A Class Of Semilinear Parabolic Problems Runge-kutta Collocation Method

This text combines Runge-Kutta methods with collocation methods to solve semi-linear parabolic problems for the first time. Utilizing midpoint Euler method namely second-order Runge-Kutta method to discretize time and combined with orthogonal collocation .we have got the fully discretized collocation scheme.Furthermore,the existence and uniqueness of the numerical solution are proved and the optimal order estimate in L2— norms is derived.Semilinear parabolic equations are widely used in many mathematical physical fields, such as chemist.ry,biology.So it is necessary to invesgate all-around at all events.Runge-Kutta method is a high-order single step method.You will notice that Runge-Kutta method requires you not to supply values of the derivatives but to calculate the function values at different points.Then it do the linear association to the function value to construct the approximate formula .Runge-Kutta method is widely used in solving ordinary differential equations.But it is seldom used to solve paratial differential equation.The collocation method is a numerical method which search for the approximate solution of the equation by way of meeting pure interpolation condition.lt has the advantage in ease of implementation and high-order accuracy.Douglas and Dupont deeply discussed the spline collocation methods for nonlinear parabolic: initial-boundary value problems in one space variable.They considered the use of orthogonal collocation method combined with various time-stepping procedures and got, some optimal results. Greenwell and Fairweather studied the spline collocation methods for parabolic and hyperbolic problems in two space variables.Then according to the above.in this dissertation a Discrete-Time Collocation Method integrated Runge-Kutta method and collocation method is given for a class of two-dimensional semilinear parabolic equation. In this paper,the space of piecewi.se Hermite bicubics is the approximation space, it gives the fully discrete collocation scheme.the existence and uniqueness of the numerical solution are proved, and optimal order L~2 error estimate is derived .The outline of this paper is as follows:In Chapter 1 ,we discuss Runge-Kutta collocation method with constant coefficient. Section 1 is introduction.We considere the following semilinear parabolic problem:It gives partitions of it in the x- and y-directions.Regard the grid points as (x,, yj), i = 0,1.....M:j = 0,1...., N. And0 = x0 < Xi < ? ■ ? < xA/ = 1; 0 = j/o < J/i < ? ? ? < J/n = 1- (11)LetClij = [x,_i,x,] x [yj_i,j/j],i = l,...,A/:j = l,...,N.Denote P3 as polynomial set of degree at most 3.SetHr = {v € Cl[0A]\v e P3{Ik)J = l,...,A/} H° = {veHx : v(0) = ?(1) = 0}HtJ = {(' € Cl[0. l]|u € P3(/,w), / = 1, ■ ? ?, A/} H°y = {v € //? : v(0) = v(l) = 0} Define the space of piecewise Heimite bicubics:and gives the collocation points.inner product and norm. Section 2 gives the fully discrete collocation scheme:{ KT6'V-.Ix(0. T; H\n)).ut 6 L3O(0.r;/f4(n)),then for sufficiently small AUhe solution for (12) is unique.and there exits a constant C to supply:omaxv \\uk - Uk\\LHU) < C(Af + /i4) (13)In Chapter 2 we further discuss changable coefficient Runge-Kutta collocation method. We considere the following semilinear parabolic problem:^ - V ? (DVu) = /(*, u). (x, y) 6fi.i€ (0,7],u(i, y,0) = uo(x: y), (i.y) G SI. (14)u(x.y.t)=0 (x,y) € 0Q, t 6 (0. T\where0 < D. < ￡?(x, y) < D* < +oc (15)Similarly we can give the fully discrete collocation scheme:r Tn + l _ t niV ? (￡>V6"'+l'2) - r+^{Un + ^-/n(^"))}(^,^) = 0 n = 0,1. n(16) ALso we can know that this scheme is equal to discrete Galerkin method.(L r'A~in - V ? (￡> W1*1'2) - fn+l/2(U" + ^-f"{Un)). z) = 0 c e H° (17)In ihe end we give convergence analysis and get the following theorm .?theorm 2.1. Suppose u is the accurate solution of the equations (14),U is the fully discrete collocation solution of (16).Suppose / is Lipschicz continuous about n.u € Lx(0. T: H(i(tt)).ut € Lx(0. T\ H\tt))t\iui for sufficiently small At.the solution for (16) is unique.nnd there exits a constant C to supply :max \\uk - Uk\\vHn) < C{At + h4) (18)0