648 research outputs found

    Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems

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    This paper studies the spatial manifestations of order reduction that occur when time-stepping 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.Comment: 41 pages, 9 figure

    A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

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    Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes

    A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems

    Get PDF
    Our study is concerned with a hybrid spectral collocation approach to solving singularly perturbed 1-D parabolic convection-diffusion problems. In this approach, discretization in time is carried out with the help of Taylor series expansions before the spectral based on novel special polynomials is applied to the spatial operator in the time step. A detailed error analysis of the presented technique is conducted with regard to the space variable. The advantages of this attempt are presented through comparison of our results in the model problems obtained by this technique and other existing schemes
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