Efficient Spectral Element Method for the Euler Equations on Unbounded Domains

Abstract

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Figures (14)

• Figure 1: Example of a finite spectral element domain $\Omega^F$ with $N_F = 16$ connected to a semi-infinite element domain $\Omega^{S}$ with $N_S = 4$.
• Figure 2: First six scaled Laguerre functions (SLFs) specified by \ref{['eq:laguerre_function']} with scaling factor $\lambda=1$.
• Figure 3: 1D wave equation: Time evolution of $u$: Top-left: $t=0~$s, top-right: $t=1.5~$s, center-left: $t=3~$s, center-right: $t=3.3~$s, bottom-left: $t=3.9~$s, bottom-right: $t=9~$s. The initial perturbation splits into two waves, one left-going wave and one right-going wave, both reach the edges of the finite domain (illustrated by the red vertical lines) and are subsequently damped as they move through the semi-infinite elements.
• Figure 4: 1D wave equation: Same as figure \ref{['fig:Wave_u']} but for $v$
• Figure 5: 1D wave train: Evolution of the height perturbation $h$ over time for $x \in \Omega^F \cup \Omega^S$: Top-left: $t=0~$s, top-right: $t=500~$s, bottom-left $t=1,000~$s, bottom-right $t=5,000~$s. The waves caused by a continuous forcing at the leftmost point of the domain are transmitted through $\Omega^F$ until reaching the semi-infinite element $\Omega^S$ (the interface between the two domains is illustrated by the red line in the plots), where they are progressively damped over time.