A Discrete-Time Collocation Method For A Class Of Nonlinear Hyperbolic Equations

A class of hyperbolic equations on a rectangle with the solution subject to homegeneous boundary condition is considered in this dissertation. It is solved approximately by the discrete-time collocation process based on having the differential equation satisfied at Gaussian points. It is proved that for sufficiently small time stepsize the solution to the scheme is unique, and of optimal-order accuracy in space in the H¹ norm.Hyperbolic equations reflect the waving phenomena in nature, the research for this class of problem is valuable and significative for theories and pratice.The collocation method is a numerical method which search for the approximation solution of the operator function by satisfying pure interpolation condiction for about thirty years,and it is widely used for solving both engineering and computing mathematics due to its ease of implementation and high-order accuracy.For these.it is widely used in many fields of engineering calculation and computation mathematics.J.Douglas. T.Dupont[1,2] have done much with solving one-dimensional linear and nonlinear parabolic equations by collocation method. R.I.Fernandes— used collocation method solving hyperbolic equations. Alternating Direction Collocation Methods and Efficient Orthogonal Spline Collocation Methods are used to solve two-dimensional const coefficients hyperbolic problem, and got L² , H¹-norm error estimate of them respectively.And there are Houstis.E.N.[7]'s research about this aspect. Collocation method of two-dimensional variable coefficients hyperbolic problem are given in Greenwell-Yanik.C.E.. Fairwearher.G.[8].Then according to the above.in this dissertation a Discrete-Time Collocation Method integrated finite collocation and finite difference is given for a class of two-dimensional nonlinear variable hyperbolic equation.It is important how to dealwith the variable coefficients and how to prove the estimate of the time part.In dissertation,the space of piecewise Hermite bicubics is the approximation space.it gives the complete discrete collocation scheme.the existence and uniqueness of the numerical solution are proved.and optimal order error estimate is derived O(h3).This dissertation discusses variable coefficients in two dimensions and is divided into five chapters.ChapterO is introduction.the hyperbolic problem is given byc(x.t.u)—^ - V â– la(x.t.u)Vu]-b{x.t. u) â–  Vu = f(x.t.u), i6fi.(€ (O.Tj:h(j-.o) = uo(x). ut(x.o) = Ul(x), xenu{x.t) = 0. x€dQ.tï¿¡(0.T):where Q = (0. 1) x (0.1), OQ is the boundary of Q , b = (6i.6o)r is a vector function. We also give the conduction .which coefficients should meet.Chaper 1 is preliminary knowledge.lt gives a partition of U , let M be the space of piecewise Hermite bicubics. which is the approximate space..U = MxsMy.\vhereMx = {v € Cl[0,1] : r U..,,,G P3, k = 1.2 â–  ? ? -Vx}.My = {?? G Cl[0.1] : v !?.,.?,€ P3.l = 1,2- -N,}:and gives the collocation points.inner product and norm.Chaper 2 is lemmas.In this chapter.gives the inequality .which will be used in later prove of convergence analysis.Also gives Lemma'2.1 2.4. it is important to prove Lemma2.3.Chapter 3 gives the complete discrete collocation scheme.proves the existence and imiquciiess of the method. Suppose the coefficients satisfy the condictiou (A) , (B) gi\'ed in instruction.Assume a(x. t. u) = Land we shall do so without loss of generalitv.Then gives the complete discrete scheme bases on ?(.r. t. it) = 1.prove this scheme is equal to discrete Galerkin method.so get the uniqueness.Chapter 4 is conference analysis and error estimate.lt is important to deal with the time part.then get //'-noun error estimate.THEOREM 4.1Suppose u is the accurate solution of the hyperbolic problem (0.1)^0.3 , U is the complete discrete collocation solution of (3.1)^3.2.suppose cofficients c,a.b,f satisfy (A) , (B) , assume a = 1, |j"H2.2.2- IMIo.o.4 defined by (4.6); If u € Lx(0,T:H5{n)), dltu € LX(O.T: H\n) n H^(Q)),, then for sufficiently small At.U is unique , and we have the error estimate:...