Assessing the Numerical Accuracy of Complex Spherical Shallow-Water Flows

2007 ◽  
Vol 135 (11) ◽  
pp. 3876-3894 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
David G. Dritschel
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Large Scale ◽  
Numerical Accuracy ◽  
Conservation Of Energy ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Low Froude Number ◽  
Contour Advection ◽  
High Reynolds

Abstract The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Quasi-two-layer finite-volume scheme for modelling shallow water flows over an arbitrary bed in the presence of external force. II. Algorithm applications and numerical results

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Kirill V. Karelsky ◽  
Arakel S. Petrosyan ◽  
Alexander G. Slavin
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
External Force ◽  
Large Scale ◽  
Lower Layer ◽  
Three Dimensional ◽  
Finite Volume Scheme ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Scale Motion

AbstractA finite-volume numerical method for studying shallow water flows over an arbitrary bed profile in the presence of external force has been proposed in [33]. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with advanced consideration of the flow features near the step. A distinctive feature of the suggested model is a separation of the studied flow into two layers in calculating the flow quantities near each step, and improving by this means the approximation of depth-averaged solutions of the initial three-dimensional Euler equations. We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Besides, the quasi-two-layer approach leads to the appearance of additional terms in the one-layer finite-difference representation of balance equations. Numerical simulations are performed based on the proposed in [33] algorithm of various physical phenomena, such as breakdown of the rectangular fluid column over an inclined plane, large-scale motion of fluid in the gravity field in the presence of Coriolis force over amounted obstacle on the underlying surface. Computations are made for the two-dimensional dam-break problem on a slope precisely conform to laboratory experiments. The interaction of the Tsunami wave with the shore line including an obstacle has been simulated to demonstrate the efficiency of the developed algorithm in domains, including partly flooded and dry regions.

scholarly journals A 1D–2D Coupled Lattice Boltzmann Model for Shallow Water Flows in Large Scale River-Lake Systems

2019 ◽  
Vol 10 (1) ◽  
pp. 108
Author(s):  
Wanwan Meng ◽  
Yongguang Cheng ◽  
Jiayang Wu ◽  
Chunze Zhang ◽  
Linsheng Xia
Keyword(s):  
Shallow Water ◽  
Lattice Boltzmann ◽  
Large Scale ◽  
Distribution Functions ◽  
Lattice Boltzmann Model ◽  
Hydropower Station ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Proposed Model ◽  
Lake Systems

Simulating shallow water flows in large scale river-lake systems is important but challenging because huge computer resources and time are needed. This paper aimed to propose a simple and efficient 1D–2D coupled model for simulating these flows. The newly developed lattice Boltzmann (LB) method was adopted to simulate 1D and 2D flows, because of its easy implementation, intrinsic parallelism, and high accuracy. The coupling strategy of the 1D–2D interfaces was implemented at the mesoscopic level, in which the unknown distribution functions at the coupling interfaces were calculated by the known distribution functions and the primitive variables from the adjacent 1D and 2D lattice nodes. To verify the numerical accuracy and stability, numerical tests, including dam-break flow and surge waves in the tailrace canal of a hydropower station, were simulated by the proposed model. The results agreed well with both analytical solutions and commercial software results, and second-order convergence was verified. The application of the proposed model in simulating the surge wave propagation and reflection phenomena in a reservoir of a run-of-river hydropower station indicated that it had a huge advantage in simulating flows in large-scale river-lake systems.

IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

2014 ◽  
Vol 16 (2) ◽  
pp. 307-347 ◽  
Author(s):  
Georgij Bispen ◽  
K. R. Arun ◽  
Mária Lukáčová-Medvid’ová ◽  
Sebastian Noelle
Keyword(s):  
Shallow Water ◽  
Froude Number ◽  
Finite Volume ◽  
Large Time ◽  
Time Step ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Nonlinear Advection ◽  
Low Froude Number ◽  
Large Time Step

AbstractWe present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

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