【作者】 刘敏；

【导师】 李订芳；

【作者基本信息】 武汉大学 ， 计算数学， 2022， 硕士

【摘要】 本文提出了一种具有和谐性、保正性、守恒性和TVD性质（Total Variation Diminishing,总变差减少）的求解双曲守恒律的非交错中心Runge-Kutta格式。与经典中心格式不同的是,在时间上采用能达到任意阶数的Runge-Kutta求解器,使高阶格式的时间复杂度更小。空间上则与经典中心格式一样采用本质非振荡线性斜率限制器达到二阶,因此本文格式继承了经典中心格式的简单、高效和无黎曼求解器的优点。本文的工作主要涉及以下两个方面:第一项工作是中心格式在求解浅水方程时如何处理干湿面处的水位和如何具有和谐性、保正性以及水量守恒性。通过在向后投影步构造水位的上、下阈值,并将交错单元的积分平均值与之比较来判断非交错单元是否有水,从而对干湿面处相邻的水位斜率纠正以保证格式在向后投影步具有和谐性和守恒性。通过定义交错节点处的底部值,将源项在非交错单元界面处分开积分,使得格式在理论上能精准地平衡数值源项和数值通量,保证了格式在更新步具有和谐性。因此,本文格式不仅在深水区域具有和谐性,在干湿面处也具有和谐性和守恒性。数值实验选取了稳态解的小扰动、静水扰动、抛物线形碗以及倾斜面溃坝等算例,展示了本文格式的干湿面的处理能力、和谐性、保正性以及水量守恒性。第二项工作是减少非交错中心格式的数值耗散。经典中心格式在计算间断解时因为数值耗散的存在导致精度降低,表现在激波和接触间断处。本文通过在向前投影步引入参数α将前一时间层的数值解与当前时间层的数值解进行组合,减少数值解在投影步积累的数值耗散。经典中心格式的数值耗散为u_xxxx（Δx）⁴/Δt,本文格式引入α后的数值耗散为αu_xxxx（Δx）⁴/Δt。因此α取得越小,格式的数值耗散就越小,但维持格式稳定的CFL数的取值范围也相应变窄。通过对格式TVD性质的理论分析发现,α、CFL（Courant-Friedrich-Lewy）数和minmod参数之间存在着非线性关系式,这有助于在实际计算的时候提供一个初步的取值范围,且表明了对于任意∈α（0,1）,总存在一个合适的CFL数使得数值解不会产生新的极值点,即格式可以保持稳定。数值实验选取了Euler方程的五个黎曼问题和几个经典的对流扩散模型,展示了本文格式的数值耗散比经典中心格式小,在间断解处的空间精度非常高。更多还原

【Abstract】 In this thesis,we present a non-staggered central Runge-Kutta scheme for solving hyperbolic conservation laws,which has the properties of well-balaning,positivity preserving,mass conservation and TVD（Total Variation Decreasing）.Unlike the classical central scheme,the presented scheme utilize the Runge Kutta solver which can reach arbitrary order in time to reduce the time complexity of the high-order scheme.On the other hand,the non-oscillatory minmod limiter is used to achieve the second-order accuracy in space like the classical central scheme.Therefore,the presented scheme inherits all advantages of the classical central scheme:simplicity,efficiency and no Riemann solvers.The works of this thesis is as follows:The first work is,when solving the shallow water equation,how to deal with the water level at the dry-wet surface and how to make the presented scheme well-balancing,positivity preserving and mass conservation.The upper and lower thresholds of water level are constructed in the backward projection step,and the integral average value of the staggered cell is compared with thresholds to determine whether there is water in the non-staggered cell,so as to correct the adjacent water level slope at the dry-wet surface and ensure the well-balancing and mass conservation of the presented scheme in the backward projectio.By defining the bottom value at the staggered node,the source term is divided into two-part integrals at the non-staggered cell,which can make the presented scheme accurately balance the numerical source term and numerical flux theoretically,ensuring the well-balancing of the presented scheme in the numerical evolution.Therefore,the presented scheme is well-balanced not only in the deep water area,but also in the dry-wet surface.In the numerical experiment,small perturbation of a steady-state solution,small perturbation of the still water,parabolic bowl and dam break over inclined planes are simulated to demonstrate the well-balancing,positivity preserving,mass conservation and the ability of solving dry-wet surface of the presented scheme.The second work is reducing the numerical dissipation of the non-staggered central scheme.The discontinuous solutions accuracy of classical central schemes is reduced due to the existence of numerical dissipation,especially at the shock waves and contact discontinuities.By introducing parametersain the forward projection,we combine the numerical solution of the previous time level with the numerical solution of the current time level to reduce the numerical dissipation accumulated in the projection.The numerical dissipation of the classical central scheme is u_xxxx（Δx）⁴/Δt,and the numerical dissipation of the presented scheme is αu_xxxx（Δx）^/4Δt.Therefore,the smaller the_ais,the smaller the numerical dissipation is.However,when the parameter_ais taken too small,the CFL（Courant-Friedrich-Lewy）restriction will become more stringent.Through the theoretical analysis of the TVD properties of the presented scheme,it is found that there is a nonlinear relationship betweena,CFL numbers and minmod parameter,which helps to provide a preliminary value range in the actual calculation.The theoretical analysis also shows that there is always an appropriate CFL numbers in any case,so that the numerical solution will not produce new extreme points,which means the presented scheme can stay stable.Five Riemann problems of Euler equations and several classical convection-diffusion models are selected in numerical experiments.It is shown that the numerical dissipation of the presented scheme is smaller than that of the classical central scheme,and the spatial accuracy of the presented scheme is very high at the discontinuous solution.更多还原