Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes

TL;DR

This paper derives analytical vortex solutions for 2D shallow water and Euler equations, enabling precise testing of high-order numerical schemes' accuracy with solutions of varying smoothness.

Contribution

It provides a new set of vortex-like analytical solutions with different smoothness levels for validating high-order numerical methods.

Findings

Solutions cover a range of smoothness, including $ ext{C}^ ext{infty}$

Useful for testing accuracy of high-order schemes

Facilitates validation of numerical methods for shallow water and Euler equations

Abstract

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.