A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography

Thanh, Nguyen Xuan; Thanh, Mai Duc; Cuong, Dao Huy

doi:10.1007/s10444-020-09832-9

Cited by 1 publication

(1 citation statement)

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“…In recent years studies have been made on the development of well-balanced numerical schemes for the Ripa model. The first work seems to be [8] [31] (a second-order positivity preserving finite volume scheme on rectangular meshes), Sánchez-Linares et al [27] (a second-order positivity preserving HLLC scheme based in path-conservative approximate Riemann solvers, for the onedimensional Ripa model), Han and Li [13] (a high-order finite difference weighted essentially non-oscillatory (WENO) scheme), Saleem et al [26] (a kinetic flux vector splitting scheme on rectangular meshes), Thanh et al [30] (a high-order scheme of van Leer's type for the one-dimensional SWEs with temperature gradient), Rehman et al [24] (a fifth-order finite volume multi-resolution WENO scheme on rectangular meshes), Britton and Xing [6] (a DG scheme for the one-dimensional Ripa model), Qian et al [23] (a DG method based on a source term approximation technique), and Li et al [20] (a DG method based on hydrostatic reconstruction on rectangular meshes). Fixed meshes are employed in the above mentioned works.…”

Section: Introductionmentioning

confidence: 99%

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

Huang¹,

Li²,

Qiu³

et al. 2022

Preprint

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A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model -a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivitypreserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.

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