Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes

Abstract

We consider the numerical approximation of the Shallow Water Equations (SWEs) in spherical geometry for oceanographic applications. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we restate all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.

Positivity-Preserving Well-Balanced Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Methods for the Shallow Water Equations

“…In [44,45], the authors introduced a new interpolation in finite volume methods to remap the flow variables to the new mesh while keeping the well-balanced property and height positive. In [1,2], finite volume methods has been coupled pre-balanced ALE form of shallow water equations. In [28], space-time discontinuous DG methods on moving meshes and a new weak form were developed to preserve the steady states.…”

An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge

“…As a final remark, we note that di↵erently from two-dimensional formulations, the right-hand side appears in Cartesian form and it is well-posed on the whole sphere. However the tangent basis in (32) is not well defined at the poles, for the latitude-longitude parametrization that we have chosen. For this reason, only to evaluate the tangent basis and its derivatives, a polar cap defined by a limiting latitude z lim , is deployed.…”

“…Convergence curves are compared in figure 1 against the results of the original [10]. To make the comparison e↵ective, only for this experiment, we have tried to stick to their same scheme, DG in the weak-form (32) with Lax-Friedrich flux. We have remarked that the use of the well balanced momentum form (43) has almost no impact on the errors.…”

“…Relative energy conservation error is reported on the right column in figure 3: it is comparable to or better than those typically reported in literature. In the same figure we compare with a second order residual distribution scheme proposed by the authors [32], with a third order FV scheme on a cubed sphere [5] and with the high resolution results by the German Weather Service (DWD) http://icon.enes.org/swm/stswm/node5.html.…”

“…The eastern boundary is a wall to let a southward propagating barotropic Kelvin wave develop. We compare (43) against the non well balanced scheme based on the weak formulation (32). In the left block of figure (6) two snapshots of upwelling/downwelling at the lateral wall show that the P1 solution of the non well balanced scheme is a↵ected by small wiggles.…”