Selective decay for the rotating shallow-water equations with a structure-preserving discretization

Abstract

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2), 495 (2013)], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [Pavlov et al., "Structure-preserving discretization of incompressible fluids," Physica D 240(6), 443-458 (2011)] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

On perturbations retaining conservation laws of difftrential equations

“…Thus the present paper as well as [5] and our previous research of the KdV solitons in nonhomogeneous media, [6], persuades that the selective decay approach is a valid and effective instrument to obtain qualitative approximations and estimates for behavior of solutions.…”

“…There is a considerable number of publication in the field, see a recent paper [5] for recent developments.…”

Casimir-Dissipation Stabilized Stochastic Rotating Shallow Water Equations on the Sphere

“…As suggested in [3], our approach is to discretize the deterministic parts (the det and −∇ • (uh) terms) with a variational discretization [1] and the stochastic terms with standard finite difference operators. Then, we add an energy preserving Casimir dissipation [2] or biharmonic Laplacian dissipation.…”

“…For a semi-discrete Lagrangian ℓ(A, h), the curves A(t), h(t) ∈ R are critical for the variational principle of Eq. ( 28) in [2] if and only if they satisfy…”

“…As a remedy to this latter point, we suggest here to use Casimir dissipation for stabilization to avoid the dissipative effect of the Laplacian on the total energy while keeping the stochastic scheme stable also in case of inhomogenous noise. The suggested combination is possible because the derivation of the Casimir dissipation for the RSW equations in [2] follows the same variational discretization principle of [1] applied to the RSW equations that form the deterministic core of the stochastic RSW-LU model of [3].…”