Nonstaggered Central Scheme Research Articles

A moving water equilibria preserving nonstaggered central scheme achieved via flux globalization for the Ripa model

A moving water equilibria preserving nonstaggered central scheme achieved via flux globalization for the Ripa model

A Modified High-Resolution Non-staggered Central Scheme with Adjustable Numerical Dissipation

A Modified High-Resolution Non-staggered Central Scheme with Adjustable Numerical Dissipation

Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem.

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

The objective of this paper is to present a nonstaggered central scheme based on hydrostatic reconstruction (HR) for the shallow water equations with Coriolis forces and nonflat bottom topography. The discretization of the bed slope source term is based on the HR method to ensure the well‐balanced property for the stationary solution. We use the cell average of the discharge on the staggered cell to discretize the source term due to the effect of Coriolis forces. The positivity‐preserving property of the water depth is obtained by providing an appropriate CFL condition with the help of a “draining” time‐step technique. Finally, several two‐dimensional numerical results of classical problems of the system are presented to test these properties of the current scheme.

Exactly well‐balanced positivity preserving nonstaggered central scheme for open‐channel flows

SummaryIn this paper, we construct and study an exactly well‐balanced positivity‐preserving nonstaggered central scheme for shallow water flows in open channels with irregular geometry and nonflat bottom topography. We introduce a novel discretization of the source term based on hydrostatic reconstruction to obtain the exactly well‐balanced property for the still water steady‐state solution even in the presence of wetting and drying transitions. The positivity‐preserving property of the cross‐sectional wet area is obtained by using a modified “draining" time‐step technique. The current scheme is also Riemann‐solver‐free. Several classical problems of open‐channel flows are used to test these properties. Numerical results confirm that the current scheme is robust, exactly well‐balanced and positivity‐preserving.

On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms

The stability of nonlinear explicit difference schemes with not, in general, open domains of the scheme operators are studied. For the case of path-connected, bounded, and Lipschitz domains, we establish the notion that a multi-level nonlinear explicit scheme is stable iff (if and only if) the corresponding scheme in variations is stable. A new modification of the central Lax–Friedrichs (LxF) scheme is developed to be of the second-order accuracy. The modified scheme is based on nonstaggered grids. A monotone piecewise cubic interpolation is used in the central scheme to give an accurate approximation for the model in question. The stability of the modified scheme is investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second-order nonstaggered central scheme based on operator-splitting techniques.

A high-order non-oscillatory central scheme with non-staggered grids for hyperbolic conservation laws

In this work, we present a scheme which is based on non-staggered grids. This scheme is a new family of non-staggered central schemes for hyperbolic conservation laws. Motivation of this work is a staggered central scheme recently introduced by A.A.I. Peer et al. [A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws, Appl. Numer. Math. 58 (2008) 674–688]. The most important properties of the technique developed in the current paper are simplicity, high-resolution and avoiding the use of staggered grids and hence is simpler to implement in frameworks which involve complex geometries and boundary conditions. Numerical implementation of the new scheme is carried out on the scalar conservation laws with linear, non-linear flux and systems of hyperbolic conservation laws. The numerical results confirm the expected accuracy and high-resolution properties of the scheme.

On Stability, Monotonicity, and Construction of Difference Schemes I: Theory

The stability of nonlinear explicit difference schemes with not, in general, open domains of the scheme operators are studied. For the case of path-connected, bounded, and Lipschitz domains, we establish the notion that a multi-level nonlinear explicit scheme is stable iff (if and only if) the corresponding scheme in variations is stable. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. The modified scheme is based on nonstaggered grids. A monotone piecewise cubic interpolation is used in the central scheme to give an accurate approximation for the model in question. The stability of the modified scheme is investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second order nonstaggered central scheme based on operator-splitting techniques.

Implementation of semi‐discrete, non‐staggered central schemes in a colocated, polyhedral, finite volume framework, for high‐speed viscous flows

AbstractWe describe the implementation of a computational fluid dynamics solver for the simulation of high‐speed flows. It comprises a finite volume (FV) discretization using semi‐discrete, non‐staggered central schemes for colocated variables prescribed on a mesh of polyhedral cells that have an arbitrary number of faces. We describe the solver in detail, explaining the choice of variables whose face interpolation is limited, the choice of limiter, and a method for limiting the interpolation of a vector field that is independent of the coordinate system. The solution of momentum and energy transport in the Navier–Stokes equations uses an operator‐splitting approach: first, we solve an explicit predictor equation for the convection of conserved variables, then an implicit corrector equation for the diffusion of primitive variables. Our solver is validated against four sets of data: (1) an analytical solution of the one‐dimensional shock tube case; (2) a numerical solution of two dimensional, transient, supersonic flow over a forward‐facing step; (3) interferogram density measurements of a supersonic jet from a circular nozzle; and (4) pressure and heat transfer measurements in hypersonic flow over a 25∘–55∘ biconic. Our results indicate that the central‐upwind scheme of Kurganov, Noelle and Petrova (SIAM J. Sci. Comput. 2001; 23:707–740) is competitive with the best methods previously published (e.g. piecewise parabolic method, Roe solver with van Leer limiting) and that it is inherently simple and well suited to a colocated, polyhedral FV framework. Copyright © 2009 John Wiley & Sons, Ltd.

Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction

The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408–463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241–282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82–104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199–224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255–283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.

High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation Laws

We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.