Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like analytical solutions in 2 dimensions for the shallow water equations (and, hence, Euler equations) that can be used to test the order of accuracy of numerical methods. These solutions have different smoothness in their derivatives (up to $\mathcal C^\infty$) and can be used accordingly to the order of accuracy of the scheme to test.

翻译：测试浅水(和Euler)方程式的(非常)高排序方法的精度顺序是一个微妙的操作,测试案例是这一操作的关键起点。我们为浅水方程式(以及,因此,Euler方程式)提供了两个层面的涡旋式分析解决方案的简短衍生物,可用于测试数字方法的精度顺序。这些解决方案的衍生物具有不同的顺畅性(最高可达$\mathcal C ⁇ infty$),因此可以用于测试计划的精度顺序。