有限点方法与数值激波不稳定研究
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摘要
本文主要研究了有限点方法和数值激波不稳定现象。主要结果有:
1.对二维光滑函数,推导与给出了在任一点上二阶方向微商的极值公式,即一阶微商的梯度公式。设给定三个互不平行方向上的二阶方向微商,或者给定两个不平行方向上的二阶微商及其混合微商,本文分别给出了二阶方向微商达到极大值和极小值的方向,以及相应的极大值公式与极小值公式。极大值方向与极小值方向是垂直的。
2.利用二阶方向微商关系式,推导与给出了第二类一阶方向微商的四点近似公式;讨论了三点公式四点公式在不同意义下的差异;给出了从两点公式到五点公式的截断误差不断递减的结论。
3.在二维非规则区域的散乱离散点集上,应用有限点公式,对多个算例,成功地和有效地数值求解了椭圆型方程的第一边值问题。数值结果表明,有限点方法具有较好的计算精度与收敛速度。一般说来,一阶微商具有接近二阶的精度,二阶微商具有接近一阶的精度。
4.对双曲守恒律方程的有限点方法,开展了选点方法,人工粘性方法与格式构造研究,获得了初步成果与较深刻的认识:对间断较弱的二维问题,获得了较满意的二维图像;对一般的二维问题,需根据流场的方向性以及间断的形态,恰当地选择邻点与粘性。
5.对二维流体力学有限体积方法应用中出现的激波不稳定现象,给出了混合格式的设计方法,取得了良好的消除激波不稳定性的效果。主要做法为:对连续性方程以及动量方程之一,采用已有的消除激波不稳定性通量,如混合HLL通量等;而对能量方程及另一动量方程仍采用原有通量。我们利用混合方法,有效地消除了Roe格式,HLLC格式和AUSMD格式等的数值激波不稳定现象。该方法计算效率高,激波分辨率好。
This thesis is about Finite Point Method (FPM) and numerical shock instability. The main results in this paper consist of the next parts as following:
1. The formulae of the second-order directional differential's extremum at an arbitrary point in a computational domain are presented for 2-D smooth function. Given three second-order directional differentials in corresponding mutual non-parallel directions, or two second-order directional differentials in corresponding non-parallel directions and their second-order mixed differential, the maximum and minimum of the second-order directional differentials and the corresponding directions are presented. A conclusion is that the maximum direction and the minimum direction are perpendicular to each other.
2. Based on the second-order directional differential formulae, the second type four-point formula about the first-order directional differential are derived; the three-point formula and four-point formula in a different sense of the difference are discussed; also, a conclusion is given that the truncation error of from two to five-point formula continues to decline.
3. By FPM, the simulation to the second-order elliptic PDE on a set of 2-D scattering points in non-regular domain is proposed. Several test problems and their numerical results show that the FPM has good accuracy and convergence rate. Generally, the approximation of first-order differential is almost second-order accurate and the approximation of second-order differential is almost first-order accurate.
4. In order to simulate the hyperbolic conservation laws by FPM, the method of choosing neighboring points, artificial viscosity and corresponding schemes are studied. The preliminary numerical results and their conclusions are given: for FPM, the satisfying results for many problems with weak discontinuities are obtained; as to the general problems, the better viscosity and better method for choosing neighboring points in accordance with the direction of the flow field and the form of discontinuities are under consideration.
5. A hybrid method to eliminate the shock instability in 2-D Euler equations is proposed. On one hand, this paper chooses the fluxes that are free of shock instability to be used in mass equation and one of momentum equation. On the other hand, this paper chooses the fluxes that can resolve full-wave in Riemann problems and suffer from shock instability in another momentum equation and energy equation. This hybrid method does help to eliminate shock instability of the Roe solver, HLLC solver and AUSMD scheme. Furthermore, the hybrid method has a high computational efficiency and good resolution of shock.
1.对二维光滑函数,推导与给出了在任一点上二阶方向微商的极值公式,即一阶微商的梯度公式。设给定三个互不平行方向上的二阶方向微商,或者给定两个不平行方向上的二阶微商及其混合微商,本文分别给出了二阶方向微商达到极大值和极小值的方向,以及相应的极大值公式与极小值公式。极大值方向与极小值方向是垂直的。
2.利用二阶方向微商关系式,推导与给出了第二类一阶方向微商的四点近似公式;讨论了三点公式四点公式在不同意义下的差异;给出了从两点公式到五点公式的截断误差不断递减的结论。
3.在二维非规则区域的散乱离散点集上,应用有限点公式,对多个算例,成功地和有效地数值求解了椭圆型方程的第一边值问题。数值结果表明,有限点方法具有较好的计算精度与收敛速度。一般说来,一阶微商具有接近二阶的精度,二阶微商具有接近一阶的精度。
4.对双曲守恒律方程的有限点方法,开展了选点方法,人工粘性方法与格式构造研究,获得了初步成果与较深刻的认识:对间断较弱的二维问题,获得了较满意的二维图像;对一般的二维问题,需根据流场的方向性以及间断的形态,恰当地选择邻点与粘性。
5.对二维流体力学有限体积方法应用中出现的激波不稳定现象,给出了混合格式的设计方法,取得了良好的消除激波不稳定性的效果。主要做法为:对连续性方程以及动量方程之一,采用已有的消除激波不稳定性通量,如混合HLL通量等;而对能量方程及另一动量方程仍采用原有通量。我们利用混合方法,有效地消除了Roe格式,HLLC格式和AUSMD格式等的数值激波不稳定现象。该方法计算效率高,激波分辨率好。
This thesis is about Finite Point Method (FPM) and numerical shock instability. The main results in this paper consist of the next parts as following:
1. The formulae of the second-order directional differential's extremum at an arbitrary point in a computational domain are presented for 2-D smooth function. Given three second-order directional differentials in corresponding mutual non-parallel directions, or two second-order directional differentials in corresponding non-parallel directions and their second-order mixed differential, the maximum and minimum of the second-order directional differentials and the corresponding directions are presented. A conclusion is that the maximum direction and the minimum direction are perpendicular to each other.
2. Based on the second-order directional differential formulae, the second type four-point formula about the first-order directional differential are derived; the three-point formula and four-point formula in a different sense of the difference are discussed; also, a conclusion is given that the truncation error of from two to five-point formula continues to decline.
3. By FPM, the simulation to the second-order elliptic PDE on a set of 2-D scattering points in non-regular domain is proposed. Several test problems and their numerical results show that the FPM has good accuracy and convergence rate. Generally, the approximation of first-order differential is almost second-order accurate and the approximation of second-order differential is almost first-order accurate.
4. In order to simulate the hyperbolic conservation laws by FPM, the method of choosing neighboring points, artificial viscosity and corresponding schemes are studied. The preliminary numerical results and their conclusions are given: for FPM, the satisfying results for many problems with weak discontinuities are obtained; as to the general problems, the better viscosity and better method for choosing neighboring points in accordance with the direction of the flow field and the form of discontinuities are under consideration.
5. A hybrid method to eliminate the shock instability in 2-D Euler equations is proposed. On one hand, this paper chooses the fluxes that are free of shock instability to be used in mass equation and one of momentum equation. On the other hand, this paper chooses the fluxes that can resolve full-wave in Riemann problems and suffer from shock instability in another momentum equation and energy equation. This hybrid method does help to eliminate shock instability of the Roe solver, HLLC solver and AUSMD scheme. Furthermore, the hybrid method has a high computational efficiency and good resolution of shock.
引文
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