| Many natural phenomena and physical laws can be described by time dependent nonlinear differential equations, such as heat conduction equation, sound wave and elastic wave equation, reaction diffusion and convection diffusion equation, fluid and aerodynamics equations, etc. It is clear that analytic solutions are difficult to obtain, therefore the numerical approximation of the solutions plays an important role for studying such equations.In this dissertation, we construct different numerical schemes to approximate the solutions of three kinds of time dependent nonlinear differential equations, respectively, and study the corresponding theoretical analysis and numerical experiments. There are three different topices in this dissertation, in which we respectively study the numerical approximation methods for the shallow-water equations with source terms, the "good" Boussinesq equation describing small amplitude and long wave transmiton in shallow water, and a fourth order nonlinear Cahn - Hilliard equation.1. Shallow-water problem and numerical methodsConsider the problem of one-dimensional shallow-water flow over an isolated obstacle in a rectangle channel, which is governed by the following equations:where g is the gravitational constant, the isolated obstacle B(x) is convex and symmetric with respect to the origin (x = 0). h denotes the height of the fluid above the obstacle andu is the horizontal velocity of the fluid. When h(x) and u(x) are smooth, the system (1) can be rewritten as a complete divergence formBoth of the steady state solutions (h(x),u(x)) of the system (1) and (2) satisfy the following balance conditionsTo study the steady state problems conveniently, we introduce the following dimension-less variableswhere ho and UQ are the initial height and initial velocity of the fluid, respectively. Hc is the height of the obstacle B(x) crest.We prove the existence and uniqueness of the continuous steady state solution (see Theorem 1) and the existence of steady state solution with hydraulic jump in typical situations (see Theorem 2 and Theorem 3).Theorem 1 Assume Q < F0 1 and Mc < Mt. The steady state problem (3) has a unique continuous solution (h(x),u(x)), x R. Moreover, the height h(x), the velocity u(x) and the free surface h(x) + B(x) of the fluid are symmetric with respect to the origin (x = 0), due to B(x) = B(-x).Some experimentations illustrate that when Mc > M , the problem hasn't any continuous steady state solution, which generally contains hydraulic jumps. When studying the steady state solution with hydraulic jumps, the following hypotheses are assumed.(i) The critical condition occurs at the obstacle crest, i.e. uc = , where C and uc denote the height and velocity of the fluid at the obstacle crest, respectively.(ii) The steady state solutions satisfy the R-H jump conditions at the jump point.Under the above hypotheses, the structure analysis of the steady state solution can be divided into upstream and downstream portions.First, we consider the flow with a hydraulic jump in upstream (- < x 0), and the steady state height and velocity on the backside of the jump line denoted by hA and uA, then the determining equations of the asymptotic solution in upstream portion consist ofTheorem 2 When Mc > M, there exists a unique solution (hA, UA, cl, hc, uc) R5 of equations (4) - (6) , which satisfies the physical conditions: hA > h0, UA < UQ. Therefore, under hypotheses (i),(ii) and when Mc >M , there exists an asymptotic solution (h(x), u(x)) with a hydraulic jump, in the upstream portion of the obstacle, which satisfies the physical conditions as well.Next, we consider the solution-structure of downstream portion in a typical situation, we extend the known upstream steady state solution to the lee side, which create a new steady state height ha and velocity us- Due to the local affection of the obstacle to the steady state solution, the state of the flow far away from the obstacle will retrieve the initial state (h0, u0). This transition is very similar to th...
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