用脉冲谱-优化法求解对流-扩散方程源项控制反问题

Numerical Solution to an Inverse Problem of Source Control for Convection-Diffusion Equations by PST-Optimization

作者曾撰文求解了对流-扩散方程边界条件控制反问题,可根据下游河道的环境容量反求上游边界的限制浓度,作为确定排放标准的参考。在此基础上,本文求解对流-扩散方程的另一类反问题——源项控制反问题,并考虑了水流自净能力;作为典型的算例,可根据下游河道的环境容量反求散布在上游河道中的若干个污染源的限制强度。本文的成果与前文的成果相结合,可基本解决—维河道的污染源控制问题。解算方法仍为作者在前文中提出的脉冲谱-优化法,但本文导出了目标泛函对未知源项的变分的解析表达式。计算结果表明,用脉冲谱-优化法求解源项控制反问题,避免了传统的“试错法”,求解所需的信息少,计算精度高、速度快,能一次解出若干个污染点源的限制强度,在环境水力学领域有广泛的实用意义,也可用于泥沙悬移质控制问题。

The inverse problem of boundary condition control for convection-diffusion equations has been solved by the authors in Reference[1].Accordingly,a limited concentration on the upstream boundary is determined,based on downstream environmental tolerance,and serves as a good reference for determining the discharge standard.In the present paper,on that basis,another class of inverse problem,i.e.source control problem,is solved with the capacity of flow self-purification being taken into account,which has not been considered in Reference[1].As a typical numerical example,the limited intensities of several pollutant sourees scattered upstream can be determined simultaneously, based on downstream environmental tolerance.Pollution control in one-dimensional channel or river can thus be fulfilled by a combination of results from this paper and reference[1].The solution method is again PST-Optimization presented by the authors in Reference[1],but an analytical expres- sion of variation for the objective function to the unknown source term is derived in this paper.Nu- merical results show that the present method is not only high in accuracy but also fast in computation, thus avoiding traditional tedious trial and error.The method can be applied in the field of environmen- ta! hydraulics and suspension-load control problem in sedimentology.