The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work we implemented an elaborated deflated continuation method, that relies on the spectral element method (SEM) and on the reduced basis (RB) one, to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.
翻译:多数最常见的物理现象可以用部分差异方程式(PDEs)来描述。然而,这些现象往往具有很强的非线性特征。这些特征导致双向理论所研究的多种解决方案共存。不幸的是,在实际假设中,人们不得不利用数字方法来计算PDE的系统解决方案,即使古典技术通常只能够计算出一个单一的解决方案,当更多的分支存在时,参数的任何值都是单一的。在这项工作中,我们采用了一种精心设计的减缩的延续方法,该方法依赖于光谱元素法(SEM)和减缩的基础(RB) 1,以高效地计算带有更多参数和更多双向点的双向图。在典型的继续法和通缩法相结合:前者用来完全跟踪图表的每个已知分支,而后者则用来发现新的分支。最后,如果考虑不止一个参数,计算效率的保证了图表可以在在线阶段中进行计算,而在离线的阶段中,在离线期间,可以有效地计算双向的双向图图中,只能用一种细的公式来测量一个图表的公式。