【作者】 尹丽；

【导师】 黄明游； 邹永魁；

【作者基本信息】 吉林大学 ， 计算数学， 2004， 博士

【摘要】 大多数的自然现象与物理规律都可归结为与时间相关的非线性偏微分方程，诸如热传导方程、声波与弹性波方程、反应扩散与对流扩散方程、流体与气体力学方程组等发展方程。绝大多数情形这些问题的解不能用解析的公式表达出来，因此研究这些发展方程的数值求解问题备受人们关注。 在本文中我们主要构造了三类与时间相关的偏微分方程的数值计算方法，并作了相应的理论分析。本文主要内容可分为三部分，分别研究了含源项的浅水波方程组，描述浅水域表面小振幅、长波传播的“good”Boussinesq(简记为GB)方程，以及四阶非线性扩散模型Cahn-Hilliard方程的数值计算方法。 1．浅水波问题及其数值计算方法 考虑在矩形通道中孤立障碍物上方的一维流体运动问题，此时的数学模型为： h_t+(hu)_x=0， (hu)_t+(hu~2+1/2gh~2)_x=-ghB’(x)，其中常数g代表重力加速度，障碍物B(x)是关于原点(x=0)对称的凸函数。h代表在障碍物上方流体的厚度，u代表流体水平方向的速度。当h(x)，u(x)充分光滑时，系统(1)可以转化成完全散度形式： h_t+(hu)_x=0， u_t+(1/2u~2+gh+gB)_x=0系统(1)与系统(2)的稳态解(又称定常解)(h(x)，u(x))都满足如下平衡条件： h(x)u(x)=const， 1/2u~2(x)+gh(x)+gB(x)=const,为了便于对定常间题进行理论分析，引进如下无量纲变量:凡一揣，M*一告“一;睽+1，Mc一会其中ho和。。分别代表来流的流体厚度与速度，Hc代表障碍物顶点的高度. 我们证明了连续定常解的存在唯一性(见定理l)以及在典型情形下含间断定常解的存在性(见定理2和定理3).定理1(连续定常解的存在唯一性)对任意给定的参数0<凡笋1，当Mc<M*时，定常问题(3)存在唯一的一条连续解曲线(h(x)，诚x))，x任R.并且由B(x)=B(一x)可知，此时流体厚度h(x)，速度可x)和自由表面侧x)十B(x)是关于原点(x=0)对称的函数. 一些实验表明:当Mc>M*时，定常解不再为连续函数，通常含有间断.对于间断定常解的分析有如下基本假设: (i)临界状态出现在障碍物的顶点位置，即tL。=两孚二，其中功。和。。分别代表流体在障碍物顶点处的厚度与速度. (ii)在间断点处，定常解满足激波间断条件. 基于上述基本假设，定常解的结构分析可按上游和下游两部分进行. 首先，考虑流体在障碍物上游(一co<x兰0)发生了激波间断，并记流体在间断线后方的稳态厚度和稳态速度分别为hA和。A，则上述上游稳态解满足下列方程: hAu注一houohA一hoc‘一。一恤了亲磊，(F卜H间斯条件)，(4)+必c+刀七uc功。=uAhA。。=斌万砚，一爵+h一凡，(平衡条件)，=怕，(5)屺一助(临界条件).(6)定理2当Mc>M*时，存在唯一的数组(hA，。A，c:，功。，恢)〔R”满足物理条件:hA>h0，。，<二。并使方程(4)一(6)成立.因此，在基本假设(i)，(11)T，当Mc>*时，在降碍物上游存在含间断的定常物理解(城x)，u(x))，一co<x兰0. 2 其次，考虑一类典型的下游定常解结构.这里，将上游定常解按平衡条件延伸到下游，由此在障碍物的后面获得一个新的稳态速度tLB与稳态厚度hB.由于障碍物对定常解的影响是局部的，故流体在远离障碍物的状态将恢复为来流状态(h。，u。).这与黎曼问题相似，我们考虑状态(hB，。B)与状态(ho，坳)通过一个激波间断和一个2一稀疏波连接，则上述下游稳态解满足下列方程:二2 .2箭+“”一Ka‘一箭+“A)，(平衡条件)，tLBhB二从(=。AhA)，(7)h刀牡刀一hxtLx hB一hx(R-H间断条件)，(8)Ct.=牡B一hx。x一2而石(2一稀疏波).(9)定理3当Mc>M*时，。标<h0且助十标了亲藉<咐ho斌薪藉.成立，肪在唯一的数组(hB，。B，hx，ux，cr)〔R“满足物理条件:hB<hA，。A<。B，ho>x>B，二x<二B，并使方程(7)一(g)成立.从而，在基本饭设(O，(词下，当Mc>*时，在降碍物下游存在一个由激波与稀硫波相连接的定常物理解(城x)，可x))，O<x<+co. 从浅水波方程组的完全散度形式(2)出发，用迎风差分格式、流矢量分裂法和基于流失量分裂法的TVD格式来模拟各种情形下的非定常解.数值试验结果与前面的理论分析相吻合.并且证明了迎风差分格式和流矢量分裂法的数值近似解在各个界面处满足具有二阶精度的平衡条件(见命题1和命题2).进而证明了流矢量分裂法保持了厚度变量h非负性的物理要求(见定理，). 从浅水波方程组的完全散度形式出发，在单元Ix，一1/2，肠+1/z]上进行积分，并对时间用向前Euler差分格式离散后，得到如下全离散格式:0典(l时畔十‘一岭 矛，夕.，，。一尸夕·价朴二竺认王一竺-其中叮为t。时刻在单元冈一1/2，xj十1/aj上的U一(h，司T的平均值，峨1/2为‘定义在单元边界xj+1/:处的数值流量F= 1。__‘甲(h“，乏“‘+gh+g万)‘3迎风差分格式中数值流量象:/2定义为:瓜:/2一叠!二(二卜;(二1)卜看s、·(，麒毛些生))!二(二1卜二(二)}·(11)其中Sign(A)=TSign(A)T一’，(Sign(A))灯=Sign(a‘)占巧，A更多还原

【Abstract】 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更多还原