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Award ID contains: 1952878

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  1. ; ; ; (, Communications in Mathematical Sciences)
    This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge–Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed. 
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  2. ; ; ; (, Communications in Applied Mathematics and Computational Science)
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  3. ; ; ; ; (, The Journal of Chemical Physics)
  4. ; ; ; (, Computer Methods in Applied Mechanics and Engineering)
    null (Ed.)
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  5. ; ; (, Mathematical Descriptions of Traffic Flow: Micro, Macro and Kinetic Models. SEMA SIMAI Springer Series)
    null; null (Ed.)
    It is known that inhomogeneous second-order macroscopic traffic models can reproduce the phantom traffic jam phenomenon: whenever the sub-characteristic condition is violated, uniform traffic flow is unstable, and small perturbations grow into nonlinear traveling waves, called jamitons. In contrast, what is essentially unstudied is the question: which jamiton solutions are dynamically stable? To understand which stop-and-go traffic waves can arise through the dynamics of the model, this question is critical. This paper first presents a computational study demonstrating which types of jamitons do arise dynamically, and which do not. Then, a procedure is presented that characterizes the stability of jamitons. The study reveals that a critical component of this analysis is the proper treatment of the perturbations to the shocks, and of the neighborhood of the sonic points. 
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  6. ; ; ; ; (, The Journal of Chemical Physics)