In an earlier study (Zhang & Shu 2010 b J. Comput. Phys. 229 , 3091–3120 ([doi:10.1016/j.jcp.2009.12.030][1])), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov–Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

 [1]: http://dx.doi.org/10.1016/j.jcp.2009.12.030

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2011.0153

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

http://www.math.utah.edu/~epshteyn/selectedpapers.html/paperSW1.pdf

Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

Springer Proceedings in Mathematics & Statistics  This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Springer Proceedings in Mathematics & Statistics

In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography The main new technique is a special reconstruction of the flow variables in wet---dry cells, which is presented in this paper for the one dimensional case We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun Math Sci, 5(1):133---160, 2007) The positivity of the computed water height is ensured following (Bollermann et al, Commun Comput Phys, 10:371---404, 2011): The outgoing fluxes are limited in case of draining cells

https://www.igpm.rwth-aachen.de/Download/reports/pdf/IGPM313_k_Neu.pdf

A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations

The aim of this paper is to present a novel monotone upstream scheme for conservation law (MUSCL) on unstructured grids. The novel edge-based MUSCL scheme is devised to construct the required values at the midpoint of cell edges in a more straightforward and effective way compared to other conventional approaches, by making better use of the geometrical property of the triangular grids. The scheme is incorporated into a two-dimensional (2D) cell-centered Godunov-type finite volume model as proposed in Hou et?al. (2013a,c) to solve the shallow water equations (SWEs). The MUSCL scheme renders the model to preserve the well-balanced property and achieve high accuracy and efficiency for shallow flow simulations over uneven terrains. Furthermore, the scheme is directly applicable to all triangular grids. Application to several numerical experiments verifies the efficiency and robustness of the current new MUSCL scheme. A novel MUSCL scheme is developed to provide more efficient and accurate numerical solutions to the 2D shallow water equations.The scheme is flexible for implementation on all common 2D unstructured grids.The scheme satisfies the well-balanced property.It can be generalized into 3D tetrahedron grids and to solve other hyperbolic PDEs.

/pdf/an-efficient-unstructured-muscl-scheme-for-solving-the-2d-1lhxxuutb6.pdf

An efficient unstructured MUSCL scheme for solving the 2D shallow water equations

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Luk\'a\v{c}ov\'a, Noelle and Kraft, J. Comp. Phys. 221, 2007), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

/pdf/finite-volume-evolution-galerkin-methods-for-the-shallow-36gn81ohgt.pdf

We give a further examination of the stabilized Residual Distribution schemes for the solution of the shallow water equations proposed in (Ricchiuto and Bollermann, J.Comp.Phys., available online 25 October 2008). Based on a non-linear variant of a Lax-Friedrichs scheme, the scheme is wellbalanced, able to handle dry areas and, for smooth regions of the solution, obtains second order of accuracy. We will analyze the accuracy when dry areas are included in the domain of computation.

Accuracy of stabilized residual distribution for shallow water flows including dry beds

A good numerical method for the Saint-Venant system of shallow water equations must be well-balanced in the sense that the method should exactly preserve \lake at rest" steady states. There are many numerical methods capable of achieving this goal, but only in the case of submerged bottom topography. When dry areas (island, shores) are contained in the computa- tional domain, most of well-balanced schemes fail to preserve \lake at rest" steady states unless the computational grid is adjusted to the wetting/drying fronts. This diculty can be overcome using a special reconstruction, which is presented in this paper. The proposed reconstruction is conservative, well-balanced and positivity preserving. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (A. Kurganov and G. Petrova, Commun. Math. Sci., 2007). The positivity of the computed water height is ensured following (A. Bollermann, S. Noelle and M. Luk a cov a, Commun. Comput. Phys., 2010): The outgoing uxes are limited in case of draining cells.

/pdf/a-well-balanced-reconstruction-for-wetting-drying-fronts-qkfi2lk8kj.pdf

A well-balanced reconstruction for wetting/drying fronts

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Papers

A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations

Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

Accuracy of stabilized residual distribution for shallow water flows including dry beds

A well-balanced reconstruction for wetting/drying fronts