Bivariate spline method for numerical solution of time evolution Navier-Stokes equations over polygons in stream function formulation

Authors

  • Ming-Jun Lai,

    Corresponding author
    1. Department of Mathematics, University of Georgia, Allentown, Georgia 30602
    2. Department of Mathematics, Penn State University, College Park, Pennsylvania 16802
    3. Department of Mathematics, University of Georgia, Allentown, Georgia
    • Department of Mathematics, University of Georgia, Athens, GA 30602
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  • Chun Liu,

    1. Department of Mathematics, University of Georgia, Allentown, Georgia 30602
    2. Department of Mathematics, Penn State University, College Park, Pennsylvania 16802
    3. Department of Mathematics, University of Georgia, Allentown, Georgia
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  • Paul Wenston

    1. Department of Mathematics, University of Georgia, Allentown, Georgia 30602
    2. Department of Mathematics, Penn State University, College Park, Pennsylvania 16802
    3. Department of Mathematics, University of Georgia, Allentown, Georgia
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Abstract

We use a bivariate spline method to solve the time evolution Navier-Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier-Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth-order equation, Crank-Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L2(0, T; H2(Ω)) ∩ L(0, T; H1(Ω)) of the 2D nonlinear fourth-order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C1 cubic splines are implemented in MATLAB for solving the Navier-Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.

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