A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

Abstract

SUMMARY On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.

Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

“…The present results are compared to the results reported in the works of Hou et al 16 and Delis and Nikolos. 29 The cell numbers in these test cases are similar to those in the present one; therefore, a comparison is meaningful. The VEQ yields less accurate results, and the VEM yields almost similar accuracy.…”

A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids

“…A recent representative multislope MUSCL scheme proposed by Touze [34] is introduced in the following, which essentially extends the original multislope MUSCL schemes [17,22], and based on that a review of the multislope MUSCL scheme is presented.…”

“…The modified Van Albada limiting function φ VA (Equation 35) [17,26,39] is employed here to turn the upwind and downwind slopes into a limited slope, which is then used to calculate the left and right Riemann states:…”

“…This method integrates characteristics of the CCFV and NCFV methods, allowing the slope to be univocally defined in each cell and enabling a straightforward inclusion of the geo-failure operator in the morphodynamic model. The Godunov-type CCFV scheme has been a widely used numerical method for solving SWEs in recent decades [7,[14][15][16][17][18][19][20], for its shock-capturing nature and desirable property of automatic conservation. Unfortunately, the conventional Godunov-type CCFV scheme provides only first-order accuracy in space that exhibits relatively large numerical diffusion and follows a low-resolution property.…”

Contradiction between the C‐property and mass conservation in adaptive grid based shallow flow models: cause and solution

“…conservation property) [10] of an SWE model. Cproperty essentially requires a model to numerically preserve a quiescent steady flow and is usually employed to reflect the well-balancedness of a numerical scheme, for example, in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The well-balancedness refers to the capability of a numerical scheme solving the SWE to balance the flux gradients and the source terms [28][29][30][31].…”