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Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation
Author(s) -
Lai MingJun,
Liu Chun,
Wenston Paul
Publication year - 2003
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.10072
Subject(s) - mathematics , discretization , nonlinear system , spline (mechanical) , uniqueness , mathematical analysis , navier–stokes equations , partial differential equation , numerical analysis , galerkin method , crank–nicolson method , physics , structural engineering , quantum mechanics , compressibility , engineering , aerospace engineering
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 3 r 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 L 2 (0, T ; H 2 (Ω)) ∩ L ∞ (0, T ; H 1 (Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C 1 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.