[Paper Review] Numerical wave propagation for the triangular $P1_{DG}$-$P2$ finite element pair

Colin J. Cotter|arXiv (Cornell University)|Jan 17, 2010

Geophysics and Gravity Measurements4 citations

TL;DR

This paper analyzes the numerical wave propagation properties of the $P1_{DG}$-$P2$ finite element pair for the linearized rotating shallow-water equations on triangular meshes. Using a discrete Helmholtz decomposition, it shows the method achieves third-order spatial accuracy for inertia-gravity and Rossby waves, with only non-propagating spurious inertial oscillations, and proposes a velocity space restriction to eliminate these modes entirely.

ABSTRACT

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the $f$-plane case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \pdgp finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency $f$, and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $β$-plane case, resulting in a Rossby wave equation which is also third-order accurate.

Motivation & Objective

• To analyze the wave propagation behavior of the $P1_{DG}$-$P2$ finite element pair on unstructured triangular meshes for the linearized rotating shallow-water equations.
• To address the challenge of spurious modes in mixed finite element methods for geophysical fluid dynamics, particularly on unstructured grids.
• To demonstrate that the $P1_{DG}$-$P2$ pair supports only non-propagating spurious inertial oscillations, which can be removed via a velocity space restriction.
• To show that the method achieves third-order spatial accuracy for inertia-gravity and Rossby waves, improving wave dispersion properties.
• To provide a framework for using the method in adaptive mesh refinement and numerical weather prediction by ensuring geostrophic balance and accurate wave representation.

Proposed method

• Applies a discrete Helmholtz decomposition of the velocity space using potentials derived from the $P2$ pressure space to decouple geostrophic, inertia-gravity, and inertial modes.
• Derives a discrete inertia-gravity wave equation equivalent to the $P2$ continuous finite element method, establishing third-order spatial accuracy.
• Analyzes the $\beta$-plane case using the quasi-geostrophic limit, showing the resulting Rossby wave equation is also third-order accurate.
• Proposes a restricted $P1_{DG}$ velocity space (denoted $H(P2)$) that eliminates spurious inertial oscillations by projecting out non-geostrophic components.
• Uses numerical dispersion analysis on equilateral triangular meshes to validate the accuracy of the dispersion relation.
• Leverages mass-lumping techniques to maintain efficiency while preserving wave accuracy and geostrophic balance.

Experimental results

Research questions

• RQ1 How accurate is the wave propagation of the $P1_{DG}$-$P2$ finite element pair for inertia-gravity and Rossby waves on unstructured triangular meshes?
• RQ2 What types of spurious modes, if any, are supported by the $P1_{DG}$-$P2$ discretization, and how do they affect wave dynamics?
• RQ3 Can the discrete Helmholtz decomposition be used to decouple physical and spurious modes and derive accurate wave equations?
• RQ4 Does the $P1_{DG}$-$P2$ method achieve third-order spatial accuracy for wave propagation, and how does this compare to standard second-order methods?
• RQ5 Can a velocity space restriction eliminate spurious inertial oscillations without compromising geostrophic balance or wave accuracy?

Key findings

• The $P1_{DG}$-$P2$ finite element pair achieves third-order spatial accuracy for both inertia-gravity and Rossby wave propagation, significantly improving dispersion accuracy over standard second-order methods.
• The only spurious modes present are non-propagating inertial oscillations at frequency $f$, which are uncoupled from physical wave modes and do not affect geostrophic balance.
• A restricted velocity space ($H(P2)$) can be defined to eliminate spurious inertial oscillations entirely, resulting in a method free of any spurious modes.
• The quasi-geostrophic limit of the $P1_{DG}$-$P2$ discretization yields a Rossby wave equation identical to the $P2$ continuous finite element method, confirming third-order accuracy.
• The method preserves steady geostrophic modes and maintains accurate wave representation even on arbitrary unstructured triangulations.
• Mass-lumping has minimal impact on dispersion relations, preserving wave accuracy while enabling efficient time integration.