Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions but rather local approximate solutions to the governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation, the present work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial.
翻译:特雷夫茨方法是接近边界和(或)初步价值问题的解决方案的数字方法,是具有特定测试和试验功能的加勒金方法,这些方法在当地解决管辖的局部差分方程(PDE),这种属性称为特雷夫茨属性。采用了基亚西-特雷夫茨方法,以利用特雷夫茨方法的优势解决可变系数PDE所管辖的问题,将特雷夫茨属性放松为所谓的准特雷夫茨属性:测试和试验功能不是精确的解决方案,而是治理PDE的局部近似解决方案。为了为形成由凝固的赫尔姆霍茨方程所管辖的丙基-亚精算问题而开发的夸斯-特雷夫茨方法,目前的工作解决了半特雷夫茨函数三个组合的定义、构造和近似属性的问题:两个基于平面波解决方案的概括化,一个是多式的。