Comparison of numerical schemes of river flood routing with an inertial approximation of the Saint Venant equations

ABSTRACT The one-dimensional flow routing inertial model, formulated as an explicit solution, has advantages over other explicit models used in hydrological models that simplify the Saint-Venant equations. The main advantage is a simple formulation with good results. However, the inertial model is restricted to a small time step to avoid numerical instability. This paper proposes six numerical schemes that modify the one-dimensional inertial model in order to increase the numerical stability of the solution. The proposed numerical schemes were compared to the original scheme in four situations of river’s slope (normal, low, high and very high) and in two situations where the river is subject to downstream effects (dam backwater and tides). The results are discussed in terms of stability, peak flow, processing time, volume conservation error and RMSE (Root Mean Square Error). In general, the schemes showed improvement relative to each type of application. In particular, the numerical scheme here called Prog Q(k+1)xQ(k+1) stood out presenting advantages with greater numerical stability in relation to the original scheme. However, this scheme was not successful in the tide simulation situation. In addition, it was observed that the inclusion of the hydraulic radius calculation without simplification in the numerical schemes improved the results without increasing the computational time.

摘要 针对用于简化圣维南方程组（Saint-Venant equations）的各类水文显式模型而言，被构建为显式求解形式的一维河道水流演进惯性模型具备显著优势，其核心优势在于模型形式简洁且求解效果优异。但该惯性模型需采用较小的时间步长以规避数值不稳定性问题。为此，本文提出六种针对一维惯性模型的改进数值格式，以提升其求解过程的数值稳定性。本文将所提的六种数值格式与原始格式进行对比验证，对比场景涵盖四种河道坡度工况（正常、缓坡、陡坡、极陡坡），以及两种受下游影响的河道工况：坝前回水与潮汐作用。研究从数值稳定性、洪峰流量、计算耗时、水量守恒误差以及均方根误差（Root Mean Square Error，RMSE）这几个维度对结果展开讨论。总体而言，各类改进格式在对应应用场景中均展现出性能提升。尤为值得注意的是，本文命名为Prog Q(k+1)xQ(k+1)的数值格式表现突出，相较于原始格式具备更优异的数值稳定性优势，但该格式在潮汐模拟场景中未能取得理想效果。此外，研究发现，在数值格式中引入未简化的水力半径（hydraulic radius）计算方式，可在不增加计算耗时的前提下提升求解结果精度。