一维Euler方程的高分辨率有限体积格式

发布时间：2018-09-11 10:32

【摘要】：Euler方程的Riemann问题是典型的一维守恒律方程组,该方程组常常难于解析求解,需要用数值计算方法进行求解。线性低阶精度格式,数值耗散太大,高阶精度格式不满足有界性容易产生非物理振荡。为解决数值解非物理振荡问题,在方程线性对流项离散过程中,结合TVD准则和CBC—BAIR条件,利用Hermite加插值,构造一种新的NVF高分辨率格式HRFVM(High Resolution Finite Volume Scheme)。时间项的离散选用三阶TVD型Runge-kutta方法以保证整体精度。文章通过典型算例来验证新格式的计算精度和逼近效果。线性对流方程的精确解算例验证新格式的精度,间断解算例中新格式与CUI格式对比,证实格式能有效抑制间断处的非物理震荡;非线性方程的一维无粘Burgers方程和Buckely-Leverett问题,进一步验证格式的逼近效果;最后求解一维Euler方程的Lax激波管问题和Shu-Osher问题,通过与WENO5格式的对比,说明新格式逼近效果良好。
[Abstract]:The Riemann problem of Euler equation is a typical one dimensional conservation law system, which is often difficult to be solved analytically, and needs to be solved by numerical method. The numerical dissipation is too large for the linear low-order precision scheme, and the non-physical oscillation is easy to occur when the high-order accuracy scheme does not satisfy the boundedness. In order to solve the non-physical oscillation problem, a new NVF high resolution scheme HRFVM (High Resolution Finite Volume Scheme). Is constructed by using Hermite plus interpolation in the linear convection discretization process of the equation, combining the TVD criterion and the CBC-BAIR condition. The third order TVD type Runge-kutta method is used to discretize the time term to ensure the overall accuracy. A typical example is given to verify the accuracy and approximation effect of the new scheme. The accuracy of the new scheme is verified by an exact solution example of linear convection equation. By comparing the new scheme with the CUI scheme in the discontinuous solution example, it is proved that the scheme can effectively suppress the nonphysical oscillation at the discontinuity, and the one-dimensional inviscid Burgers equation and the Buckely-Leverett problem for the nonlinear equation. Finally, the Lax shock tube problem and Shu-Osher problem of one-dimensional Euler equation are solved. The comparison with WENO5 scheme shows that the new scheme has good approximation effect.
【学位授予单位】：内蒙古大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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