53,907 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

    Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation

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    The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Ampere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added coment

    Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

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    In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O(p2d)\mathcal{O}(p^{2d}) storage and O(p3d)\mathcal{O}(p^{3d}) computational work, where pp is the degree of basis polynomials used, and dd is the spatial dimension. Our SVD-based tensor-product preconditioner requires O(pd+1)\mathcal{O}(p^{d+1}) storage, O(pd+1)\mathcal{O}(p^{d+1}) work in two spatial dimensions, and O(pd+2)\mathcal{O}(p^{d+2}) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in pp per degree of freedom in 2D, and reduce the computational complexity from O(p9)\mathcal{O}(p^9) to O(p5)\mathcal{O}(p^5) in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to p=15p=15. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees pp.Comment: 40 pages, 15 figure

    Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

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    The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

    Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

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    Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations
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