Numerical Method Of Combining Orthogonal Spline Collocation Method And Alternating Direction Implicit Method For Two-dimensional Parabolic Problems On Rectangles

The collocation method is one numerical method which searches for the approximatesolution of the operator function by satisfying pure interpolation condition, and it was invented in the 1970s. Collocation methods are essentially involved in determiningan approximate solution by piecewise polynomial by requiring it to satisfy the differential equation and boundary condition exactly at certain points. The collocationmethod needn't compute numerical integral, so it has easier implementation and higher convergence rate than the finite element method, and it is widely used for solving both engineering and computing mathematics. Spline collocation method using the nodes of Gauss quadrature formula(Gauss points) as collocation points is named orthogonal spline collocation method(OSC), which has better precision and faster than the normal spline collocation methods.For higher-dimensional equations, implicit schemes are absolutely stable, but increasing the arid workload. While explicit schemes are easily computed, but it is difficult to establish the stable condition. Therefore, some researchers establish one absolutely stable scheme - alternating direction implicit method. And it is easy to make a transform from higher-dimensional equations to lower-dimensional ones and decrease the workload by using alternating direction implicit method. And it is widely used in the engineering fields.One method combining the orthogonal spline method and alternating direction implicit method is given in this paper. We establish one two-level piecewise Hermite bicubic orthogonal spline collocation scheme. And the existence and uniqueness of numerical solution are proved, optimal error estimate and the stability result are also derived.This paper is divided into four sections.Section 1 is introduction. The two-dimensional parabolic problems is given bywhereΩ=(0,1)×(0,1), (?) is the boundary ofΩ, and 0＜a_min≤a₁(x,y.t),a₂(x,y,t)≤a_max,(x,y,t)∈Q.Section 2 is preliminaries.In section 3, the collocation scheme is given and the solution will be proved uniquely.LetLet {t_n}_n=0^N_t a partition of [0, T] such that t_n=nÏ„, whereÏ„=T/N_t, and let t_n+1/2=t_n+Ï„/2,n = 0,…, N_t-1. Let L_iⁿ and L_i^n+1/2(i=1,2) be the differential operators given by, respectively, with t_n and t_n+1/2≡(n+1/2)Ï„. The collocation scheme is given by The functions u_h⁰,(?),n=1,…,N_t, can be prescribed by approximating the initial and boundary conditions of the equation using Hermite interpolants.For adequately smallτ＞0, the collocation method possess a unique solution.In the fourth section the L²- norm error estimate is derived and the stability result is also proved.