High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

Cueto‐Felgueroso, Luis; Colominas, Ignasi; Fe, J.; Navarrina, F.; Casteleiro, Manuel

doi:10.1002/nme.1442

Cited by 45 publications

(47 citation statements)

References 42 publications

(54 reference statements)

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“…Figure 15 shows the carpet view of the water level at 7.2 t = seconds. The 1 p solution qualitatively compares well with that in [26] where a Riemann-solver-based method is used.…”

Section: Shallow Water Equationsmentioning

confidence: 59%

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A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

2015

AJCM

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This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.

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“…Figure 15 shows the carpet view of the water level at 7.2 t = seconds. The 1 p solution qualitatively compares well with that in [26] where a Riemann-solver-based method is used.…”

Section: Shallow Water Equationsmentioning

confidence: 59%

“…This is in contrast to the results reported in [25] showing that the least-square finite element method is not able to capture the strong shock in the right location. The second case is another classical benchmark problem widely used to verify shallow water equation solvers [26]. Figure 15 (left) depicts the domain geometry.…”

Section: Shallow Water Equationsmentioning

confidence: 99%

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

2015

AJCM

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“…) and the derivatives of [17,16,18]. A wide variety of kernel functions appear in the literature, most of them being spline or exponential functions.…”

Section: General Formulationmentioning

confidence: 99%

“…Once the shape functions and their derivatives have been evaluated at a certain location x x x x x x x x x x x x x x, the flow variables and their succesive derivatives are readily computed [17,16]. Note that, using fixed clouds, the MLS shape functions do not change in time and, therefore, they need to be computed only once at the preprocessing phase.…”

Section: Practical Computation Of the Mls Shape Functions On Unstructmentioning

confidence: 99%

“…However, they are perfectly suitable for the evaluation of the diffusive fluxes in Navier-Stokes computations. Given their centered character and accuracy, MLS approximants also possess nice properties to be used in the development of centered schemes with added dissipation [16]. A current research line pursued by the authors relates to the posibility of using MLS approximants for the direct reconstruction of the convective fluxes, employing adaptive stencils or adaptive kernel functions to provide the necessary upwinding.…”

Section: Introductionmentioning

confidence: 99%

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Finite volume solvers and Moving Least-Squares approximations for the compressible Navier–Stokes equations on unstructured grids

Cueto‐Felgueroso

Colominas²,

Nogueira³

et al. 2007

Computer Methods in Applied Mechanics and Engineering

105

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This paper introduces the use of Moving Least-Squares (MLS) approximations for the development of high order upwind schemes on unstructured grids, applied to the numerical solution of the compressible Navier-Stokes equations. This meshfree interpolation technique is designed to reproduce arbitrary functions and their succesive derivatives from scattered, pointwise data, which is precisely the case of unstructured-grid finite volume discretizations. The Navier-Stokes solver presented in this study follows the ideas of the generalized Godunov scheme, using Roe's approximate Riemann solver for the inviscid fluxes. Linear, quadratic and cubic polynomial reconstructions are developed using MLS to compute high order derivatives of the field variables. The diffusive fluxes are computed using MLS as a global reconstruction procedure. Various examples of inviscid and viscous flow are presented and discussed.

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

Delis

Nikolos

2012

Numerical Methods in Fluids

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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.

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High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

Cited by 45 publications

References 42 publications

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

Finite volume solvers and Moving Least-Squares approximations for the compressible Navier–Stokes equations on unstructured grids

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

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