带源项双曲守恒律方程的高精度和谐算法

【作者】 钱守国；

【导师】 邵峰晶；

【作者基本信息】 青岛大学 ， 系统理论， 2018， 博士

【摘要】 带源项双曲守恒律方程是描述流体运动的一类重要模型,其数值模拟的一个主要困难是不管它们的初始条件是否光滑,它们的解都可能会出现激波、旋涡、接触间断等问题,低阶精度数值算法在处理这类问题时容易在间断处过多地抹平,因此无法准确模拟解的状态,而有限差分WENO算法和间断有限元算法等高阶数值算法得到的数值解在光滑区域能达到任意阶精度,在间断区域能保持数值稳定,且不会产生数值伪震荡,因此引起了人们广泛的研究兴趣。带源项双曲守恒律方程数值模拟的另一个主要困难是关于源项的处理,这是由于当源项与流通量的梯度保持平衡时,模型保持某种定常解,如果不采用特殊的源项处理,即使采用常规高精度数值算法也不能保持定常解,甚至会产生数值伪震荡,而和谐算法由于能在较粗糙网格划分以及考虑计算机舍入误差情形下近似精确保持定常解,因此成为非常活跃的研究方向。论文的主要研究工作如下:1.设计了带几何源项血液方程的高精度有限差分WENO和谐算法,该算法具有保持静血液定常解的和谐性。首先对原血液方程进行重构,这样做的目的是当血液方程处于静血液定常解时,源项与流通量具有相同的结构;然后构造了特殊的源项近似以及和谐数值流通量,并给出了该算法保持静血液定常解的和谐性证明;最后通过经典数值算例验证了该算法对光滑解能达到预期的高阶精度,具有保持静血液定常解的和谐性,能有效解决静血液定常解小扰动、血液流的波动、脉冲传播以及粘性阻尼运动等问题。2.设计了带几何源项沟渠浅水波方程的高精度DG和谐算法,该算法具有保持静水定常解的和谐性。首先利用静水定常解将沟渠浅水波方程进行重构;然后构造了特殊的源项近似以及和谐数值流通量,并设计了简单的保正算法,从而保证潮湿区域的水深非负;最后通过经典数值算例验证了该算法具有保持静水定常解的和谐性,能有效处理静水定常解的小扰动、跨临界流和动水定常解等问题,且保正算法能有效处理不平底部的沟渠震荡和排水等问题。3.设计了带几何源项温度场影响下的浅水波方程的高精度DG和谐算法,该算法具有保持静水定常解的和谐性。首先考虑一维情形,利用静水定常解将一维温度场影响浅水波方程进行重构,构造了特殊的源项近似以及和谐数值流通量,并证明了该算法具有保持静水定常解的和谐性;然后将一维算法类似推广到二维情形;最后通过经典数值算例验证了该算法具有保持静水定常解的和谐性,对连续光滑解能达到预期的高阶精度,能准确捕捉静水定常解的小扰动,以及有效模拟多种类型的溃坝问题。4.设计了带重力源项欧拉方程的高精度DG和谐算法,该算法具有保持等熵定常解的和谐性。首先考虑一维情形,利用等熵定常解将一维欧拉方程进行重构,构造了特殊的源项近似以及和谐数值流通量,并证明了该算法具有保持等熵定常解的和谐性;然后将一维算法类似推广到多维情形;最后通过经典数值算例验证了该算法对连续光滑解能达到预期的高阶精度,具有保持等熵定常解的和谐性,并能准确模拟激波、等熵定常解小扰动以及间断界面等问题。更多还原

【Abstract】 The hyperbolic conservation laws equations with source terms are important models which describe the fluid motions.One main difficulty in solving them is that whether their initial conditions are smooth or not,their solutions may appear discontinuous problems,such as shock wave,vortex and contact discontinuities.The low order schemes are easy to level off when dealing with such problems,so they can not accurately simulate the states in the discontinuities.Therefore it becomes a research focus to design high order schemes to solve such problems.Among all the high order schemes,the finite difference WENO schemes and the discontinuous Galerkin schemes are commonly used to deal with the convection-diffusion equations,which are especially suitable for numerical simulation of the hyperbolic conservation laws equations with source terms.The advantages of these two high order schemes consist that they can achieve arbitrary high order precision in smooth region,maintain numerical stability in the discontinuous area,and effectively deal with various steep problems.Another main difficulty in solving the hyperbolic conservation laws equations with source terms is the treatment of source terms,which need to be balanced by the flux gradient at the steady state.Standard numerical methods may not satisfy the discrete version of this balance exactly at(or near)the steady state,and may introduce spurious oscillations,unless the mesh size is extremely refined.But this numerical procedure of mesh refinement is not applicable for the high dimensional cases due to too high computational cost.In order to save the computational cost,well-balanced schemes are designed to preserve exactly these steady state solutions up to the machine accuracy,and have been active research areas in the past two decades.The main contents of this dissertation can be summarized as follows:1.We present a high order well-balanced finite difference WENO scheme for the blood flow equation with a geometrical source term,which maintains the still blood steady state.In order to maintain the well-balanced property,we propose to reformulate the blood flow model in order that the source term has the same structure as the flux at the still blood steady state.Moreover,we apply a novel source term approximation as well as well-balanced numerical fluxes.We prove that the proposed scheme for the blood flow equation can maintain well-balanced property.Extensive numerical experiments are preformed to verify the performances of the proposed scheme such as the maintenance of well-balanced property,the genuine high order accuracy for smooth solutions,the ability to capture the perturbations of still blood steady state,and effective simulations of wave equation,propagation of a pulse and viscous damping problems.2.We present high order DG scheme for the shallow water flows through channels with irregular geometry,which maintains the still water steady state.We reformulate the shallow water equation through channels by the still steady state,and propose to construct a novel source term approximation as well as well-balanced numerical fluxes.In addition,we design a simple positivity-preserving limiter,which can ensure the resulting methods maintain the non-negativity of the cross sectional wet area.We have carried out extensive numerical simulations,which demonstrate that the proposed methods are well-balanced,efficient for the small perturbation test near the steady state solutions,positivity-preserving near the wetting and drying front,high order accurate and also good for both continuous and discontinuous solutions.3.We present high order DG scheme for the shallow water equation under temperature fields with a geometrical source term,which maintains the still water steady state.For one-dimensional case,we reformulate the shallow water equation under temperature by the still steady state,and propose to construct a novel source term approximation as well as well-balanced numerical fluxes.In addition,we prove that the proposed scheme for the one-dimensional shallow water equation under temperature can maintain well-balanced property.Then we generalized the one-dimensional scheme to the two-dimensional case.Extensive numerical results all verify that the proposed methods are well-balanced for the still steady state,keep genuinely high order accuracy in smooth regions for smooth solutions,capture the perturbations of the still water steady state,and efficiently simulate some type of dam break problems.4.We construct high order DG scheme for the Euler equations under gravitational fields,which are well-balanced for the isentropic type hydrostatic equilibrium state.For one-dimensional case,we reformulate the Euler equation in its equivalent form by the isentropic type hydrostatic equilibrium state,and propose a novel source term approximation as well as well-balanced numerical fluxes.In addition,we prove that the proposed scheme for the one-dimensional Euler equation can maintain well-balanced property.Then we generalized the one-dimensional scheme to the multi-dimensional case.Extensive numerical results are performed to test the genuine high order accuracy in smooth regions,the maintenance of well-balanced property,and the ability to capture small perturbation of the isentropic hydrostatic solution state.更多还原