette flow and Stokes second problem, where the adaptive spacetime parallel algorithm demonstrates up to a 13.7Ã— speedup over a timesequential algorithm. However, the traditional multigrid methods employed by MGRIT have difficulties when applied to hyperbolic systems and, therefore, many practical fluid flows. To overcome this in a turbulent flow, multigrid operations are applied in a novel way where temporal parallelization is instead achieved by exploiting the spacetime localization of fine turbulent scales. This leads to rapid convergence of the bulk flow, which is important for computing macroscopic properties useful for engineering purposes. The novel multigrid operations are applied to the compressible inviscid TaylorGreen vortex flow where convergence of the lowfrequency modes is achieved within a few iterations. Future work will be focused on a speedup study for practical, highly turbulent flows. Host: CCS2 