Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation

Ming Jun Lai, Chun Liu, Paul Wenston

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3 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)776-827
Number of pages52
JournalNumerical Methods for Partial Differential Equations
Volume19
Issue number6
DOIs
StatePublished - Nov 1 2003

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Bivariate Splines
Stream Function
Splines
Navier Stokes equations
Polygon
Evolution Equation
Navier-Stokes Equations
Numerical Solution
Weak Solution
Formulation
Quadrangulation
Fourth-order Equations
Cubic Spline
Galerkin Method
Newton Methods
Spline
MATLAB
Fourth Order
Smoothness
Nonlinear Equations

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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