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  1. Abstract This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε are independently adapted. The resulting algorithms are supported by analysis and numerical tests. The first and second-order methods are embedded. As a result, the computational, cognitive, and space complexities of the adaptive ε , k algorithms are negligibly greater than that of the simplest, first-order, constant ε , constant k artificial compression method.
  2. This paper develops, analyzes and tests a time-accurate partitioned method for the Stokes-Darcy equations. The method combines a time filter and Backward Euler scheme, is second order accurate and provide, at no extra complexity, an estimated the temporal error. This approach post-processes the solutions of Backward Euler scheme by adding three lines to original codes to increase the time accuracy from first order to second order. We prove long time stability and error estimates of Backward Euler plus time filter with constant time stepsize. Moreover, we extend the approach to variable time stepsize and construct adaptive algorithms. Numerical tests show convergence of our method and support the theoretical analysis.
  3. This report presents a low computational and cognitive complexity, stable, time accurate and adaptive method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on the fully implicit / backward Euler time discretization, does not add to the computational complexity, and is conceptually simple. The backward Euler approximation is simply post-processed with a two-step, linear time filter. The time filter additionally removes the overdamping of Backward Euler while remaining unconditionally energy stable, proven herein. Even for constant stepsizes, the method does not reduce to a standard / named time stepping method but is related to a known 2-parameter family of A-stable, two step, second order methods. Numerical tests confirm the predicted convergence rates and the improved predictions of flow quantities such as drag and lift.
  4. This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε are independently adapted. The resulting algorithms are supported by analysis and numerical tests. The first and second-order methods are embedded. As a result, the computational, cognitive and space complexities of the adaptive ε,k algorithms are negligibly greater than that of the simplest, first-order, constant ε, constant k artificial compression method.