High Order Orthogonal Collocation Methods For Time-Dependent Equations

The collocation method is a numerical method which search for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years, and it is widely used for solving both engineering and computing mathematics due to its ease of implementation, high-order accuracy and no integrals need be not evaluated. Collocation methods essentially involves determining an approximate solution by a piecewise polynomial by requiring it to satisfy the differential equation and boundary conditions exactly at certain points. Original spine collocation methods collocate at the nodes by cubic spline functions, but the precision isn't good. For improving the convergence rate, collocation points usually use the nodes of Gauss quadrature formula, and choose piecewise Hermite bicubics polynomial as the approximative space, convergence rate can reach h~4, spline collocation at Gauss points is named orthogonal spline collocation method(OSC).Orthogonal collocation was first introduced by C.deBoor and Swartz [2] for mth order ODE. In one space variable, Douglas and Dupont[3] give C~1 finite element method (r ≥ 3), Robinson and Fairweather [4] consider OSC for Schrodinger-type equation OSC, Lu[9] proposed the characteristics collocation method for convection diffusion equation, Wang[82]proposed the single-node characteristics collocation method for convection diffusion equation. Houstis[74]gives OSC method for hyperbolic equation. In two space variables, Prenter and Ruscll[6] gives OSC for elliptic equation. Bialecki and Cai [11] consider two kinds of interpolation for the boundary conditions of elliptic equation, i.e. Hermite interpolation and Gauss interpolation, optimal estimate can be get. Percell and Wheeler [5] consider the elliptic problems (r ≥ 3). Bialeki[12] extends...