High order interpolation methods for semi‐Lagrangian models of mobile‐bed hydrodynamics on Cartesian grids with cut cells

SUMMARYKernel-based reconstruction methods are applied to obtain highly accurate approximations of local vector fields from normal components assigned at the edges of a computational mesh. The theoretical background of kernel-based reconstructions for vector-valued functions is first reviewed, before the reconstruction method is adapted to the specific requirements of relevant applications in computational fluid dynamics. To this end, important computational aspects concerning the design of the reconstruction scheme like the selection of suitable stencils are explained in detail. Extensive numerical examples and comparisons concerning hydrodynamic models show that the proposed kernel-based reconstruction improves the accuracy of standard finite element discretizations, including Raviart-Thomas (RT) elements, quite significantly, while retaining discrete conservation properties of important physical quantities, such as mass, vorticity, or potential enstrophy.

Section: Comparisons Between Gaussian Kernels and Rt Elementsmentioning

confidence: 93%

Kernel‐based vector field reconstruction in computational fluid dynamic models

Bonaventura

Iske

Miglio

2011

“…It can be shown that the iterates m , m = 1, 2, … generated by (12) are monotonically decreasing and converge to the exact solution of system (10) in a finite number of iterations, actually very few. 13,[28][29][30] In summary, assuming the knowledge of u n , n i , and V i ( n i ) from the previous time level t n , each time step is advanced by preliminarily determining the wet cross-section areas a n and the wet lengths n from (5)- (6). Then, the piecewise linear system (10) is assembled and solved iteratively to obtain simultaneously the new free-surface elevations n+1 i and the corresponding fluid volumes V i ( n+1 i ).…”

Section: Finite Difference-finite Volume Approximationmentioning

confidence: 99%

“…Solving the two-dimensional and three-dimensional shallow water equations with semi-implicit numerical methods on uniform Cartesian grids have shown to be simple, very efficient, and rather accurate. [1][2][3][4][5][6][7] Semi-implicit methods have been generalized to unstructured orthogonal grids in order to fit arbitrary geometries. [8][9][10][11][12] In both uniform Cartesian grids and unstructured orthogonal grids, the discrete flow variables are defined at staggered locations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computational grid, subgrid, and pixels

Casulli

2019

Summary Free‐surface flows in rivers, estuaries, and coastal areas are strongly dominated by the geometrical details of the study area. Nowadays, accurate bathymetric data are easily available on raster‐based digital elevation models with an impressive spatial resolution. These data are often accessible as large two‐dimensional arrays containing several millions of pixel values. Recent numerical methods are very efficient and rather accurate but far from being able to solve the governing differential equations on a computational grid with such a fine spatial resolution. In the present investigation, the unaltered pixel values from a digital elevation model are clustered to form subgrids of a coarser computational grid. Artificial cross‐flow between disconnected areas is inhibited by introducing cell clones and edge clones. Each clone consists of directly connected pixels. It is shown how the resulting computational grid is able to resolve geometrical details of complex study areas to pixel resolution and for any grid size. As an example, the performance of the proposed algorithm is tested to simulate a typical tidal flow in the San Francisco Bay and the Sacramento‐San Joaquin Delta area by using an extreme subgrid resolution given by a digital elevation model containing 196 000 000 pixels with 10 m pixel size.

“…poor conservation of advected properties and/or high numerical dissipations [7,17]. During the past decades, many researchers have tried to solve the problems by developing better discretization schemes [18,19], high-order interpolation schemes [20,21], and hybrid methods combining interpolation schemes with different orders [7]. Zerroukat [17] proposed a semi-Lagrangian method equipped with conservative remapping scheme.…”

Section: Introductionmentioning

confidence: 99%

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Mortezazadeh

Wang

2017

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.