人工粘性对线性双曲型方程组数值边界反馈控制的影响
The effect of artificial viscosity on numerical boundary feedback control of linear hyperbolic systems
摘要：
为了了解数值扩散对数值边界反馈控制的影响，对人工粘性的影响进行了数值分析。分析进行的线性双曲型系统离散使用迎风格式。迎风格式求解平流-扩散方程，精度可达二阶。分析表明，CFL等于1的迎风格式给出了预期的理论衰减到一阶。另一方面，CFL小于1的迎风格式根据数据的二阶导数和CFL数给出衰减。此外，如果解的二阶导数很小，则衰减速率恶化。因此，由数值方案计算的衰变率往往比理论预测的要高。对包括等温欧拉方程和圣维南方程的算例的计算证实了分析结果。
英文摘要：
A numerical analysis of the effect of artificial viscosity is undertaken in order to understand the effect of numerical diffusion on numerical boundary feedback control. The analysis is undertaken on the linear hyperbolic systems discretised using the upwind scheme. The upwind scheme solves the advection-diffusion equation with up to second-order accuracy. The analysis shows that the upwind scheme with CFL equal to one gives the expected theoretical decay up to first-order. On the other hand the upwind scheme with CFL less than one gives decay depending on the second derivative of the data and the CFL number. Further the decay rates deteriorate if the second derivatives of the solution are small. Thus the decay rates computed by the numerical schemes tend to be higher in comparison to the theoretical prediction. Computations on test cases which include isothermal Euler and the St Venant Equations confirm the analytical results.
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