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Alternating Direction Implicit orthogonal spline collocation on Non-Rectangular Regions

Advances in Applied Mathematics and Mechanics 2013年第4期5卷 461-476页

作者： Bernard Bialecki Ryan I.Fernandes Department of Applied Mathematics and Statistics Colorado School of MinesGoldenCO 80401USA Department of Mathematics The Petroleum InstituteAbu DhabiUnited Arab Emirates

The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on *** the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high ***[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such *** approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.

关键词： Alternating direction implicitmethod orthogonal spline collocation two point boundary value problem Crank Nicolson parabolic equation non-rectangular region

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The Crank-Nicolson Hermite Cubic orthogonal spline collocation Method for the Heat Equation with Nonlocal Boundary Conditions

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Advances in Applied Mathematics and Mechanics 2013年第4期5卷 442-460页

作者： B.Bialecki G.Fairweather J.C.Lopez-Marcos Department of AppliedMathematics and Statistics Colorado School of MinesGoldenCO 80401USA Mathematical Reviews American Mathematical Society416 Fourth StreetAnn ArborMI 48103USA Departamento de Matem´atica Aplicada Universidad de ValladolidValladolidSpain

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial *** an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in *** discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear *** present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

关键词： Heat equation nonlocal boundary conditions orthogonal spline collocation Hermite cubic splines convergence analysis superconvergence

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